From 0345245e860375a32c9a437c4a9d9cae807134e9 Mon Sep 17 00:00:00 2001 From: Shashank Date: Mon, 29 May 2017 12:40:26 +0530 Subject: CMSCOPE changed --- modules/linear_algebra/help/en_US/addchapter.sce | 11 + modules/linear_algebra/help/en_US/eigen/CHAPTER | 2 + modules/linear_algebra/help/en_US/eigen/balanc.xml | 106 +++ modules/linear_algebra/help/en_US/eigen/bdiag.xml | 106 +++ modules/linear_algebra/help/en_US/eigen/gschur.xml | 99 +++ modules/linear_algebra/help/en_US/eigen/gspec.xml | 45 ++ modules/linear_algebra/help/en_US/eigen/hess.xml | 97 +++ modules/linear_algebra/help/en_US/eigen/pbig.xml | 125 ++++ .../linear_algebra/help/en_US/eigen/projspec.xml | 90 +++ modules/linear_algebra/help/en_US/eigen/psmall.xml | 122 ++++ modules/linear_algebra/help/en_US/eigen/schur.xml | 386 +++++++++++ modules/linear_algebra/help/en_US/eigen/spec.xml | 301 +++++++++ modules/linear_algebra/help/en_US/eigen/sva.xml | 84 +++ modules/linear_algebra/help/en_US/eigen/svd.xml | 126 ++++ 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100755 modules/linear_algebra/help/pt_BR/subspaces/spantwo.xml create mode 100755 modules/linear_algebra/help/ru_RU/addchapter.sce (limited to 'modules/linear_algebra/help') diff --git a/modules/linear_algebra/help/en_US/addchapter.sce b/modules/linear_algebra/help/en_US/addchapter.sce new file mode 100755 index 000000000..a33120d03 --- /dev/null +++ b/modules/linear_algebra/help/en_US/addchapter.sce @@ -0,0 +1,11 @@ +// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +// Copyright (C) 2009 - DIGITEO +// +// This file must be used under the terms of the CeCILL. +// This source file is licensed as described in the file COPYING, which +// you should have received as part of this distribution. The terms +// are also available at +// http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt + +add_help_chapter("Linear Algebra",SCI+"/modules/linear_algebra/help/en_US",%T); + diff --git a/modules/linear_algebra/help/en_US/eigen/CHAPTER b/modules/linear_algebra/help/en_US/eigen/CHAPTER new file mode 100755 index 000000000..88f8bc42b --- /dev/null +++ b/modules/linear_algebra/help/en_US/eigen/CHAPTER @@ -0,0 +1,2 @@ +title = Eigenvalue and Singular Value + diff --git a/modules/linear_algebra/help/en_US/eigen/balanc.xml b/modules/linear_algebra/help/en_US/eigen/balanc.xml new file mode 100755 index 000000000..3f79b79c8 --- /dev/null +++ b/modules/linear_algebra/help/en_US/eigen/balanc.xml @@ -0,0 +1,106 @@ + + + + + balanc + matrix or pencil balancing + + + Calling Sequence + [Ab,X]=balanc(A) + [Eb,Ab,X,Y]=balanc(E,A) + + + + Arguments + + + A: + + a real square matrix + + + + X: + + a real square invertible matrix + + + + E: + + + a real square matrix (same dimension as A) + + + + + Y: + + a real square invertible matrix. + + + + + + Description + + Balance a square matrix to improve + its condition number. + + + [Ab,X] = balanc(A) finds a similarity transformation + X such that + + + Ab = inv(X)*A*X has approximately equal row and column norms. + + + For matrix pencils,balancing is done for improving the + generalized eigenvalue problem. + + + [Eb,Ab,X,Y] = balanc(E,A) returns left and right transformations X and Y + such that Eb=inv(X)*E*Y, Ab=inv(X)*A*Y + + + + Remark + + Balancing is made in the functions bdiag and spec. + + + + Examples + + + + See Also + + + bdiag + + + spec + + + schur + + + + diff --git a/modules/linear_algebra/help/en_US/eigen/bdiag.xml b/modules/linear_algebra/help/en_US/eigen/bdiag.xml new file mode 100755 index 000000000..4cc6433e8 --- /dev/null +++ b/modules/linear_algebra/help/en_US/eigen/bdiag.xml @@ -0,0 +1,106 @@ + + + + + bdiag + block diagonalization, generalized eigenvectors + + + Calling Sequence + [Ab [,X [,bs]]]=bdiag(A [,rmax]) + + + Arguments + + + A + + real or complex square matrix + + + + rmax + + real number + + + + Ab + + real or complex square matrix + + + + X + + real or complex non-singular matrix + + + + bs + + vector of integers + + + + + + Description + + + performs the block-diagonalization of matrix A. bs + gives the structure of the blocks (respective sizes of the + blocks). X is the change of basis i.e + Ab = inv(X)*A*Xis block diagonal. + + + rmax controls the conditioning of X; the + default value is the l1 norm of A. + + + To get a diagonal form (if it exists) choose a large value for + rmax (rmax=1/%eps for example). + Generically (for real random A) the blocks are (1x1) and (2x2) and + X is the matrix of eigenvectors. + + + + Examples + + + + See Also + + + schur + + + sylv + + + spec + + + sysdiag + + + + diff --git a/modules/linear_algebra/help/en_US/eigen/gschur.xml b/modules/linear_algebra/help/en_US/eigen/gschur.xml new file mode 100755 index 000000000..2c9d2ae15 --- /dev/null +++ b/modules/linear_algebra/help/en_US/eigen/gschur.xml @@ -0,0 +1,99 @@ + + + + + gschur + + generalized Schur form. This function is obsolete. + + + + Calling Sequence + [As,Es]=gschur(A,E) + [As,Es,Q,Z]=gschur(A,E) + [As,Es,Z,dim] = gschur(A,E,flag) + [As,Es,Z,dim]= gschur(A,E,extern) + + + + Description + + This function is obsolete and is now included in the schur + function. In most cases the gschur function will still work as + before, but it will be removed in the future release. + + + The first three syntaxes can be replaced by + + + + The last syntax requires little more adaptations: + + + + if + + extern is a scilab function the new calling sequence + should be [As,Es,Z,dim]= schur(A,E,Nextern) with + Nextern defined as follow: + + + + + + if + + extern is the name of an external function coded in Fortran or C + the new calling sequence should be [As,Es,Z,dim]= schur(A,E,'nextern') with nextern defined as follow: + + + + + + + + See Also + + + external + + + schur + + + + diff --git a/modules/linear_algebra/help/en_US/eigen/gspec.xml b/modules/linear_algebra/help/en_US/eigen/gspec.xml new file mode 100755 index 000000000..a5c5043a0 --- /dev/null +++ b/modules/linear_algebra/help/en_US/eigen/gspec.xml @@ -0,0 +1,45 @@ + + + + + gspec + + eigenvalues of matrix pencil. This function is obsolete. + + + + Calling Sequence + [al,be]=gspec(A,E) + [al,be,Z]=gspec(A,E) + + + + Description + + This function is now included in the spec function. + the calling syntax must be replaced by + + + + + See Also + + + spec + + + + diff --git a/modules/linear_algebra/help/en_US/eigen/hess.xml b/modules/linear_algebra/help/en_US/eigen/hess.xml new file mode 100755 index 000000000..5f6add49f --- /dev/null +++ b/modules/linear_algebra/help/en_US/eigen/hess.xml @@ -0,0 +1,97 @@ + + + + + hess + Hessenberg form + + + Calling Sequence + H = hess(A) + [U,H] = hess(A) + + + + Arguments + + + A + + real or complex square matrix + + + + H + + real or complex square matrix + + + + U + + orthogonal or unitary square matrix + + + + + + Description + + [U,H] = hess(A) produces a unitary matrix + U and a Hessenberg matrix H so that + A = U*H*U' and U'*U = + Identity. By itself, hess(A) returns H. + + + The Hessenberg form of a matrix is zero below the first + subdiagonal. If the matrix is symmetric or Hermitian, the form is + tridiagonal. + + + + References + + hess function is based on the Lapack routines + DGEHRD, DORGHR for real matrices and ZGEHRD, ZORGHR for the complex case. + + + + Examples + + + + See Also + + + qr + + + contr + + + schur + + + + + Used Functions + + hess function is based on the Lapack routines + DGEHRD, DORGHR for real matrices and ZGEHRD, ZORGHR for the + complex case. + + + diff --git a/modules/linear_algebra/help/en_US/eigen/pbig.xml b/modules/linear_algebra/help/en_US/eigen/pbig.xml new file mode 100755 index 000000000..0d7f8a772 --- /dev/null +++ b/modules/linear_algebra/help/en_US/eigen/pbig.xml @@ -0,0 +1,125 @@ + + + + + pbig + eigen-projection + + + Calling Sequence + [Q,M]=pbig(A,thres,flag) + + + Arguments + + + A + + real square matrix + + + + thres + + real number + + + + flag + + + character string ('c' or 'd') + + + + + Q,M + + real matrices + + + + + + Description + + Projection on eigen-subspace associated with eigenvalues with real + part >= thres (flag='c') or + with magnitude >= thres + (flag='d'). + + + The projection is defined by Q*M, Q is + full column rank, M is full row rank and + M*Q=eye. + + + If flag='c', the eigenvalues of + M*A*Q = eigenvalues of A with real part + >= thres. + + + If flag='d', the eigenvalues of + M*A*Q = eigenvalues of A with magnitude + >= thres. + + + If flag='c' and if [Q1,M1] = + full rank factorization (fullrf) of + eye()-Q*M then eigenvalues of M1*A*Q1 = + eigenvalues of A with real part < + thres. + + + If flag='d' and if [Q1,M1] = + full rank factorization (fullrf) of + eye()-Q*M then eigenvalues of M1*A*Q1 = + eigenvalues of A with magnitude < + thres. + + + + Examples + + + + See Also + + + psmall + + + projspec + + + fullrf + + + schur + + + + + Used Functions + + pbig is based on the ordered schur form (scilab + function schur). + + + diff --git a/modules/linear_algebra/help/en_US/eigen/projspec.xml b/modules/linear_algebra/help/en_US/eigen/projspec.xml new file mode 100755 index 000000000..d79f4573f --- /dev/null +++ b/modules/linear_algebra/help/en_US/eigen/projspec.xml @@ -0,0 +1,90 @@ + + + + + projspec + spectral operators + + + Calling Sequence + [S,P,D,i]=projspec(A) + + + Arguments + + + A + + square matrix + + + + S, P, D + + square matrices + + + + i + + + integer (index of the zero eigenvalue of A). + + + + + + + Description + + Spectral characteristics of A at 0. + + + S = reduced resolvent at 0 (S = -Drazin_inverse(A)). + + + P = spectral projection at 0. + + + D = nilpotent operator at 0. + + + index = index of the 0 eigenvalue. + + + One has (s*eye()-A)^(-1) = D^(i-1)/s^i +... + D/s^2 + P/s - S - s*S^2 -... + around the singularity s=0. + + + + Examples + + + + See Also + + + coff + + + + diff --git a/modules/linear_algebra/help/en_US/eigen/psmall.xml b/modules/linear_algebra/help/en_US/eigen/psmall.xml new file mode 100755 index 000000000..c3823df65 --- /dev/null +++ b/modules/linear_algebra/help/en_US/eigen/psmall.xml @@ -0,0 +1,122 @@ + + + + + psmall + spectral projection + + + Calling Sequence + [Q,M]=psmall(A,thres,flag) + + + Arguments + + + A + + real square matrix + + + + thres + + real number + + + + flag + + + character string ('c' or 'd') + + + + + Q,M + + real matrices + + + + + + Description + + Projection on eigen-subspace associated with eigenvalues with real + part < thres (flag='c') or + with modulus < thres + (flag='d'). + + + The projection is defined by Q*M, Q is + full column rank, M is full row rank and + M*Q=eye. + + + If flag='c', the eigenvalues of + M*A*Q = eigenvalues of A with real part + < thres. + + + If flag='d', the eigenvalues of + M*A*Q = eigenvalues of A with magnitude + < thres. + + + If flag='c' and if [Q1,M1] = + full rank factorization (fullrf) of + eye()-Q*M then eigenvalues of M1*A*Q1 = + eigenvalues of A with real part >= + thres. + + + If flag='d' and if [Q1,M1] = + full rank factorization (fullrf) of + eye()-Q*M then eigenvalues of M1*A*Q1 = + eigenvalues of A with magnitude >= + thres. + + + + Examples + + + + See Also + + + pbig + + + proj + + + projspec + + + + + Used Functions + + This function is based on the ordered schur form (scilab + function schur). + + + diff --git a/modules/linear_algebra/help/en_US/eigen/schur.xml b/modules/linear_algebra/help/en_US/eigen/schur.xml new file mode 100755 index 000000000..fe17a979c --- /dev/null +++ b/modules/linear_algebra/help/en_US/eigen/schur.xml @@ -0,0 +1,386 @@ + + + + + schur + [ordered] Schur decomposition of matrix and pencils + + + Calling Sequence + [U,T] = schur(A) + [U,dim [,T] ]=schur(A,flag) + [U,dim [,T] ]=schur(A,extern1) + + [As,Es [,Q,Z]]=schur(A,E) + [As,Es [,Z,dim]] = schur(A,E,flag) + [Z,dim] = schur(A,E,flag) + [As,Es [,Z,dim]]= schur(A,E,extern2) + [Z,dim]= schur(A,E,extern2) + + + + Arguments + + + A + + real or complex square matrix. + + + + E + + + real or complex square matrix with same dimensions as A. + + + + + flag + + + character string ('c' or 'd') + + + + + extern1 + + an ``external'', see below + + + + extern2 + + an ``external'', see below + + + + U + + orthogonal or unitary square matrix + + + + Q + + orthogonal or unitary square matrix + + + + Z + + orthogonal or unitary square matrix + + + + T + + upper triangular or quasi-triangular square matrix + + + + As + + upper triangular or quasi-triangular square matrix + + + + Es + + upper triangular square matrix + + + + dim + + integer + + + + + + Description + + Schur forms, ordered Schur forms of matrices and pencils + + + + MATRIX SCHUR FORM + + + + Usual schur form: + + + [U,T] = schur(A) produces a Schur matrix + T and a unitary matrix U so that + A = U*T*U' and U'*U = eye(U). By itself, schur(A) returns + T. If A is complex, the Complex + Schur Form is returned in matrix + T. The Complex Schur Form is upper triangular with + the eigenvalues of A on the diagonal. If + A is real, the Real Schur Form is returned. The Real + Schur Form has the real eigenvalues on the diagonal and the + complex eigenvalues in 2-by-2 blocks on the diagonal. + + + + + Ordered Schur forms + + + [U,dim]=schur(A,'c') returns an unitary + matrix U which transforms A into schur + form. In addition, the dim first columns of U make + a basis of the eigenspace of A associated with + eigenvalues with negative real parts (stable "continuous + time" eigenspace). + + + [U,dim]=schur(A,'d') returns an unitary + matrix U which transforms A into schur + form. In addition, the dim first columns of + U span a basis of the eigenspace of A + associated with eigenvalues with magnitude lower than 1 (stable + "discrete time" eigenspace). + + + [U,dim]=schur(A,extern1) returns an unitary matrix + U which transforms A into schur form. + In addition, the dim first columns of + U span a basis of the eigenspace of A + associated with the eigenvalues which are selected by the + external function extern1 (see external for + details). This external can be described by a Scilab function + or by C or Fortran procedure: + + + + a Scilab function + + + If extern1 is described by a Scilab function, it + should have the following calling sequence: + s=extern1(Ev), where Ev is an eigenvalue and + s a boolean. + + + + + a C or Fortran procedure + + + If extern1 is described by a C or Fortran function it + should have the following calling sequence: + int extern1(double *EvR, double *EvI) + where EvR and EvI are eigenvalue real and complex parts. + a true or non zero returned value stands for selected eigenvalue. + + + + + + + + + + + PENCIL SCHUR FORMS + + + + Usual Pencil Schur form + + + [As,Es] = schur(A,E) produces a quasi triangular + As matrix and a triangular Es matrix + which are the generalized Schur form of the pair A, E. + + + [As,Es,Q,Z] = schur(A,E) + returns in addition two unitary matrices + Q and Z such that + As=Q'*A*Z and Es=Q'*E*Z. + + + + + Ordered Schur forms: + + + [As,Es,Z,dim] = schur(A,E,'c') + returns the real generalized + Schur form of the pencil s*E-A. In addition, the dim first columns + of Z span a basis of the right eigenspace associated with + eigenvalues with negative real parts (stable "continuous + time" generalized eigenspace). + + + [As,Es,Z,dim] = schur(A,E,'d') + + + returns the real generalized + Schur form of the pencil s*E-A. In addition, the dim first columns + of Z make a basis of the right eigenspace associated with + eigenvalues with magnitude lower than 1 (stable "discrete + time" generalized eigenspace). + + + [As,Es,Z,dim] = schur(A,E,extern2) + + + returns the real generalized Schur form of the pencil s*E-A. + In addition, the dim first columns + of Z make a basis of the right eigenspace associated with + eigenvalues of the pencil which are selected according to a + rule which is given by the function extern2. (see external + for details). This external can be described by a Scilab + function or by C or Fortran procedure: + + + + A Scilab function + + + If extern2 is described by a Scilab function, it should + have the following calling sequence: + s=extern2(Alpha,Beta), where Alpha and + Beta defines a generalized eigenvalue and + s a boolean. + + + + + C or Fortran procedure + + + if external extern2 is described by a C or a + Fortran procedure, it should have the following calling + sequence: + + + int extern2(double *AlphaR, double *AlphaI, double *Beta) + + + if A and E are real and + + + int extern2(double *AlphaR, double *AlphaI, double *BetaR, double *BetaI) + + + if A or E are complex. + Alpha, and Beta defines the generalized eigenvalue. + a true or non zero returned value stands for selected generalized eigenvalue. + + + + + + + + + + + + + References + + Matrix schur form computations are based on the Lapack routines DGEES and ZGEES. + + + Pencil schur form computations are based on the Lapack routines DGGES and ZGGES. + + + + Examples + + + + See Also + + + spec + + + bdiag + + + ricc + + + pbig + + + psmall + + + + diff --git a/modules/linear_algebra/help/en_US/eigen/spec.xml b/modules/linear_algebra/help/en_US/eigen/spec.xml new file mode 100755 index 000000000..4565a68fd --- /dev/null +++ b/modules/linear_algebra/help/en_US/eigen/spec.xml @@ -0,0 +1,301 @@ + + + + + spec + eigenvalues of matrices and pencils + + + Calling Sequence + evals=spec(A) + [R,diagevals]=spec(A) + + evals=spec(A,B) + [alpha,beta]=spec(A,B) + [alpha,beta,Z]=spec(A,B) + [alpha,beta,Q,Z]=spec(A,B) + + + + Arguments + + + A + + real or complex square matrix + + + + B + + real or complex square matrix with same dimensions as + A + + + + + evals + + real or complex vector, the eigenvalues + + + + diagevals + + real or complex diagonal matrix (eigenvalues along the + diagonal) + + + + + alpha + + real or complex vector, al./be gives the eigenvalues + + + + beta + + real vector, al./be gives the eigenvalues + + + + R + + real or complex invertible square matrix, matrix right + eigenvectors. + + + + + L + + real or complex invertible square matrix, pencil left + eigenvectors. + + + + + R + + real or complex invertible square matrix, pencil right + eigenvectors. + + + + + + + Description + + + evals=spec(A) + + + returns in vector evals the + eigenvalues. + + + + + [R,diagevals] =spec(A) + + + returns in the diagonal matrix evals the + eigenvalues and in R the right + eigenvectors. + + + + + evals=spec(A,B) + + returns the spectrum of the matrix pencil A - s B, i.e. the + roots of the polynomial matrix s B - A. + + + + + [alpha,beta] = spec(A,B) + + + returns the spectrum of the matrix pencil A- s + B + + ,i.e. the roots of the polynomial matrix A - s + B + + .Generalized eigenvalues alpha and beta are so that the + matrix A - alpha./beta B is a singular matrix. + The eigenvalues are given by al./be and if + beta(i) = 0 the ith eigenvalue is at infinity. + (For B = eye(A), alpha./beta is + spec(A)). It is usually represented as the pair + (alpha,beta), as there is a reasonable interpretation for beta=0, + and even for both being zero. + + + + + [alpha,beta,R] = spec(A,B) + + + returns in addition the matrix R of + generalized right eigenvectors of the pencil. + + + + + [al,be,L,R] = spec(A,B) + + + returns in addition the matrix L and + R of generalized left and right eigenvectors of + the pencil. + + + + + [al,be,Z] = spec(A,E) + + + returns the matrix Z of right + generalized eigen vectors. + + + + + [al,be,Q,Z] = spec(A,E) + + + returns the matrices Q + and Z of right and left generalized + eigen vectors. + + + + + For big full / sparse matrix, you can use the Arnoldi module. + + + References + Matrix eigenvalues computations are based on the Lapack + routines + + + + DGEEV and ZGEEV when the matrix are not symmetric, + + + DSYEV and ZHEEV when the matrix are symmetric. + + + A complex symmetric matrix has conjugate offdiagonal terms and real + diagonal terms. + + Pencil eigenvalues computations are based on the Lapack routines + DGGEV and ZGGEV. + + + + Real and complex matrices + It must be noticed that the type of the output variables, such as + evals or R for example, is not necessarily the same as the type of the + input matrices A and B. In the following paragraph, we analyse the type of + the output variables in the case where one computes the eigenvalues and + eigenvectors of one single matrix A. + + + + Real A matrix + + + Symetric + The eigenvalues and the eigenvectors are real. + + + Not symmetric + The eigenvalues and eigenvectors are complex. + + + + + Complex A matrix + + + Symetric + The eigenvalues are real but the eigenvectors are + complex. + + + + Not symmetric + The eigenvalues and the eigenvectors are complex. + + + + + + + Examples + + + + See Also + + + poly + + + det + + + schur + + + bdiag + + + colcomp + + + dsaupd + + + dnaupd + + + + diff --git a/modules/linear_algebra/help/en_US/eigen/sva.xml b/modules/linear_algebra/help/en_US/eigen/sva.xml new file mode 100755 index 000000000..4fd62c45c --- /dev/null +++ b/modules/linear_algebra/help/en_US/eigen/sva.xml @@ -0,0 +1,84 @@ + + + + + sva + singular value approximation + + + Calling Sequence + [U,s,V]=sva(A,k) + [U,s,V]=sva(A,tol) + + + + Arguments + + + A + + real or complex matrix + + + + k + + integer + + + + tol + + nonnegative real number + + + + + + Description + + Singular value approximation. + + + [U,S,V]=sva(A,k) with k an integer + >=1, returns U,S and V such that + B=U*S*V' is the best L2 approximation of + A with rank(B)=k. + + + [U,S,V]=sva(A,tol) with tol a real + number, returns U,S and V such that + B=U*S*V' such that L2-norm of A-B + is at most tol. + + + + Examples + + + + See Also + + + svd + + + + diff --git a/modules/linear_algebra/help/en_US/eigen/svd.xml b/modules/linear_algebra/help/en_US/eigen/svd.xml new file mode 100755 index 000000000..24c626e38 --- /dev/null +++ b/modules/linear_algebra/help/en_US/eigen/svd.xml @@ -0,0 +1,126 @@ + + + + + svd + singular value decomposition + + + Calling Sequence + s=svd(X) + [U,S,V]=svd(X) + [U,S,V]=svd(X,0) (obsolete) + [U,S,V]=svd(X,"e") + [U,S,V,rk]=svd(X [,tol]) + + + + Arguments + + + X + + a real or complex matrix + + + + s + + real vector (singular values) + + + + S + + real diagonal matrix (singular values) + + + + U,V + + orthogonal or unitary square matrices (singular vectors). + + + + tol + + real number + + + + + + Description + + [U,S,V] = svd(X) produces a diagonal matrix + S , of the same dimension as X and with + nonnegative diagonal elements in decreasing order, and unitary + matrices U and V so that X = U*S*V'. + + + [U,S,V] = svd(X,0) produces the "economy + size" decomposition. If X is m-by-n with m > + n, then only the first n columns of U are computed + and S is n-by-n. + + + s= svd(X) by itself, returns a vector s + containing the singular values. + + + [U,S,V,rk]=svd(X,tol) gives in addition rk, the numerical rank of X i.e. the number of + singular values larger than tol. + + + The default value of tol is the same as in rank. + + + + Examples + + + + See Also + + + rank + + + qr + + + colcomp + + + rowcomp + + + sva + + + spec + + + + + Used Functions + + svd decompositions are based on the Lapack routines DGESVD for + real matrices and ZGESVD for the complex case. + + + diff --git a/modules/linear_algebra/help/en_US/factorization/CHAPTER b/modules/linear_algebra/help/en_US/factorization/CHAPTER new file mode 100755 index 000000000..e6daeb8eb --- /dev/null +++ b/modules/linear_algebra/help/en_US/factorization/CHAPTER @@ -0,0 +1,2 @@ +title = Factorization + diff --git a/modules/linear_algebra/help/en_US/factorization/givens.xml b/modules/linear_algebra/help/en_US/factorization/givens.xml new file mode 100755 index 000000000..355899ae0 --- /dev/null +++ b/modules/linear_algebra/help/en_US/factorization/givens.xml @@ -0,0 +1,86 @@ + + + + + givens + Givens transformation + + + Calling Sequence + U=givens(xy) + U=givens(x,y) + [U,c]=givens(xy) + [U,c]=givens(x,y) + + + + Arguments + + + x,y + + two real or complex numbers + + + + xy + + real or complex size 2 column vector + + + + U + + 2x2 unitary matrix + + + + c + + real or complex size 2 column vector + + + + + + Description + + U= givens(x, y) or U = givens(xy) with xy = [x;y] + returns a 2x2 unitary matrix U such that: + + + U*xy=[r;0]=c. + + + + Note that givens(x,y) and givens([x;y]) are equivalent. + + + + + Examples + + + + See Also + + + qr + + + + diff --git a/modules/linear_algebra/help/en_US/factorization/householder.xml b/modules/linear_algebra/help/en_US/factorization/householder.xml new file mode 100755 index 000000000..c664395a9 --- /dev/null +++ b/modules/linear_algebra/help/en_US/factorization/householder.xml @@ -0,0 +1,70 @@ + + + + + householder + Householder orthogonal reflexion matrix + + + Calling Sequence + u=householder(v [,w]) + + + Arguments + + + v + + real or complex column vector + + + + w + + + real or complex column vector with same size as v. Default value is eye(v) + + + + + u + + real or complex column vector + + + + + + Description + + given 2 column vectors v, w of same size, householder(v,w) returns a unitary + column vector u, such that (eye()-2*u*u')*v is proportional to w. + (eye()-2*u*u') is the orthogonal Householder reflexion matrix . + + + w default value is eye(v). In this case vector (eye()-2*u*u')*v is the + vector eye(v)*norm(v). + + + + See Also + + + qr + + + givens + + + + diff --git a/modules/linear_algebra/help/en_US/factorization/sqroot.xml b/modules/linear_algebra/help/en_US/factorization/sqroot.xml new file mode 100755 index 000000000..e248970f3 --- /dev/null +++ b/modules/linear_algebra/help/en_US/factorization/sqroot.xml @@ -0,0 +1,62 @@ + + + + + sqroot + W*W' hermitian factorization + + + Calling Sequence + sqroot(X) + + + Arguments + + + X + + symmetric non negative definite real or complex matrix + + + + + + Description + + returns W such that X=W*W' (uses SVD). + + + + Examples + + + + See Also + + + chol + + + svd + + + + diff --git a/modules/linear_algebra/help/en_US/kernel/CHAPTER b/modules/linear_algebra/help/en_US/kernel/CHAPTER new file mode 100755 index 000000000..be67920e1 --- /dev/null +++ b/modules/linear_algebra/help/en_US/kernel/CHAPTER @@ -0,0 +1,2 @@ +title = Kernel + diff --git a/modules/linear_algebra/help/en_US/kernel/colcomp.xml b/modules/linear_algebra/help/en_US/kernel/colcomp.xml new file mode 100755 index 000000000..7d658352e --- /dev/null +++ b/modules/linear_algebra/help/en_US/kernel/colcomp.xml @@ -0,0 +1,106 @@ + + + + + colcomp + column compression, kernel, nullspace + + + Calling Sequence + [W,rk]=colcomp(A [,flag] [,tol]) + + + Arguments + + + A + + real or complex matrix + + + + flag + + character string + + + + tol + + real number + + + + W + + square non-singular matrix (change of basis) + + + + rk + + + integer (rank of A) + + + + + + + Description + + Column compression of A: Ac = A*W is + column compressed i.e + + + Ac=[0,Af] with Af full column rank, + rank(Af) = rank(A) = rk. + + + flag and tol are optional parameters: flag = 'qr' + or 'svd' (default is 'svd'). + + + tol = tolerance parameter (of order %eps as default value). + + + The ma-rk first columns of W span the kernel of A + when size(A)=(na,ma) + + + + Examples + + + + See Also + + + rowcomp + + + fullrf + + + fullrfk + + + kernel + + + + diff --git a/modules/linear_algebra/help/en_US/kernel/fullrf.xml b/modules/linear_algebra/help/en_US/kernel/fullrf.xml new file mode 100755 index 000000000..0343f81dc --- /dev/null +++ b/modules/linear_algebra/help/en_US/kernel/fullrf.xml @@ -0,0 +1,101 @@ + + + + + fullrf + full rank factorization + + + Calling Sequence + [Q,M,rk]=fullrf(A,[tol]) + + + Arguments + + + A + + real or complex matrix + + + + tol + + real number (threshold for rank determination) + + + + Q,M + + real or complex matrix + + + + rk + + + integer (rank of A) + + + + + + + Description + + Full rank factorization : fullrf returns Q and M such + that A = Q*M + with range(Q)=range(A) and ker(M)=ker(A), + Q full column rank , M full row rank, + rk = rank(A) = #columns(Q) = #rows(M). + + + tol is an optional real parameter (default value is sqrt(%eps)). + The rank rk of A is defined as the number of singular values + larger than norm(A)*tol. + + + If A is symmetric, fullrf returns M=Q'. + + + + Examples + + + + See Also + + + svd + + + qr + + + fullrfk + + + rowcomp + + + colcomp + + + + diff --git a/modules/linear_algebra/help/en_US/kernel/fullrfk.xml b/modules/linear_algebra/help/en_US/kernel/fullrfk.xml new file mode 100755 index 000000000..f060add2e --- /dev/null +++ b/modules/linear_algebra/help/en_US/kernel/fullrfk.xml @@ -0,0 +1,74 @@ + + + + + fullrfk + full rank factorization of A^k + + + Calling Sequence + [Bk,Ck]=fullrfk(A,k) + + + Arguments + + + A + + real or complex matrix + + + + k + + integer + + + + Bk,Ck + + real or complex matrices + + + + + + Description + + This function computes the full rank factorization of A^k i.e. + Bk*Ck=A^k where Bk is full column rank and Ck full row rank. + One has range(Bk)=range(A^k) and ker(Ck)=ker(A^k). + + + For k=1, fullrfk is equivalent to fullrf. + + + + Examples + + + + See Also + + + fullrf + + + range + + + + diff --git a/modules/linear_algebra/help/en_US/kernel/im_inv.xml b/modules/linear_algebra/help/en_US/kernel/im_inv.xml new file mode 100755 index 000000000..2e288855e --- /dev/null +++ b/modules/linear_algebra/help/en_US/kernel/im_inv.xml @@ -0,0 +1,107 @@ + + + + + im_inv + inverse image + + + Calling Sequence + [X,dim]=im_inv(A,B [,tol]) + [X,dim,Y]=im_inv(A,B, [,tol]) + + + + Arguments + + + A,B + + two real or complex matrices with equal number of columns + + + + X + + + orthogonal or unitary square matrix of order equal to the number of columns of A + + + + + dim + + integer (dimension of subspace) + + + + Y + + + orthogonal matrix of order equal to the number of rows of A and B. + + + + + + + Description + + [X,dim]=im_inv(A,B) computes (A^-1)(B) + i.e vectors whose image through A are in + range(B) + + + The dim first columns of X span + (A^-1)(B) + + + tol is a threshold used to test if subspace inclusion; + default value is tol = 100*%eps. + If Y is returned, then [Y*A*X,Y*B] is partitioned as follows: + [A11,A12;0,A22],[B1;0] + + + where B1 has full row rank (equals + rank(B)) and A22 has full column rank + and has dim columns. + + + + Examples + + + + See Also + + + rowcomp + + + spaninter + + + spanplus + + + linsolve + + + + diff --git a/modules/linear_algebra/help/en_US/kernel/kernel.xml b/modules/linear_algebra/help/en_US/kernel/kernel.xml new file mode 100755 index 000000000..f891512d1 --- /dev/null +++ b/modules/linear_algebra/help/en_US/kernel/kernel.xml @@ -0,0 +1,93 @@ + + + + + kernel + kernel, null space + + + Calling Sequence + W=kernel(A [,tol,[,flag]) + + + Arguments + + + A + + full real or complex matrix or real sparse matrix + + + + flag + + + character string 'svd' (default) or 'qr' + + + + + tol + + real number + + + + W + + full column rank matrix + + + + + + Description + + W=kernel(A) returns the kernel (null space) of A, and size(W,2) is the nullity of A. + If A has full column rank then an empty matrix [] is returned. + + + flag and tol are optional parameters: flag = 'qr' + or 'svd' (default is 'svd'). + + + tol = tolerance parameter (of order %eps as default value). + + + + Examples + + + + See Also + + + colcomp + + + fullrf + + + fullrfk + + + linsolve + + + + diff --git a/modules/linear_algebra/help/en_US/kernel/range.xml b/modules/linear_algebra/help/en_US/kernel/range.xml new file mode 100755 index 000000000..b8def609f --- /dev/null +++ b/modules/linear_algebra/help/en_US/kernel/range.xml @@ -0,0 +1,94 @@ + + + + + range + range (span) of A^k + + + Calling Sequence + [X,dim]=range(A,k) + + + Arguments + + + A + + real square matrix + + + + k + + integer + + + + X + + orthonormal real matrix + + + + dim + + integer (dimension of subspace) + + + + + + Description + + Computation of Range A^k ; the first dim rows of X span the + range of A^k. The last rows of X span the + orthogonal complement of the range. X*X' is the Identity matrix + + + + Examples + + + + See Also + + + fullrfk + + + rowcomp + + + + + Used Functions + + The range function is based on the rowcomp function + which uses the svd decomposition. + + + diff --git a/modules/linear_algebra/help/en_US/kernel/rowcomp.xml b/modules/linear_algebra/help/en_US/kernel/rowcomp.xml new file mode 100755 index 000000000..f012c21f6 --- /dev/null +++ b/modules/linear_algebra/help/en_US/kernel/rowcomp.xml @@ -0,0 +1,123 @@ + + + + + rowcomp + row compression, range + + + Calling Sequence + [W,rk]=rowcomp(A [,flag [,tol]]) + + + Arguments + + + A + + real or complex matrix + + + + flag + + optional character string, with possible values + 'svd' or 'qr'. The default value is 'svd'. + + + + + tol + + optional real non negative number. The default value is + sqrt(%eps)*norm(A,1). + + + + + W + + square non-singular matrix (change of basis) + + + + rk + + + integer (rank of A) + + + + + + + Description + + Row compression of A. Ac = W*A is a row compressed matrix: i.e. + Ac=[Af;0] with Af full row rank. + + + flag and tol are optional parameters: flag='qr' + or 'svd' (default 'svd'). + + + tol is a tolerance parameter. + + + The rk first columns of W' span the range of + A. + + + The rk first (top) rows of W span the row + range of A. + + + A non zero vector x belongs to range(A) iff + W*x is row compressed in accordance with Ac + i.e the norm of its last components is small w.r.t its first + components. + + + + Examples + + + + See Also + + + colcomp + + + fullrf + + + fullrfk + + + + + Used Functions + + The rowcomp function is based on the svd or + qr decompositions. + + + diff --git a/modules/linear_algebra/help/en_US/linear/CHAPTER b/modules/linear_algebra/help/en_US/linear/CHAPTER new file mode 100755 index 000000000..7d9d9cf49 --- /dev/null +++ b/modules/linear_algebra/help/en_US/linear/CHAPTER @@ -0,0 +1,2 @@ +title = Linear Equations + diff --git a/modules/linear_algebra/help/en_US/linear/aff2ab.xml b/modules/linear_algebra/help/en_US/linear/aff2ab.xml new file mode 100755 index 000000000..756469256 --- /dev/null +++ b/modules/linear_algebra/help/en_US/linear/aff2ab.xml @@ -0,0 +1,150 @@ + + + + + aff2ab + linear (affine) function to A,b conversion + + + Calling Sequence + [A,b]=aff2ab(afunction,dimX,D [,flag]) + + + Arguments + + + afunction + + + a scilab function Y =fct(X,D) where X, D, Y are list of matrices + + + + + dimX + + + a p x 2 integer matrix (p is the number of matrices in X) + + + + + D + + + a list of real matrices (or any other valid Scilab object). + + + + + flag + + + optional parameter (flag='f' or flag='sp') + + + + + A + + a real matrix + + + + b + + + a real vector having same row dimension as A + + + + + + + Description + + aff2ab returns the matrix representation of an affine + function (in the canonical basis). + + + afunction is a function with imposed syntax: + Y=afunction(X,D) where X=list(X1,X2,...,Xp) is + a list of p real matrices, and Y=list(Y1,...,Yq) is + a list of q real real matrices which depend linearly of + the Xi's. The (optional) input D contains + parameters needed to compute Y as a function of X. + (It is generally a list of matrices). + + + dimX is a p x 2 matrix: dimX(i)=[nri,nci] + is the actual number of rows and columns of matrix Xi. + These dimensions determine na, the column dimension of + the resulting matrix A: na=nr1*nc1 +...+ nrp*ncp. + + + If the optional parameter flag='sp' the resulting A + matrix is returned as a sparse matrix. + + + This function is useful to solve a system of linear equations + where the unknown variables are matrices. + + + + Examples + + + + See Also + + + linsolve + + + + diff --git a/modules/linear_algebra/help/en_US/linear/chol.xml b/modules/linear_algebra/help/en_US/linear/chol.xml new file mode 100755 index 000000000..a0a37f1de --- /dev/null +++ b/modules/linear_algebra/help/en_US/linear/chol.xml @@ -0,0 +1,81 @@ + + + + + chol + Cholesky factorization + + + Calling Sequence + [R]=chol(X) + + + Arguments + + + X + + a symmetric positive definite real or complex matrix. + + + + + + Description + + If X is positive definite, then R = chol(X) produces an upper + triangular matrix R such that R'*R = X. + + + chol(X) uses only the diagonal and upper triangle of X. + The lower triangular is assumed to be the (complex conjugate) + transpose of the upper. + + + + References + + Cholesky decomposition is based on the Lapack routines + DPOTRF for real matrices and ZPOTRF for the complex case. + + + + Examples + + + + See Also + + + spchol + + + qr + + + svd + + + bdiag + + + fullrf + + + + diff --git a/modules/linear_algebra/help/en_US/linear/inv.xml b/modules/linear_algebra/help/en_US/linear/inv.xml new file mode 100755 index 000000000..1589972e7 --- /dev/null +++ b/modules/linear_algebra/help/en_US/linear/inv.xml @@ -0,0 +1,105 @@ + + + + + inv + matrix inverse + + + Calling Sequence + inv(X) + + + Arguments + + + X + + real or complex square matrix, polynomial matrix, rational matrix in transfer or state-space representation. + + + + + + Description + + inv(X) is the inverse of the square matrix X. A warning + message is printed if X is badly scaled or nearly singular. + + + For polynomial matrices or rational matrices in transfer representation, + inv(X) is equivalent to invr(X). + + + For linear systems in state-space representation (syslin list), + invr(X) is equivalent to invsyslin(X). + + + + References + + inv function for matrices of numbers is based on the Lapack routines + DGETRF, DGETRI for real matrices and ZGETRF, ZGETRI for the complex case. + For polynomial matrix and rational function matrix inv is based on the invr + Scilab function. + + + + Examples + + + + See Also + + + slash + + + backslash + + + pinv + + + qr + + + lufact + + + lusolve + + + invr + + + coff + + + coffg + + + + diff --git a/modules/linear_algebra/help/en_US/linear/linsolve.xml b/modules/linear_algebra/help/en_US/linear/linsolve.xml new file mode 100755 index 000000000..01df2dc47 --- /dev/null +++ b/modules/linear_algebra/help/en_US/linear/linsolve.xml @@ -0,0 +1,121 @@ + + + + + linsolve + linear equation solver + + + Calling Sequence + [x0,kerA]=linsolve(A,b [,x0]) + + + Arguments + + + A + + + a na x ma real matrix (possibly sparse) + + + + + b + + + a na x 1 vector (same row dimension as A) + + + + + x0 + + a real vector + + + + kerA + + + a ma x k real matrix + + + + + + + Description + + linsolve computes all the solutions to A*x+b=0. + + + x0 is a particular solution (if any) and kerA= nullspace + of A. Any x=x0+kerA*w with arbitrary w satisfies + A*x+b=0. + + + If compatible x0 is given on entry, x0 is returned. If not + a compatible x0, if any, is returned. + + + + Examples + + + + See Also + + + inv + + + pinv + + + colcomp + + + im_inv + + + umfpack + + + backslash + + + + diff --git a/modules/linear_algebra/help/en_US/linear/lsq.xml b/modules/linear_algebra/help/en_US/linear/lsq.xml new file mode 100755 index 000000000..1060d605b --- /dev/null +++ b/modules/linear_algebra/help/en_US/linear/lsq.xml @@ -0,0 +1,113 @@ + + + + + lsq + linear least square problems. + + + Calling Sequence + X=lsq(A,B [,tol]) + + + Arguments + + + A + + Real or complex (m x n) matrix + + + + B + + real or complex (m x p) matrix + + + + tol + + positive scalar, used to determine the effective rank of A + (defined as the order of the largest leading triangular + submatrix R11 in the QR factorization with pivoting of A, + whose estimated condition number <= 1/tol. The tol default value is + set to sqrt(%eps). + + + + + X + + real or complex (n x p) matrix + + + + + + Description + + X=lsq(A,B) computes the minimum norm least square solution of + the equation A*X=B, while X=A \ B compute a least square + solution with at at most rank(A) nonzero components per column. + + + + References + + lsq function is based on the LApack functions DGELSY for + real matrices and ZGELSY for complex matrices. + + + + Examples + + + + See Also + + + backslash + + + inv + + + pinv + + + rank + + + + diff --git a/modules/linear_algebra/help/en_US/linear/lu.xml b/modules/linear_algebra/help/en_US/linear/lu.xml new file mode 100755 index 000000000..2da9ad568 --- /dev/null +++ b/modules/linear_algebra/help/en_US/linear/lu.xml @@ -0,0 +1,154 @@ + + + + + lu + LU factorization with pivoting + + + Calling Sequence + [L,U]= lu(A) + [L,U,E]= lu(A) + + + + Arguments + + + A + + real or complex matrix (m x n). + + + + L + + real or complex matrices (m x min(m,n)). + + + + U + + real or complex matrices (min(m,n) x n ). + + + + E + + a (n x n) permutation matrix. + + + + + + Description + + [L,U]= lu(A) produces two matrices L and + U such that A = L*U with U + upper triangular and L a general matrix without any particular + structure. In fact, the matrix A is factored as E*A=B*U + where the matrix B is lower triangular + and the matrix L is computed from L=E'*B. + + + If A has rank k, rows k+1 to + n of U are zero. + + + [L,U,E]= lu(A) produces three matrices L, U and + E such that E*A = L*U with + U upper triangular and E*L lower + triangular for a permutation matrix E. + + + If A is a real matrix, using the function + lufact and luget it is possible to obtain + the permutation matrices and also when A is not full + rank the column compression of the matrix L. + + + + Example #1 + + In the following example, we create the Hilbert matrix of size 4 and + factor it with A=LU. Notice that the matrix L is not lower triangular. + To get a lower triangular L matrix, we should have given the + output argument E to Scilab. + + + + + Example #2 + + In the following example, we create the Hilbert matrix of size 4 and + factor it with EA=LU. Notice that the matrix L is lower triangular. + + + + + Example #3 + + The following example shows how to use the lufact and luget functions. + + + + + See Also + + + lufact + + + luget + + + lusolve + + + qr + + + svd + + + + + Used Functions + + lu decompositions are based on the Lapack routines DGETRF for real + matrices and ZGETRF for the complex case. + + + diff --git a/modules/linear_algebra/help/en_US/linear/pinv.xml b/modules/linear_algebra/help/en_US/linear/pinv.xml new file mode 100755 index 000000000..af87b920b --- /dev/null +++ b/modules/linear_algebra/help/en_US/linear/pinv.xml @@ -0,0 +1,83 @@ + + + + + pinv + pseudoinverse + + + Calling Sequence + pinv(A,[tol]) + + + Arguments + + + A + + real or complex matrix + + + + tol + + real number + + + + + + Description + + X= pinv(A) produces a matrix X of the + same dimensions as A' such that: + + + A*X*A = A, X*A*X = X and both + A*X and X*A are Hermitian . + + + The computation is based on SVD and any singular values + lower than a tolerance are treated as zero: this tolerance + is accessed by X=pinv(A,tol). + + + + Examples + + + + See Also + + + rank + + + svd + + + qr + + + + + Used Functions + + pinv function is based on the singular value decomposition + (Scilab function svd). + + + diff --git a/modules/linear_algebra/help/en_US/linear/qr.xml b/modules/linear_algebra/help/en_US/linear/qr.xml new file mode 100755 index 000000000..52b3bb433 --- /dev/null +++ b/modules/linear_algebra/help/en_US/linear/qr.xml @@ -0,0 +1,184 @@ + + + + + qr + QR decomposition + + + Calling Sequence + [Q,R]=qr(X [,"e"]) + [Q,R,E]=qr(X [,"e"]) + [Q,R,rk,E]=qr(X [,tol]) + + + + Arguments + + + X + + real or complex matrix + + + + tol + + nonnegative real number + + + + Q + + square orthogonal or unitary matrix + + + + R + + + matrix with same dimensions as X + + + + + E + + permutation matrix + + + + rk + + + integer (QR-rank of X) + + + + + + + Description + + + [Q,R] = qr(X) + + + produces an upper triangular matrix R of the same dimension as X and an orthogonal (unitary in the complex case) matrix Q so that X = Q*R. [Q,R] = qr(X,"e") produces an "economy size": If X is m-by-n with m > n, then only the first n columns of Q are computed as well as the first n rows of R. + + + From Q*R = X , it follows that + the kth column of the matrix X, is expressed as a linear combination + of the k first columns of Q (with coefficients R(1,k), ..., R(k,k) ). The k first columns of Q make an orthogonal basis + of the subspace spanned by the k first comumns of X. If column k + of X (i.e. X(:,k) ) is a linear combination of the first + p columns of X, then the entries R(p+1,k), ..., R(k,k) + are zero. It this situation, R is upper trapezoidal. If X has + rank rk, rows R(rk+1,:), R(rk+2,:), ... are zeros. + + + + + [Q,R,E] = qr(X) + + + produces a (column) permutation matrix E, an upper + triangular R with decreasing diagonal elements and an + orthogonal (or unitary) Q so that X*E = Q*R. + If rk is the rank of X, the + rk first entries along the main diagonal of + R, i.e. R(1,1), R(2,2), ..., R(rk,rk) + are all different from zero. [Q,R,E] = qr(X,"e") + produces an "economy size": + If X is m-by-n with m > n, then only the first n + columns of Q are computed as well as the first n + rows of R. + + + + + [Q,R,rk,E] = qr(X ,tol) + + + returns rk = rank estimate of X i.e. rk is the number of diagonal elements in R which are larger than a given threshold tol. + + + + + [Q,R,rk,E] = qr(X) + + + returns rk = rank estimate of X + i.e. rk is the number of diagonal elements in + R which are larger than + tol=R(1,1)*%eps*max(size(R)). See rankqr + for a rank revealing QR factorization, using the condition number + of R. + + + + + + + Examples + rk first +//diagonal entries of R are non zero : +A=rand(5,2)*rand(2,5); +[Q,R,rk,E] = qr(A,1.d-10); +norm(Q'*A-R) +svd([A,Q(:,1:rk)]) //span(A) =span(Q(:,1:rk)) + ]]> + + + See Also + + + rankqr + + + rank + + + svd + + + rowcomp + + + colcomp + + + + + Used Functions + + qr decomposition is based the Lapack routines DGEQRF, DGEQPF, + DORGQR for the real matrices and ZGEQRF, ZGEQPF, ZORGQR for the + complex case. + + + diff --git a/modules/linear_algebra/help/en_US/linear/rankqr.xml b/modules/linear_algebra/help/en_US/linear/rankqr.xml new file mode 100755 index 000000000..3532b3daa --- /dev/null +++ b/modules/linear_algebra/help/en_US/linear/rankqr.xml @@ -0,0 +1,150 @@ + + + + + rankqr + rank revealing QR factorization + + + Calling Sequence + [Q,R,JPVT,RANK,SVAL]=rankqr(A, [RCOND,JPVT]) + + + Arguments + + + A + + real or complex matrix + + + + RCOND + + real number used to determine the effective rank of + A, which is defined as the order of the largest leading + triangular submatrix R11 in the QR factorization with + pivoting ofA, whose estimated condition number < + 1/RCOND. + + + + + JPVT + + + integer vector on entry, if JPVT(i) is not 0, the + i-th column of A is permuted to the front + of AP, otherwise column i is a free + column. On exit, if JPVT(i) = k, then the + i-th column of A*P was the + k-th column of A. + + + + + RANK + + + the effective rank of A, i.e., the order of the + submatrix R11. This is the same as the order of the + submatrix T1 in the complete orthogonal factorization + of A. + + + + + SVAL + + real vector with 3 components; The estimates of some of the + singular values of the triangular factor R. + + + SVAL(1) is the largest singular value of + R(1:RANK,1:RANK); + + + SVAL(2) is the + smallest singular value of R(1:RANK,1:RANK); + + + SVAL(3) is the smallest singular value of + R(1:RANK+1,1:RANK+1), if RANK < MIN(M,N), + or of R(1:RANK,1:RANK), otherwise. + + + + + + + Description + + To compute (optionally) a rank-revealing QR factorization of a real + general M-by-N real or complex matrix A, which may be + rank-deficient, and estimate its effective rank using incremental + condition estimation. + + + The routine uses a QR factorization with column pivoting: + + + + with R11 defined as the largest leading submatrix whose + estimated condition number is less than 1/RCOND. The + order of R11, RANK, is the effective rank of + A. + + + If the triangular factorization is a rank-revealing one (which will be + the case if the leading columns were well- conditioned), then + SVAL(1) will also be an estimate for the largest singular + value of A, and SVAL(2) and + SVAL(3) will be estimates for the RANK-th + and (RANK+1)-st singular values of A, + respectively. + + + By examining these values, one can confirm that the + rank is well defined with respect to the chosen value of + RCOND. The ratio SVAL(1)/SVAL(2) is an + estimate of the condition number of R(1:RANK,1:RANK). + + + + Examples + + + + See Also + + + qr + + + rank + + + + + Used Functions + + Slicot library routines MB03OD, ZB03OD. + + + diff --git a/modules/linear_algebra/help/en_US/markov/CHAPTER b/modules/linear_algebra/help/en_US/markov/CHAPTER new file mode 100755 index 000000000..c29eb913c --- /dev/null +++ b/modules/linear_algebra/help/en_US/markov/CHAPTER @@ -0,0 +1,2 @@ +title = Markov Matrices + diff --git a/modules/linear_algebra/help/en_US/markov/classmarkov.xml b/modules/linear_algebra/help/en_US/markov/classmarkov.xml new file mode 100755 index 000000000..a80312508 --- /dev/null +++ b/modules/linear_algebra/help/en_US/markov/classmarkov.xml @@ -0,0 +1,93 @@ + + + + + classmarkov + recurrent and transient classes of Markov matrix + + + Calling Sequence + [perm,rec,tr,indsRec,indsT]=classmarkov(M) + + + Arguments + + + M + + real N x N Markov matrix. Sum of entries in each row should add to one. + + + + perm + + integer permutation vector. + + + + rec, tr + + integer vector, number (number of states in each recurrent classes, number of transient states). + + + + indsRec,indsT + + integer vectors. (Indexes of recurrent and transient states). + + + + + + Description + + Returns a permutation vector perm such that + + + + Each Mii is a Markov matrix of dimension rec(i) i=1,..,r. + Q is sub-Markov matrix of dimension tr. + States 1 to sum(rec) are recurrent and states from r+1 to n + are transient. + One has perm=[indsRec,indsT] where indsRec is a vector of size sum(rec) + and indsT is a vector of size tr. + + + + Examples + + + + See Also + + + genmarkov + + + eigenmarkov + + + + diff --git a/modules/linear_algebra/help/en_US/markov/eigenmarkov.xml b/modules/linear_algebra/help/en_US/markov/eigenmarkov.xml new file mode 100755 index 000000000..950937188 --- /dev/null +++ b/modules/linear_algebra/help/en_US/markov/eigenmarkov.xml @@ -0,0 +1,81 @@ + + + + + eigenmarkov + normalized left and right Markov eigenvectors + + + Calling Sequence + [M,Q]=eigenmarkov(P) + + + Arguments + + + P + + real N x N Markov matrix. Sum of entries in each row should add to one. + + + + M + + real matrix with N columns. + + + + Q + + real matrix with N rows. + + + + + + Description + + Returns normalized left and right eigenvectors + associated with the eigenvalue 1 of the Markov transition matrix P. + If the multiplicity of this eigenvalue is m and P + is N x N, M is a m x N matrix and Q a N x m matrix. + M(k,:) is the probability distribution vector associated with the kth + ergodic set (recurrent class). M(k,x) is zero if x is not in the + k-th recurrent class. + Q(x,k) is the probability to end in the k-th recurrent class starting + from x. If P^k converges for large k (no eigenvalues on the + unit circle except 1), then the limit is Q*M (eigenprojection). + + + + Examples + + + + See Also + + + genmarkov + + + classmarkov + + + + diff --git a/modules/linear_algebra/help/en_US/markov/genmarkov.xml b/modules/linear_algebra/help/en_US/markov/genmarkov.xml new file mode 100755 index 000000000..edf5baeeb --- /dev/null +++ b/modules/linear_algebra/help/en_US/markov/genmarkov.xml @@ -0,0 +1,83 @@ + + + + + genmarkov + generates random markov matrix with recurrent and transient classes + + + Calling Sequence + M=genmarkov(rec,tr) + M=genmarkov(rec,tr,flag) + + + + Arguments + + + rec + + integer row vector (its dimension is the number of recurrent classes). + + + + tr + + integer (number of transient states) + + + + M + + real Markov matrix. Sum of entries in each row should add to one. + + + + flag + + + string 'perm'. If given, a random permutation of the states is done. + + + + + + + Description + + Returns in M a random Markov transition probability matrix + with size(rec,1) recurrent classes with rec(1),...rec($) + entries respectively and tr transient states. + + + + Examples + + + + See Also + + + classmarkov + + + eigenmarkov + + + + diff --git a/modules/linear_algebra/help/en_US/matrix/CHAPTER b/modules/linear_algebra/help/en_US/matrix/CHAPTER new file mode 100755 index 000000000..bb89125cd --- /dev/null +++ b/modules/linear_algebra/help/en_US/matrix/CHAPTER @@ -0,0 +1,2 @@ +title = Matrix Analysis + diff --git a/modules/linear_algebra/help/en_US/matrix/cond.xml b/modules/linear_algebra/help/en_US/matrix/cond.xml new file mode 100755 index 000000000..4c85c76c1 --- /dev/null +++ b/modules/linear_algebra/help/en_US/matrix/cond.xml @@ -0,0 +1,158 @@ + + + + + cond + condition number + + + Calling Sequence + + c = cond(X) + c = cond(X, p) + + + + Arguments + + + X + + real or complex matrix. If c = cond(X, p), X must be real or complex square matrix. + + + + p + + scalar or string (type of norm) (default value = 2). + + + + c + + real scalar, the condition number. + + + + + + Description + + + c = cond(X) + + + returns condition number in 2-norm.cond(X) is the ratio of the + largest singular value of X to the smallest. + + + + + c = cond(X, p) + + + returns condition number in p-norm : norm(X, p) * norm(inv(X), p). + If p is specified, p can be equal to : + + + + + p = 1. cond(X, p) returns condition number in 1-norm. + + + + + p = 2. cond(X, p) returns condition number in 2-norm. + + + + + p = %inf or 'inf'. cond(X, p) returns condition number in infinity norm. + + + + + p = 'fro'. cond(X, p) returns condition number in Frobenius norm. + + + + + + + + + Examples + + + + + + See Also + + + rcond + + + svd + + + norm + + + + + History + + + 5.4.0 + + + Calling cond(X), where X is + non square matrix, is now managed. For example: + + + + + + Calling cond(X, p) allows to calculate p-norm + condition number. For example: + + + + + + + + + diff --git a/modules/linear_algebra/help/en_US/matrix/det.xml b/modules/linear_algebra/help/en_US/matrix/det.xml new file mode 100755 index 000000000..10d6965a0 --- /dev/null +++ b/modules/linear_algebra/help/en_US/matrix/det.xml @@ -0,0 +1,104 @@ + + + + + det + determinant + + + Calling Sequence + det(X) + [e,m]=det(X) + + + + Arguments + + + X + + real or complex square matrix, polynomial or rational matrix. + + + + m + + real or complex number, the determinant base 10 mantissae + + + + e + + integer, the determinant base 10 exponent + + + + + + Description + + det(X) ( m*10^e is the determinant of the square matrix X. + + + For polynomial matrix det(X) is equivalent to determ(X). + + + For rational matrices det(X) is equivalent to detr(X). + + + + The det and detr functions don't use the same algorithm. + For a rational fraction, det(X) is overloaded by %r_det(X) which is based on the determ function. + detr() uses the Leverrier method. + + + Sometimes the det and detr functions may return different values for rational matrices. + In such cases you should set rational simplification mode off by using simp_mode(%f) to get the same result. + + + + + References + + det computations are based on the Lapack routines + DGETRF for real matrices and ZGETRF for the complex case. + + + Concerning sparse matrices, the determinant is obtained from LU factorization of umfpack library. + + + + Examples + + + + See Also + + + detr + + + determ + + + simp_mode + + + + diff --git a/modules/linear_algebra/help/en_US/matrix/orth.xml b/modules/linear_algebra/help/en_US/matrix/orth.xml new file mode 100755 index 000000000..a0c7c36eb --- /dev/null +++ b/modules/linear_algebra/help/en_US/matrix/orth.xml @@ -0,0 +1,76 @@ + + + + + orth + orthogonal basis + + + Calling Sequence + Q=orth(A) + + + Arguments + + + A + + real or complex matrix + + + + Q + + real or complex matrix + + + + + + Description + + Q=orth(A) returns Q, an orthogonal + basis for the span of A. Range(Q) = + Range(A) and Q'*Q=eye. + + + The number of columns of Q is the rank of + A as determined by the QR algorithm. + + + + Examples + + + + See Also + + + qr + + + rowcomp + + + colcomp + + + range + + + + diff --git a/modules/linear_algebra/help/en_US/matrix/rank.xml b/modules/linear_algebra/help/en_US/matrix/rank.xml new file mode 100755 index 000000000..ec651a774 --- /dev/null +++ b/modules/linear_algebra/help/en_US/matrix/rank.xml @@ -0,0 +1,87 @@ + + + + + rank + rank + + + Calling Sequence + [i]=rank(X) + [i]=rank(X,tol) + + + + Arguments + + + X + + real or complex matrix + + + + tol + + nonnegative real number + + + + + + Description + + rank(X) is the numerical rank of X + i.e. the number of singular values of X that are larger than + norm(size(X),'inf') * norm(X) * %eps. + + + rank(X,tol) is the number of singular values of + X that are larger than tol. + + + + Note that the default value of tol is proportional to + norm(X). As a consequence + rank([1.d-80,0;0,1.d-80]) is 2 !. + + + + + Examples + + + + See Also + + + svd + + + qr + + + rowcomp + + + colcomp + + + lu + + + + diff --git a/modules/linear_algebra/help/en_US/matrix/rcond.xml b/modules/linear_algebra/help/en_US/matrix/rcond.xml new file mode 100755 index 000000000..4d95cb28d --- /dev/null +++ b/modules/linear_algebra/help/en_US/matrix/rcond.xml @@ -0,0 +1,82 @@ + + + + + rcond + inverse condition number + + + Calling Sequence + rcond(X) + + + Arguments + + + X + + real or complex square matrix + + + + + + Description + + rcond(X) is an estimate for the reciprocal of the + condition of X in the 1-norm. + + + If X is well conditioned, rcond(X) is close to 1. + If not, rcond(X) is close to 0. + + + We compute the 1-norm of A with Lapack/DLANGE, compute its LU decomposition with Lapack/DGETRF + and finally estimate the condition with Lapack/DGECON. + + + + Examples + + + Estimating the 1-norm inverse condition number with rcond is + much faster than computing the 2-norm condition number with cond. + As a trade-off, rcond may be less reliable. + + + + + See Also + + + svd + + + cond + + + inv + + + + diff --git a/modules/linear_algebra/help/en_US/matrix/rref.xml b/modules/linear_algebra/help/en_US/matrix/rref.xml new file mode 100755 index 000000000..d2d1f7774 --- /dev/null +++ b/modules/linear_algebra/help/en_US/matrix/rref.xml @@ -0,0 +1,68 @@ + + + + + rref + computes matrix row echelon form by lu transformations + + + Calling Sequence + R=rref(A) + + + Arguments + + + A + + m x n matrix with scalar entries + + + + R + + m x n matrix,row echelon form of a + + + + + + Description + + rref computes the row echelon form of the given matrix by left lu + decomposition. If ones need the transformation used just call + X=rref([A,eye(m,m)]) the row echelon form R is X(:,1:n) and + the left transformation L is given by X(:,n+1:n+m) such as L*A=R + + + + Examples + + + + See Also + + + lu + + + qr + + + + diff --git a/modules/linear_algebra/help/en_US/matrix/trace.xml b/modules/linear_algebra/help/en_US/matrix/trace.xml new file mode 100755 index 000000000..84ba1e20c --- /dev/null +++ b/modules/linear_algebra/help/en_US/matrix/trace.xml @@ -0,0 +1,57 @@ + + + + + trace + trace + + + Calling Sequence + trace(X) + + + Arguments + + + X + + real or complex square matrix, polynomial or rational matrix. + + + + + + Description + + trace(X) is the trace of the matrix X. + + + Same as sum(diag(X)). + + + + Examples + + + + See Also + + + det + + + + diff --git a/modules/linear_algebra/help/en_US/pencil/CHAPTER b/modules/linear_algebra/help/en_US/pencil/CHAPTER new file mode 100755 index 000000000..86d1da116 --- /dev/null +++ b/modules/linear_algebra/help/en_US/pencil/CHAPTER @@ -0,0 +1,2 @@ +title = Matrix Pencil + diff --git a/modules/linear_algebra/help/en_US/pencil/companion.xml b/modules/linear_algebra/help/en_US/pencil/companion.xml new file mode 100755 index 000000000..0509be671 --- /dev/null +++ b/modules/linear_algebra/help/en_US/pencil/companion.xml @@ -0,0 +1,77 @@ + + + + + companion + companion matrix + + + Calling Sequence + A=companion(p) + + + Arguments + + + p + + polynomial or vector of polynomials + + + + A + + square matrix + + + + + + Description + + Returns a matrix A with characteristic polynomial equal + to p if p is monic. If p is not monic + the characteristic polynomial of A is equal to + p/c where c is the coefficient of largest degree + in p. + + + If p is a vector of monic polynomials, A is block diagonal, + and the characteristic polynomial of the ith block is p(i). + + + + Examples + + + + See Also + + + spec + + + poly + + + randpencil + + + + diff --git a/modules/linear_algebra/help/en_US/pencil/ereduc.xml b/modules/linear_algebra/help/en_US/pencil/ereduc.xml new file mode 100755 index 000000000..40a83b693 --- /dev/null +++ b/modules/linear_algebra/help/en_US/pencil/ereduc.xml @@ -0,0 +1,114 @@ + + + + + ereduc + computes matrix column echelon form by qz transformations + + + Calling Sequence + [E,Q,Z [,stair [,rk]]]=ereduc(X,tol) + + + Arguments + + + X + + m x n matrix with real entries. + + + + tol + + real positive scalar. + + + + E + + column echelon form matrix + + + + Q + + m x m unitary matrix + + + + Z + + n x n unitary matrix + + + + stair + + vector of indexes, + + + * + + + ISTAIR(i) = + j if the boundary element E(i,j) is a corner point. + + + + + * + + + ISTAIR(i) = - j if the boundary element E(i,j) is not a corner point. + + + + + + (i=1,...,M) + + + + + rk + + integer, estimated rank of the matrix + + + + + + Description + + Given an m x n matrix X (not necessarily regular) the function + ereduc computes a unitary transformed matrix E=Q*X*Z which is in + column echelon form (trapezoidal form). Furthermore the rank of + matrix X is determined. + + + + Examples + + + + See Also + + + fstair + + + + diff --git a/modules/linear_algebra/help/en_US/pencil/fstair.xml b/modules/linear_algebra/help/en_US/pencil/fstair.xml new file mode 100755 index 000000000..9d43f915b --- /dev/null +++ b/modules/linear_algebra/help/en_US/pencil/fstair.xml @@ -0,0 +1,157 @@ + + + + + fstair + computes pencil column echelon form by qz transformations + + + Calling Sequence + [AE,EE,QE,ZE,blcks,muk,nuk,muk0,nuk0,mnei]=fstair(A,E,Q,Z,stair,rk,tol) + + + Arguments + + + A + + m x n matrix with real entries. + + + + tol + + real positive scalar. + + + + E + + column echelon form matrix + + + + Q + + m x m unitary matrix + + + + Z + + n x n unitary matrix + + + + stair + + vector of indexes (see ereduc) + + + + rk + + integer, estimated rank of the matrix + + + + AE + + m x n matrix with real entries. + + + + EE + + column echelon form matrix + + + + QE + + m x m unitary matrix + + + + ZE + + n x n unitary matrix + + + + nblcks + + + is the number of submatrices having full row rank >= 0 detected in matrix A. + + + + + muk: + + integer array of dimension (n). Contains the column dimensions mu(k) (k=1,...,nblcks) of the submatrices having full column rank in the pencil sE(eps)-A(eps) + + + + nuk: + + integer array of dimension (m+1). Contains the row dimensions nu(k) (k=1,...,nblcks) of the submatrices having full row rank in the pencil sE(eps)-A(eps) + + + + muk0: + + integer array of dimension (n). Contains the column dimensions mu(k) (k=1,...,nblcks) of the submatrices having full column rank in the pencil sE(eps,inf)-A(eps,inf) + + + + nuk: + + integer array of dimension (m+1). Contains the row dimensions nu(k) (k=1,...,nblcks) of the submatrices having full row rank in the pencil sE(eps,inf)-A(eps,inf) + + + + mnei: + + integer array of dimension (4). mnei(1) = row dimension of sE(eps)-A(eps) + + + + + + Description + + Given a pencil sE-A where matrix E is in column echelon form the + function fstair computes according to the wishes of the user a + unitary transformed pencil QE(sEE-AE)ZE which is more or less similar + to the generalized Schur form of the pencil sE-A. + The function yields also part of the Kronecker structure of + the given pencil. + + + Q,Z are the unitary matrices used to compute the pencil where E + is in column echelon form (see ereduc) + + + + See Also + + + quaskro + + + ereduc + + + + diff --git a/modules/linear_algebra/help/en_US/pencil/glever.xml b/modules/linear_algebra/help/en_US/pencil/glever.xml new file mode 100755 index 000000000..cd3d577df --- /dev/null +++ b/modules/linear_algebra/help/en_US/pencil/glever.xml @@ -0,0 +1,118 @@ + + + + + glever + inverse of matrix pencil + + + Calling Sequence + [Bfs,Bis,chis]=glever(E,A [,s]) + + + Arguments + + + E, A + + two real square matrices of same dimensions + + + + s + + + character string (default value 's') + + + + + Bfs,Bis + + two polynomial matrices + + + + chis + + polynomial + + + + + + Description + + Computation of + + + (s*E-A)^-1 + + + by generalized Leverrier's algorithm for a matrix pencil. + + + + chis = characteristic polynomial (up to a multiplicative constant). + + + Bfs = numerator polynomial matrix. + + + Bis + = polynomial matrix ( - expansion of (s*E-A)^-1 at infinity). + + + Note the - sign before Bis. + + + + Caution + + This function uses cleanp to simplify Bfs,Bis and chis. + + + + Examples + + + + See Also + + + rowshuff + + + det + + + invr + + + coffg + + + pencan + + + penlaur + + + + diff --git a/modules/linear_algebra/help/en_US/pencil/kroneck.xml b/modules/linear_algebra/help/en_US/pencil/kroneck.xml new file mode 100755 index 000000000..68488e0e7 --- /dev/null +++ b/modules/linear_algebra/help/en_US/pencil/kroneck.xml @@ -0,0 +1,159 @@ + + + + + kroneck + Kronecker form of matrix pencil + + + Calling Sequence + [Q,Z,Qd,Zd,numbeps,numbeta]=kroneck(F) + [Q,Z,Qd,Zd,numbeps,numbeta]=kroneck(E,A) + + + + Arguments + + + F + + + real matrix pencil F=s*E-A + + + + + E,A + + two real matrices of same dimensions + + + + Q,Z + + two square orthogonal matrices + + + + Qd,Zd + + two vectors of integers + + + + numbeps,numeta + + two vectors of integers + + + + + + Description + + Kronecker form of matrix pencil: kroneck computes two + orthogonal matrices Q, Z which put the pencil F=s*E -A into + upper-triangular form: + + + + The dimensions of the four blocks are given by: + + + eps=Qd(1) x Zd(1), inf=Qd(2) x Zd(2), + f = Qd(3) x Zd(3), eta=Qd(4)xZd(4) + + + The inf block contains the infinite modes of + the pencil. + + + The f block contains the finite modes of + the pencil + + + The structure of epsilon and eta blocks are given by: + + + numbeps(1) = # of eps blocks of size 0 x 1 + + + numbeps(2) = # of eps blocks of size 1 x 2 + + + numbeps(3) = # of eps blocks of size 2 x 3 etc... + + + numbeta(1) = # of eta blocks of size 1 x 0 + + + numbeta(2) = # of eta blocks of size 2 x 1 + + + numbeta(3) = # of eta blocks of size 3 x 2 etc... + + + The code is taken from T. Beelen (Slicot-WGS group). + + + + Examples + + + + See Also + + + gschur + + + gspec + + + systmat + + + pencan + + + randpencil + + + trzeros + + + + diff --git a/modules/linear_algebra/help/en_US/pencil/lyap.xml b/modules/linear_algebra/help/en_US/pencil/lyap.xml new file mode 100755 index 000000000..46874bc8e --- /dev/null +++ b/modules/linear_algebra/help/en_US/pencil/lyap.xml @@ -0,0 +1,79 @@ + + + + + lyap + Lyapunov equation + + + Calling Sequence + [X]=lyap(A,C,'c') + [X]=lyap(A,C,'d') + + + + Arguments + + + A, C + + + real square matrices, C must be symmetric + + + + + + + Description + + X= lyap(A,C,flag) solves the continuous time or + discrete time matrix Lyapunov matrix equation: + + + + + Note that a unique solution exist if and only if an eigenvalue of A is + not an eigenvalue of -A (flag='c') or 1 over an eigenvalue of A + (flag='d'). + + + + + Examples + + + + See Also + + + sylv + + + ctr_gram + + + obs_gram + + + + diff --git a/modules/linear_algebra/help/en_US/pencil/pencan.xml b/modules/linear_algebra/help/en_US/pencil/pencan.xml new file mode 100755 index 000000000..77a19a64e --- /dev/null +++ b/modules/linear_algebra/help/en_US/pencil/pencan.xml @@ -0,0 +1,105 @@ + + + + + pencan + canonical form of matrix pencil + + + Calling Sequence + [Q,M,i1]=pencan(Fs) + [Q,M,i1]=pencan(E,A) + + + + Arguments + + + Fs + + + a regular pencil s*E-A + + + + + E,A + + two real square matrices + + + + Q,M + + two non-singular real matrices + + + + i1 + + integer + + + + + + Description + + Given the regular pencil Fs=s*E-A, pencan returns matrices Q + and M + such than M*(s*E-A)*Q is in "canonical" form. + + + M*E*Q is a block matrix + + + + with N nilpotent and i1 = size of the I matrix above. + + + M*A*Q is a block matrix: + + + + + Examples + + + + See Also + + + glever + + + penlaur + + + rowshuff + + + + diff --git a/modules/linear_algebra/help/en_US/pencil/penlaur.xml b/modules/linear_algebra/help/en_US/pencil/penlaur.xml new file mode 100755 index 000000000..00392641b --- /dev/null +++ b/modules/linear_algebra/help/en_US/pencil/penlaur.xml @@ -0,0 +1,114 @@ + + + + + penlaur + Laurent coefficients of matrix pencil + + + Calling Sequence + [Si,Pi,Di,order]=penlaur(Fs) + [Si,Pi,Di,order]=penlaur(E,A) + + + + Arguments + + + Fs + + + a regular pencil s*E-A + + + + + E, A + + two real square matrices + + + + Si,Pi,Di + + three real square matrices + + + + order + + integer + + + + + + Description + + penlaur computes the first Laurent coefficients of (s*E-A)^-1 at + infinity. + + + (s*E-A)^-1 = ... + Si/s - Pi - s*Di + ... at s = infinity. + + + order = order of the singularity (order=index-1). + + + The matrix pencil Fs=s*E-A should be invertible. + + + For a index-zero pencil, Pi, Di,... are zero and Si=inv(E). + + + For a index-one pencil (order=0),Di =0. + + + For higher-index pencils, the terms -s^2 Di(2), -s^3 Di(3),... are given by: + + + Di(2)=Di*A*Di, Di(3)=Di*A*Di*A*Di (up + to Di(order)). + + + + Remark + + Experimental version: troubles when bad conditioning of so*E-A + + + + Examples + + + + See Also + + + glever + + + pencan + + + rowshuff + + + + diff --git a/modules/linear_algebra/help/en_US/pencil/quaskro.xml b/modules/linear_algebra/help/en_US/pencil/quaskro.xml new file mode 100755 index 000000000..d16e81dcd --- /dev/null +++ b/modules/linear_algebra/help/en_US/pencil/quaskro.xml @@ -0,0 +1,134 @@ + + + + + quaskro + quasi-Kronecker form + + + Calling Sequence + [Q,Z,Qd,Zd,numbeps,numbeta]=quaskro(F) + [Q,Z,Qd,Zd,numbeps,numbeta]=quaskro(E,A) + [Q,Z,Qd,Zd,numbeps,numbeta]=quaskro(F,tol) + [Q,Z,Qd,Zd,numbeps,numbeta]=quaskro(E,A,tol) + + + + Arguments + + + F + + + real matrix pencil F=s*E-A (s=poly(0,'s')) + + + + + E,A + + two real matrices of same dimensions + + + + tol + + a real number (tolerance, default value=1.d-10) + + + + Q,Z + + two square orthogonal matrices + + + + Qd,Zd + + two vectors of integers + + + + numbeps + + vector of integers + + + + + + Description + + Quasi-Kronecker form of matrix pencil: quaskro computes two + orthogonal matrices Q, Z which put the pencil F=s*E -A into + upper-triangular form: + + + + The dimensions of the blocks are given by: + + + eps=Qd(1) x Zd(1), inf=Qd(2) x Zd(2), + r = Qd(3) x Zd(3) + + + The inf block contains the infinite modes of + the pencil. + + + The f block contains the finite modes of + the pencil + + + The structure of epsilon blocks are given by: + + + numbeps(1) = # of eps blocks of size 0 x 1 + + + numbeps(2) = # of eps blocks of size 1 x 2 + + + numbeps(3) = # of eps blocks of size 2 x 3 etc... + + + The complete (four blocks) Kronecker form is given by + the function kroneck which calls quaskro on + the (pertransposed) pencil sE(r)-A(r). + + + The code is taken from T. Beelen + + + + See Also + + + kroneck + + + gschur + + + gspec + + + + diff --git a/modules/linear_algebra/help/en_US/pencil/randpencil.xml b/modules/linear_algebra/help/en_US/pencil/randpencil.xml new file mode 100755 index 000000000..c60dd7a9a --- /dev/null +++ b/modules/linear_algebra/help/en_US/pencil/randpencil.xml @@ -0,0 +1,110 @@ + + + + + randpencil + random pencil + + + Calling Sequence + F=randpencil(eps,infi,fin,eta) + + + Arguments + + + eps + + vector of integers + + + + infi + + vector of integers + + + + fin + + real vector, or monic polynomial, or vector of monic polynomial + + + + eta + + vector of integers + + + + F + + + real matrix pencil F=s*E-A (s=poly(0,'s')) + + + + + + + Description + + Utility function. + F=randpencil(eps,infi,fin,eta) returns a random pencil F + with given Kronecker structure. The structure is given by: + eps=[eps1,...,epsk]: structure of epsilon blocks (size eps1x(eps1+1),....) + fin=[l1,...,ln] set of finite eigenvalues (assumed real) (possibly []) + infi=[k1,...,kp] size of J-blocks at infinity + ki>=1 (infi=[] if no J blocks). + eta=[eta1,...,etap]: structure ofeta blocks (size eta1+1)xeta1,...) + + + epsi's should be >=0, etai's should be >=0, infi's should + be >=1. + + + If fin is a (monic) polynomial, the finite block admits the roots of + fin as eigenvalues. + + + If fin is a vector of polynomial, they are the finite elementary + divisors of F i.e. the roots of p(i) are finite + eigenvalues of F. + + + + Examples + + + + See Also + + + kroneck + + + pencan + + + penlaur + + + + diff --git a/modules/linear_algebra/help/en_US/pencil/rowshuff.xml b/modules/linear_algebra/help/en_US/pencil/rowshuff.xml new file mode 100755 index 000000000..713c0a70f --- /dev/null +++ b/modules/linear_algebra/help/en_US/pencil/rowshuff.xml @@ -0,0 +1,103 @@ + + + + + rowshuff + shuffle algorithm + + + Calling Sequence + [Ws,Fs1]=rowshuff(Fs, [alfa]) + + + Arguments + + + Fs + + + square real pencil Fs = s*E-A + + + + + Ws + + polynomial matrix + + + + Fs1 + + + square real pencil F1s = s*E1 -A1 with E1 non-singular + + + + + alfa + + + real number (alfa = 0 is the default value) + + + + + + + Description + + Shuffle algorithm: Given the pencil Fs=s*E-A, returns Ws=W(s) + (square polynomial matrix) such that: + + + Fs1 = s*E1-A1 = W(s)*(s*E-A) is a pencil with non singular E1 matrix. + + + This is possible iff the pencil Fs = s*E-A is regular (i.e. invertible). + The degree of Ws is equal to the index of the pencil. + + + The poles at infinity of Fs are put to alfa and the zeros of Ws are at alfa. + + + Note that (s*E-A)^-1 = (s*E1-A1)^-1 * W(s) = (W(s)*(s*E-A))^-1 *W(s) + + + + Examples + + + + See Also + + + pencan + + + glever + + + penlaur + + + + diff --git a/modules/linear_algebra/help/en_US/pencil/sylv.xml b/modules/linear_algebra/help/en_US/pencil/sylv.xml new file mode 100755 index 000000000..4b6d85939 --- /dev/null +++ b/modules/linear_algebra/help/en_US/pencil/sylv.xml @@ -0,0 +1,90 @@ + + + + + sylv + Sylvester equation. + + + Calling Sequence + sylv(A, B, C, flag) + + + Arguments + + + A,B,C + + three real matrices of appropriate dimensions. + + + + flag + + + character string ('c' or 'd') + + + + + + + Description + + X= sylv(A, B, C, 'c') computes X, solution + of the "continuous time" Sylvester equation + + + + X=sylv(A, B, C, 'd') computes X, solution + of the modified "discrete time" Sylvester equation + + + + X=-sylv(-A, B, C, 'd') computes X, solution + of the real "discrete time" Sylvester equation + + + + + Examples + + + + See Also + + + lyap + + + + diff --git a/modules/linear_algebra/help/en_US/proj.xml b/modules/linear_algebra/help/en_US/proj.xml new file mode 100755 index 000000000..0797774a1 --- /dev/null +++ b/modules/linear_algebra/help/en_US/proj.xml @@ -0,0 +1,72 @@ + + + + + proj + projection + + + Calling Sequence + P = proj(X1,X2) + + + Arguments + + + X1,X2 + + two real matrices with equal number of columns + + + + P + + + real projection matrix (P^2=P) + + + + + + + Description + + P is the projection on X2 parallel to X1. + + + + Examples + + + + See Also + + + projspec + + + orth + + + fullrf + + + + diff --git a/modules/linear_algebra/help/en_US/state_space/CHAPTER b/modules/linear_algebra/help/en_US/state_space/CHAPTER new file mode 100755 index 000000000..a0b62cdee --- /dev/null +++ b/modules/linear_algebra/help/en_US/state_space/CHAPTER @@ -0,0 +1,2 @@ +title = State-Space Matrices + diff --git a/modules/linear_algebra/help/en_US/state_space/coff.xml b/modules/linear_algebra/help/en_US/state_space/coff.xml new file mode 100755 index 000000000..2944ddac1 --- /dev/null +++ b/modules/linear_algebra/help/en_US/state_space/coff.xml @@ -0,0 +1,97 @@ + + + + + coff + resolvent (cofactor method) + + + Calling Sequence + [N,d]=coff(M [,var]) + + + Arguments + + + M + + square real matrix + + + + var + + character string + + + + N + + + polynomial matrix (same size as M) + + + + + d + + + polynomial (characteristic polynomial poly(A,'s')) + + + + + + + Description + + coff computes R=(s*eye()-M)^-1 for M a real matrix. + R is given by N/d. + + + N = numerator polynomial matrix. + + + d = common denominator. + + + var character string ('s' if omitted) + + + + Examples + + + + See Also + + + coffg + + + ss2tf + + + nlev + + + poly + + + + diff --git a/modules/linear_algebra/help/en_US/state_space/nlev.xml b/modules/linear_algebra/help/en_US/state_space/nlev.xml new file mode 100755 index 000000000..5d8bd89ad --- /dev/null +++ b/modules/linear_algebra/help/en_US/state_space/nlev.xml @@ -0,0 +1,88 @@ + + + + + nlev + Leverrier's algorithm + + + Calling Sequence + [num,den]=nlev(A,z [,rmax]) + + + Arguments + + + A + + real square matrix + + + + z + + character string + + + + rmax + + + optional parameter (see bdiag) + + + + + + + Description + + [num,den]=nlev(A,z [,rmax]) computes + (z*eye()-A)^(-1) + + + by block diagonalization of A followed by Leverrier's algorithm + on each block. + + + This algorithm is better than the usual leverrier algorithm but + still not perfect! + + + + Examples + + + + See Also + + + coff + + + coffg + + + glever + + + ss2tf + + + + diff --git a/modules/linear_algebra/help/en_US/subspaces/CHAPTER b/modules/linear_algebra/help/en_US/subspaces/CHAPTER new file mode 100755 index 000000000..d87d9ca5e --- /dev/null +++ b/modules/linear_algebra/help/en_US/subspaces/CHAPTER @@ -0,0 +1,3 @@ +title = Subspaces + + diff --git a/modules/linear_algebra/help/en_US/subspaces/spaninter.xml b/modules/linear_algebra/help/en_US/subspaces/spaninter.xml new file mode 100755 index 000000000..650b9dba2 --- /dev/null +++ b/modules/linear_algebra/help/en_US/subspaces/spaninter.xml @@ -0,0 +1,91 @@ + + + + + spaninter + subspace intersection + + + Calling Sequence + [X,dim]=spaninter(A,B [,tol]) + + + Arguments + + + A, B + + two real or complex matrices with equal number of rows + + + + X + + orthogonal or unitary square matrix + + + + dim + + + integer, dimension of subspace range(A) inter range(B) + + + + + + + Description + + computes the intersection of range(A) and range(B). + + + The first dim columns of X span this intersection i.e. + X(:,1:dim) is an orthogonal basis for + + + range(A) inter range(B) + + + In the X basis A and B are respectively represented by: + + + X'*A and X'*B. + + + tol is a threshold (sqrt(%eps) is the default value). + + + + Examples + + + + See Also + + + spanplus + + + spantwo + + + + diff --git a/modules/linear_algebra/help/en_US/subspaces/spanplus.xml b/modules/linear_algebra/help/en_US/subspaces/spanplus.xml new file mode 100755 index 000000000..276ccdb50 --- /dev/null +++ b/modules/linear_algebra/help/en_US/subspaces/spanplus.xml @@ -0,0 +1,100 @@ + + + + + spanplus + sum of subspaces + + + Calling Sequence + [X,dim,dima]=spanplus(A,B[,tol]) + + + Arguments + + + A, B + + two real or complex matrices with equal number of rows + + + + X + + orthogonal or unitary square matrix + + + + dim, dima + + integers, dimension of subspaces + + + + tol + + nonnegative real number + + + + + + Description + + computes a basis X such that: + + + the first dima columns of X span Range(A) + and the following (dim-dima) columns make a basis of A+B + relative to A. + + + The dim first columns of X make a basis for A+B. + + + One has the following canonical form for [A,B]: + + + + tol is an optional argument (see function code). + + + + Examples + + + + See Also + + + spaninter + + + im_inv + + + spantwo + + + + diff --git a/modules/linear_algebra/help/en_US/subspaces/spantwo.xml b/modules/linear_algebra/help/en_US/subspaces/spantwo.xml new file mode 100755 index 000000000..7be745d6d --- /dev/null +++ b/modules/linear_algebra/help/en_US/subspaces/spantwo.xml @@ -0,0 +1,110 @@ + + + + + spantwo + sum and intersection of subspaces + + + Calling Sequence + [Xp,dima,dimb,dim]=spantwo(A,B, [tol]) + + + Arguments + + + A, B + + two real or complex matrices with equal number of rows + + + + Xp + + square non-singular matrix + + + + dima, dimb, dim + + integers, dimension of subspaces + + + + tol + + nonnegative real number + + + + + + Description + + Given two matrices A and B with same number of rows, + returns a square matrix Xp (non singular but not necessarily orthogonal) + such that : + + + + The first dima columns of inv(Xp) span range(A). + + + Columns dim-dimb+1 to dima of inv(Xp) span the + intersection of range(A) and range(B). + + + The dim first columns of inv(Xp) span + range(A)+range(B). + + + Columns dim-dimb+1 to dim of inv(Xp) span + range(B). + + + Matrix [A1;A2] has full row rank (=rank(A)). Matrix [B2;B3] has + full row rank (=rank(B)). Matrix [A2,B2] has full row rank (=rank(A inter B)). Matrix [A1,0;A2,B2;0,B3] has full row rank (=rank(A+B)). + + + + Examples + + + + See Also + + + spanplus + + + spaninter + + + + diff --git a/modules/linear_algebra/help/fr_FR/addchapter.sce b/modules/linear_algebra/help/fr_FR/addchapter.sce new file mode 100755 index 000000000..6da99e965 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/addchapter.sce @@ -0,0 +1,11 @@ +// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +// Copyright (C) 2009 - DIGITEO +// +// This file must be used under the terms of the CeCILL. +// This source file is licensed as described in the file COPYING, which +// you should have received as part of this distribution. The terms +// are also available at +// http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt + +add_help_chapter("Algèbre Lineaire",SCI+"/modules/linear_algebra/help/fr_FR",%T); + diff --git a/modules/linear_algebra/help/fr_FR/eigen/CHAPTER b/modules/linear_algebra/help/fr_FR/eigen/CHAPTER new file mode 100755 index 000000000..88f8bc42b --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/eigen/CHAPTER @@ -0,0 +1,2 @@ +title = Eigenvalue and Singular Value + diff --git a/modules/linear_algebra/help/fr_FR/eigen/bdiag.xml b/modules/linear_algebra/help/fr_FR/eigen/bdiag.xml new file mode 100755 index 000000000..a7dc9f342 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/eigen/bdiag.xml @@ -0,0 +1,111 @@ + + + + + bdiag + bloc-diagonalisation, vecteurs propres généralisés + + + Séquence d'appel + [Ab [,X [,bs]]]=bdiag(A [,rmax]) + + + Paramètres + + + A + + matrice carrée réelle ou complexe + + + + + rmax + + nombre réel + + + + + Ab + + matrice carrée réelle ou complexe + + + + + X + + matrice régulière, réelle ou complexe + + + + + bs + + vecteur d'entiers + + + + + + + Description + + + [Ab [,X [,bs]]]=bdiag(A [,rmax]) calcule la forme + bloc-diagonale de A. bs précise la structure des + blocs (tailles respectives des blocs). X est la + matrice de changement de base, c'est à dire que Ab = + inv(X)*A*X + + est bloc-diagonale. + + + rmax contrôle le conditionnement de X; + la valeur par défaut est la norme l1 de A. + + + Pour obtenir une forme diagonale (si celle-ci existe) choisissez + une valeur élevée de rmax (rmax=1/%eps + par exemple). Pour une matrice réelle quelconque, les blocs sont + de taille (1x1) ou (2x2) et X est la matrice des + vecteurs propres. + + + + Exemples + + + + Voir aussi + + + schur + + + sylv + + + spec + + + + diff --git a/modules/linear_algebra/help/fr_FR/eigen/gspec.xml b/modules/linear_algebra/help/fr_FR/eigen/gspec.xml new file mode 100755 index 000000000..d7e856080 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/eigen/gspec.xml @@ -0,0 +1,69 @@ + + + + + gspec + valeurs propres d'un faisceau de matrices (obsolete) + + + Séquence d'appel + [al,be]=gspec(A,E) + [al,be,Z]=gspec(A,E) + + + + Paramètres + + + A, E + + matrices carrées réelles de mêmes dimensions + + + + + al, be + + vecteurs réels + + + + + Z + + matrice carrée régulière + + + + + + + Description + + Cette fonction est maintenant un cas particulier de la fonction + spec. La syntaxe d'appel doit être remplacée par + + + + + Voir aussi + + + spec + + + + diff --git a/modules/linear_algebra/help/fr_FR/eigen/hess.xml b/modules/linear_algebra/help/fr_FR/eigen/hess.xml new file mode 100755 index 000000000..6ae439c64 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/eigen/hess.xml @@ -0,0 +1,94 @@ + + + + + hess + Forme de Hessenberg + + + Séquence d'appel + H = hess(A) + [U,H] = hess(A) + + + + Paramètres + + + A + + matrice carrée réelle ou complexe + + + + + H + + matrice carrée réelle ou complexe + + + + + U + + matrice carrée unitaire + + + + + + + Description + + [U,H] = hess(A) Calcule une matrice unitaire U + et une matrice de Hessenberg H telles que A = + U*H*U' + + et U'*U = Identité. La syntaxe + H=hess(A) ne renvoie que la matrice de Hessenberg. + + + Les coefficients d'une matrice sous forme de Hessenberg sont nuls + sous la première sous-diagonale. Si la matrice est symétrique ou + hermitienne, la forme est tridiagonale. + + + + Exemples + + + + Voir aussi + + + qr + + + contr + + + schur + + + + + Fonctions Utilisées + + Le calcul de la forme de Hessenberg determinant est basé sur les routines Lapack : + DGEHRD, DORGHR pour les matrices réelles et ZGEHRD, ZORGHR pour le cas complexe. + + + diff --git a/modules/linear_algebra/help/fr_FR/eigen/pbig.xml b/modules/linear_algebra/help/fr_FR/eigen/pbig.xml new file mode 100755 index 000000000..724cf59f4 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/eigen/pbig.xml @@ -0,0 +1,128 @@ + + + + + pbig + projection sur des sous-espaces propres + + + Séquence d'appel + [Q,M]=pbig(A,thres,flag) + + + Paramètres + + + A + + matrice réelle carrée + + + + + thres + + nombre réel + + + + + flag + + + chaîne de caractères ('c' ou 'd') + + + + + Q,M + + matrices réelles + + + + + + + Description + + Projection sur des sous-espaces propres de A associés aux valeurs + propres avec partie réelle >= thres + (flag='c') ou avec module >= + thres (flag='d'). + + + La projection est définie par Q*M, où Q + est de rang maximal, les lignes de M sont + linéairement indépendantes et M*Q=eye. + + + Si flag='c', les valeurs propres de + M*A*Q = valeurs propres de A avec partie + réelle >= thres. + + + Si flag='d', les valeurs propres de + M*A*Q = valeurs propres de A avec module + >= thres. + + + Si flag='c' et si [Q1,M1] = + factorisation de rang maximal (fullrf) de + eye()-Q*M alors les valeurs propres de + M1*A*Q1 = valeurs propres de A avec + partie réelle < thres. + + + Si flag='d' et si [Q1,M1] = + factorisation de rang maximal (fullrf) de + eye()-Q*M alors les valeurs propres de + M1*A*Q1 = valeurs propres de A avec + module < thres. + + + + Exemples + + + + Voir aussi + + + psmall + + + projspec + + + fullrf + + + schur + + + + + Fonctions Utilisées + + pbig est basée sur la forme de Schur ordonnée + (fonction Scilab schur). + + + diff --git a/modules/linear_algebra/help/fr_FR/eigen/spec.xml b/modules/linear_algebra/help/fr_FR/eigen/spec.xml new file mode 100755 index 000000000..de3027bc2 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/eigen/spec.xml @@ -0,0 +1,211 @@ + + + + + spec + valeurs propres d'une matrice + + + Séquence d'appel + evals=spec(A) + [X,diagevals]=spec(A) + + evals=spec(A,E) + [al,be]=spec(A,E) + [al,be,Z]=spec(A,E) + [al,be]=spec(A,E) + [al,be,Q,Z]=spec(A,E) + + + + Paramètres + + + A + + matrice carrée réelle ou complexe + + + + + E + + + matrice carrée réelle ou complexe de même dimensions que A + + + + + evals + + vecteur réel ou complexe + + + + + diagevals + + matrice carrée diagonale réelle ou complexe (les éléments + diagonaux sont les valeurs propres) + + + + + al + + vecteur réel ou complexe, al./be donnes les valeurs propres + + + + + be + + vecteur réel ou complexe, al./be donnes les valeurs propres + + + + + X + + matrice carrée inversible réelle ou complexe, matrices des + vecteurs propres. + + + + + Q + + matrice carrée inversible réelle ou complexe, matrices des + vecteurs propres à gauche. + + + + + Z + + atrice carrée inversible réelle ou complexe, matrices des + vecteurs propres à droite. + + + + + + + Description + + + spec(A) + + + evals=spec(A) retourne dans le vecteur + evals les valeurs propres de A. + + + [evals,X] =spec(A) retourne de plus les vecteurs + propres (s'ils existent). Voir Aussi bdiag + + + + + spec(A,B) + + + evals=spec(A,E) retourne le spectre du faisceau + s E - A, c'est à dire les racines du déterminant de + la matrice de polynômes s E - A. + + + [al,be] = spec(A,E) retourne le spectre du faisceau + s E - A, c'est à dire les racines du déterminant de + la matrice de polynômes s E - A. Les valeurs propres + sont données par al./be. Si be(i) = 0 la + iième valeur propres est à l'infini. (Pour E = + eye(A), al./be + + est spec(A)). + + + [al,be,Z] = spec(A,E) retourne de plus la matrice + Z des vecteurs propres généralisés à droite. + + + [al,be,Q,Z] = spec(A,E) retourne de plus les matrices + Q et Z des vecteurs propres généralisés à + droite et à gauche. + + Pour les grosses matrices pleines / creuses, vous + pouvez utiliser le module Arnoldi. + + + + + + + Exemples + + + + Voir aussi + + + poly + + + det + + + gspec + + + schur + + + bdiag + + + colcomp + + + dsaupd + + + dnaupd + + + + + Fonctions Utilisées + + Le calcul des valeurs propres des matrices est basé sur les + routines Lapack DGEEV and ZGEEV. + + + diff --git a/modules/linear_algebra/help/fr_FR/eigen/sva.xml b/modules/linear_algebra/help/fr_FR/eigen/sva.xml new file mode 100755 index 000000000..91c66ed91 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/eigen/sva.xml @@ -0,0 +1,87 @@ + + + + + sva + approximation de valeurs singulières + + + Séquence d'appel + [U,s,V]=sva(A,k) + [U,s,V]=sva(A,tol) + + + + Paramètres + + + A + + matrice réelle ou complexe + + + + + k + + entier + + + + + tol + + nombre réel positif + + + + + + + Description + + Approximation de valeurs singulières. + + + [U,S,V]=sva(A,k) avec k un entier + >=1, renvoie U,S et V telles que + B=U*S*V' est la meilleure approximation au sens + l_2 de A avec rang(B)=k. + + + [U,S,V]=sva(A,tol) où tol est un réel + positif, renvoie U,S et V tels que + B=U*S*V' et la norme l_2 de A-B est + inférieure à tol. + + + + Exemples + + + + Voir aussi + + + svd + + + + diff --git a/modules/linear_algebra/help/fr_FR/eigen/svd.xml b/modules/linear_algebra/help/fr_FR/eigen/svd.xml new file mode 100755 index 000000000..a78b9dda0 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/eigen/svd.xml @@ -0,0 +1,132 @@ + + + + + svd + décomposition en valeurs singulières + + + Séquence d'appel + s=svd(X) + [U,S,V]=svd(X) + [U,S,V]=svd(X,0) (obsolete) + [U,S,V]=svd(X,"e") + [U,S,V,rk]=svd(X [,tol]) + + + + Paramètres + + + X + + matrice réelle ou complexe + + + + + s + + vecteur réel (valeurs singulières) + + + + + S + + matrice réelle diagonale (valeurs singulières sur la diagonale) + + + + + U,V + + matrices carrées unitaires (vecteurs singuliers). + + + + + tol + + nombre réel positif + + + + + + + Description + + [U,S,V]=svd(X) renvoie une matrice diagonale S, de même + dimension que X avec des éléments diagonaux positifs classés + par ordre décroissant, ainsi que deux matrices unitaires U + et V telles que + X = U*S*V'.[U,S,V]=svd(X,"e") + renvoie la décomposition réduite : si X est une + matrice m x n et que m > n alors + seulement les n premières colonnes de U sont + calculées et S est n x n . + + + s=svd(X) renvoie un vecteur s contenant + les valeurs singulières. + + + [U,S,V,rk]=svd(X [,tol]) renvoie de plus + rk, le rang "numérique" de X + c'est à dire le nombre de valeurs singulières plus grandes + que tol. + + + La valeur par défaut de tol est la même que pour la fonction rank. + + + + Exemples + + + + Voir aussi + + + rank + + + qr + + + colcomp + + + rowcomp + + + sva + + + spec + + + + + Fonctions Utilisées + + la décomposition svd est basée sur les routines DGESVD pour les + matrices réelles et ZGESVD pour le cas complexe. + + + diff --git a/modules/linear_algebra/help/fr_FR/factorization/CHAPTER b/modules/linear_algebra/help/fr_FR/factorization/CHAPTER new file mode 100755 index 000000000..e6daeb8eb --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/factorization/CHAPTER @@ -0,0 +1,2 @@ +title = Factorization + diff --git a/modules/linear_algebra/help/fr_FR/factorization/givens.xml b/modules/linear_algebra/help/fr_FR/factorization/givens.xml new file mode 100755 index 000000000..233d0cd94 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/factorization/givens.xml @@ -0,0 +1,90 @@ + + + + + givens + Transformation de Givens + + + Séquence d'appel + U=givens(xy) + U=givens(x,y) + [U,c]=givens(xy) + [U,c]=givens(x,y) + + + + Paramètres + + + x,y + + deux nombres réels ou complexes + + + + + xy + + vecteur colonne réel ou complexe à deux composantes + + + + + U + + matrice unitaire 2 x 2 + + + + + c + + vecteur colonne réel ou complexe à deux composantes + + + + + + + Description + + U= givens(x, y) ou U = givens(xy) avec xy = [x;y] + renvoie U une matrice unitaire 2x2 telle que : + + + U*xy=[r;0]=c. + + + + Notez que givens(x,y) et givens([x;y]) sont équivalents. + + + + + Exemples + + + + Voir aussi + + + qr + + + + diff --git a/modules/linear_algebra/help/fr_FR/factorization/householder.xml b/modules/linear_algebra/help/fr_FR/factorization/householder.xml new file mode 100755 index 000000000..f81b8b9ad --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/factorization/householder.xml @@ -0,0 +1,71 @@ + + + + + householder + Matrice de Householder + + + Séquence d'appel + u=householder(v [,w]) + + + Paramètres + + + v + + vecteur colonne réel ou complexe + + + + + w + + + vecteur colonne réel ou complexe de même taille que v (la valeur par défaut est eye(v)) + + + + + u + + vecteur colonne réel ou complexe + + + + + + + Description + + Etant donnés deux vecteurs colonnes v et w de même taille, householder(v,w) renvoie un vecteur normé u, tel que + (eye()-2*u*u')*v est colinéaire à w. + (eye()-2*u*u') est la matrice de la transformation de Householder correspondante. + + + La valeur par défaut de w est eye(v). Dans ce cas le vecteur (eye()-2*u*u')*v est égal à eye(v)*norm(v). + + + + Voir aussi + + + qr + + + givens + + + + diff --git a/modules/linear_algebra/help/fr_FR/factorization/sqroot.xml b/modules/linear_algebra/help/fr_FR/factorization/sqroot.xml new file mode 100755 index 000000000..55f751b12 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/factorization/sqroot.xml @@ -0,0 +1,63 @@ + + + + + sqroot + factorisation hermitienne W*W' + + + Séquence d'appel + sqroot(X) + + + Paramètres + + + X + + matrice complexe ou réelle, symétrique définie non-négative + + + + + + + Description + + renvoie W telle que X=W*W' (en utilisant svd). + + + + Exemples + + + + Voir aussi + + + chol + + + svd + + + + diff --git a/modules/linear_algebra/help/fr_FR/kernel/CHAPTER b/modules/linear_algebra/help/fr_FR/kernel/CHAPTER new file mode 100755 index 000000000..be67920e1 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/kernel/CHAPTER @@ -0,0 +1,2 @@ +title = Kernel + diff --git a/modules/linear_algebra/help/fr_FR/kernel/colcomp.xml b/modules/linear_algebra/help/fr_FR/kernel/colcomp.xml new file mode 100755 index 000000000..4f674ac87 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/kernel/colcomp.xml @@ -0,0 +1,108 @@ + + + + + colcomp + compression de colonnes, noyau + + + Séquence d'appel + [W,rk]=colcomp(A [,flag] [,tol]) + + + Paramètres + + + A + + matrice réelle ou complexe + + + + + flag + + chaîne de caractères + + + + + tol + + nombre réel + + + + + W + + matrice carré régulière (matrice de changement de base) + + + + + rk + + + entier (rang de"A) + + + + + + + Description + + Compression des colonnes de A : Ac = A*W est à colonnes compressées, c'est à dire + + + Ac=[0,Af] et Af est de rang maximal + rank(Af) = rank(A) = rk. + + + flag et tol sont des paramètres optionnels : flag = 'qr' + ou 'svd' ('svd' par défaut). + + + tol = paramètre de tolérance (de l'ordre de %eps par défaut). + + + Les ma-rk premières colonnes de W forment une base du noyau de A quand size(A)=[na,ma]. + + + + Exemples + + + + Voir aussi + + + rowcomp + + + fullrf + + + fullrfk + + + kernel + + + + diff --git a/modules/linear_algebra/help/fr_FR/kernel/fullrf.xml b/modules/linear_algebra/help/fr_FR/kernel/fullrf.xml new file mode 100755 index 000000000..07b037630 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/kernel/fullrf.xml @@ -0,0 +1,102 @@ + + + + + fullrf + factorisation de rang plein + + + Séquence d'appel + [Q,M,rk]=fullrf(A,[tol]) + + + Paramètres + + + A + + matrice réelle ou complexe + + + + + tol + + nombre réel (tolérance pour le calcul du rang) + + + + + Q,M + + matrices réelles ou complexes + + + + + rk + + + entier (rang de A) + + + + + + + Description + + Cette fonction calcule la factorisation de rang plein de A : fullrf renvoie Q et M telles que A = Q*M + avec Im(Q)=Im(A) et ker(M)=ker(A), + Q de rang maximal, et les lignes de M sont linéairement indépendantes, + rk = rank(A) = nombre de colonnes de Q = nombre de lignes de M. + + + tol = paramètre de tolérance (de l'ordre de %eps par défaut). + Le rang rk de A est considéré égal au nombre de ses valeurs singulières plus grandes que norm(A)*tol. + + + Si A est symétrique, fullrf renvoie M=Q'. + + + + Exemples + + + + Voir aussi + + + svd + + + qr + + + fullrfk + + + rowcomp + + + colcomp + + + + diff --git a/modules/linear_algebra/help/fr_FR/kernel/fullrfk.xml b/modules/linear_algebra/help/fr_FR/kernel/fullrfk.xml new file mode 100755 index 000000000..12336d7bb --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/kernel/fullrfk.xml @@ -0,0 +1,77 @@ + + + + + fullrfk + factorisation de rang plein de A^k + + + Séquence d'appel + [Q,M]=fullrfk(A,k) + + + Paramètres + + + A + + matrice réelle ou complexe + + + + + k + + entier + + + + + Q,M + + matrices réelles ou complexes + + + + + + + Description + + Cette fonction calcule la factorisation de rang plein de A^k : fullrfk renvoie Q et M telles que A^k = Q*M + avec Im(Q)=Im(A^k) et ker(M)=ker(A^k), + Q de rang maximal, et les lignes de M sont linéairement indépendantes, + + + Pour k=1, fullrfk est équivalent à fullrf. + + + + Exemples + + + + Voir aussi + + + fullrf + + + range + + + + diff --git a/modules/linear_algebra/help/fr_FR/kernel/kernel.xml b/modules/linear_algebra/help/fr_FR/kernel/kernel.xml new file mode 100755 index 000000000..9cf413739 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/kernel/kernel.xml @@ -0,0 +1,96 @@ + + + + + kernel + noyau + + + Séquence d'appel + W=kernel(A [,tol,[,flag]) + + + Paramètres + + + A + + matrice réelle ou complexe (pleine ou creuse) + + + + + flag + + chaîne de caractères + + + + + tol + + nombre réel + + + + + W + + matrice régulière + + + + + + + Description + + W=kernel(A) calcule le noyau de A, et size(W,2) est la nullité de A. + Les colonnes de W forment une base du noyau de A. + Si A est régulière, alors W=[]. + + + flag et tol sont des paramètres optionnels : flag = 'qr' + or 'svd' ('svd' par défaut). + + + tol = paramètre de tolérance (de l'ordre de %eps par défaut). + + + + Exemples + + + + Voir aussi + + + colcomp + + + fullrf + + + fullrfk + + + linsolve + + + + diff --git a/modules/linear_algebra/help/fr_FR/kernel/range.xml b/modules/linear_algebra/help/fr_FR/kernel/range.xml new file mode 100755 index 000000000..e7b6aa6ef --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/kernel/range.xml @@ -0,0 +1,95 @@ + + + + + range + Image de A^k + + + Séquence d'appel + [X,dim]=range(A,k) + + + Paramètres + + + A + + matrice réelle carrée + + + + k + + entier non négatif, La valeur par défaut est 1 + + + + X + + matrice réelle orthonormale. + + + + dim + + entier (dimension du sous-espace image) + + + + + + Description + + Calcul de l'image de A^k; les dim + premières colonnes de X forment une base de + A^k. Les dernières lignes de X forment une + base de l'orthogonal de l'image. + + + + Exemples + + + + Voir aussi + + + fullrfk + + + rowcomp + + + + + Fonctions Utilisées + + La fonction range est basée sue la fonction rowcomp + qui utilise la décomposition svd. + + + diff --git a/modules/linear_algebra/help/fr_FR/kernel/rowcomp.xml b/modules/linear_algebra/help/fr_FR/kernel/rowcomp.xml new file mode 100755 index 000000000..bb071feae --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/kernel/rowcomp.xml @@ -0,0 +1,124 @@ + + + + + rowcomp + compression de lignes, image + + + Séquence d'appel + [W,rk]=colcomp(A [,flag [,tol]]) + + + Paramètres + + + A + + matrice réelle ou complexe + + + + flag + + chaîne de caractères optionnelle qui peut prendre les valeurs + 'svd' ou 'qr'. La valeur par + défaut est sqrt(%eps)*norm(A,1). + + + + + tol + + nombre réel non négatif. La valeur par + défaut est sqrt(%eps)*norm(A,1). + + + + + W + + matrice carrée régulière (matrice de changement de base) + + + + rk + + + entier (rang de"A). + + + + + + + Description + + Compression des colonnes de A. Ac = W*A est à + lignes compressées, c'est à dire + Ac=[Af;0] et les lignes de Af sont linéairement + indépendantes. + + + flag et tol sont des paramètres optionnels : + flag = 'qr' ou 'svd' + ('svd' par défaut). + + + tol = paramètre de tolérance (de l'ordre de + %eps par défaut). + + + Les rk premières colonnes de W' forment + une base de l'image de A. + + + Un vecteur non nul x appartient à Im(A) si + W*x est à lignes compressées en accord avec Ac + c'est à dire que la norme de ses dernières composantes est nulle (à + la précision machine) par rapport à ses rk premières composantes. + + + + Exemples + + + + Voir aussi + + + colcomp + + + fullrf + + + fullrfk + + + + + Fonctions Utilisées + + La fonction rowcomp est basée sur les décompositions + svd ou qr. + + + diff --git a/modules/linear_algebra/help/fr_FR/linear/CHAPTER b/modules/linear_algebra/help/fr_FR/linear/CHAPTER new file mode 100755 index 000000000..7d9d9cf49 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/linear/CHAPTER @@ -0,0 +1,2 @@ +title = Linear Equations + diff --git a/modules/linear_algebra/help/fr_FR/linear/chol.xml b/modules/linear_algebra/help/fr_FR/linear/chol.xml new file mode 100755 index 000000000..f155bb42c --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/linear/chol.xml @@ -0,0 +1,80 @@ + + + + + chol + Factorisation de Cholesky + + + Séquence d'appel + [R]=chol(X) + + + Paramètres + + + X + + matrice réelle ou complexe + + + + + + + Description + + Si X est hermitienne (symétrique dans le cas réel) définie positive, alors R = chol(X) renvoie une matrice triangulaire supérieure R telle que R'*R = X. + + + chol(X) utilise uniquement la partie triangulaire supérieure de X dont la + partie triangulaire inférieure est supposée être la transposée (transposée conjuguée dans le cas complexe). + + + + Exemples + + + + Voir aussi + + + spchol + + + qr + + + svd + + + bdiag + + + fullrf + + + + + Fonctions Utilisées + + La décomposition de Cholesky est basée sur les routines Lapack + DPOTRF pour les matrices réelles et ZPOTRF pour le cas complexe. + + + diff --git a/modules/linear_algebra/help/fr_FR/linear/inv.xml b/modules/linear_algebra/help/fr_FR/linear/inv.xml new file mode 100755 index 000000000..2441ede3a --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/linear/inv.xml @@ -0,0 +1,112 @@ + + + + + inv + inverse d'une matrice + + + Séquence d'appel + inv(X) + + + Paramètres + + + X + + matrice carrée réelle, complexe, polynomiale ou rationnelle, + liste de type "syslin" + + + + + + + Description + + inv(X) est l'inverse de la matrice carrée + X. Un message de mise en garde est affiché si X + est mal équilibrée (termes très petits et termes très grands) ou + singulière à la précision machine. + + + Pour les matrices polynomiales ou rationnelles, inv(X) est + équivalent à invr(X). + + + Pour les systèmes dynamiques linéaires sous forme de leur représentation + d'état (liste de type syslin), inv(X) est + équivalent à invsyslin(X). + + + + Exemples + + + + Voir aussi + + + slash + + + backslash + + + pinv + + + qr + + + lufact + + + lusolve + + + invr + + + coff + + + coffg + + + + + Fonctions Utilisées + + La fonction inv pour les matrices de nombres est basée + sur les routines Lapack : + DGETRF, DGETRI pour les matrices réelles et ZGETRF, ZGETRI pour le + cas complexe. + Pour les matrices de polynomes et de fractions rationnelles + inv est basée sur la fonction Scilab invr. + + + diff --git a/modules/linear_algebra/help/fr_FR/linear/linsolve.xml b/modules/linear_algebra/help/fr_FR/linear/linsolve.xml new file mode 100755 index 000000000..d7d5c3bc2 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/linear/linsolve.xml @@ -0,0 +1,121 @@ + + + + + linsolve + solveur d'équation linéaire + + + Séquence d'appel + [x0,kerA]=linsolve(A,b [,x0]) + + + Paramètres + + + A + + + une matrice réelle na x ma (éventuellement creuse) + + + + + b + + + un vecteur na x 1 + + + + + x0 + + un vecteur réel + + + + + kerA + + + une matrice réelle ma x k + + + + + + + Description + + linsolve donne toutes les solutions de A*x+b=0. + + + x0 est une solution particulière (s'il en existe une) et kerA est le noyau de A. Tout vecteur de la forme x=x0+kerA*w avec w quelconque vérifie + A*x+b=0. + + + Si un x0 compatible est donné en entrée, x0 est renvoyé. Dans le cas contraire un x0 compatible, s'il en existe un, est renvoyé. + + + + Exemples + + + + Voir aussi + + + inv + + + pinv + + + colcomp + + + im_inv + + + umfpack + + + backslash + + + + diff --git a/modules/linear_algebra/help/fr_FR/linear/lu.xml b/modules/linear_algebra/help/fr_FR/linear/lu.xml new file mode 100755 index 000000000..498ac9713 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/linear/lu.xml @@ -0,0 +1,119 @@ + + + + + lu + factorisation LU + + + Séquence d'appel + [L,U]= lu(A) + [L,U,E]= lu(A) + + + + Paramètres + + + A + + matrice carrée réelle ou complexe (m x n). + + + + + L,U + + matrices carrées réelles ou complexes (n x n). + + + + + E + + une matrice de permutation. + + + + + + + Description + + [L,U]= lu(A) calcule deux matrices L et + U telles que A = L*U avec U + triangulaire supérieure et L triangulaire inférieure + à une permutation des lignes près. + + + Si A est de rang k, les lignes + k+1 à n de U sont nulles. + + + + + [L,U,E]= lu(A) calcule trois matrices L, + U et E telles que E*A = L*U + avec U triangulaire supérieure, L + triangulaire inférieure et E une matrice de + permutation. + + + Si A est une matrice réelle, il est possible en + utilisant lufact et luget + d'obtenir les matrices de permutations et quand + A n'est pas inversible la compression des + colonnes de la matrice L. + + + + Exemples + + + + Voir aussi + + + lufact + + + luget + + + lusolve + + + qr + + + svd + + + + + Fonctions Utilisées + La décomposition LU est basée sur les routines Lapack DGETRF pour + les matrices réelles et ZGETRF pour le cas complexe. + + + diff --git a/modules/linear_algebra/help/fr_FR/linear/pinv.xml b/modules/linear_algebra/help/fr_FR/linear/pinv.xml new file mode 100755 index 000000000..70b1d11ec --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/linear/pinv.xml @@ -0,0 +1,85 @@ + + + + + pinv + pseudo-inverse + + + Séquence d'appel + pinv(A,[tol]) + + + Paramètres + + + A + + matrice réelle ou complexe + + + + + tol + + nombre réel + + + + + + + Description + + X= pinv(A) renvoie une matrice X de mêmes dimensions que A' telle que : + + + A*X*A = A, X*A*X = X avec + A*X et X*A Hermitiennes. + + + Le calcul est basé sur une décomposition en valeurs singulières et + les valeurs singulières plus petites qu'une tolérance donnée + sont considérées comme nulles : pour cela utiliser la syntaxe + X=pinv(A,tol). + + + + Exemples + + + + Voir aussi + + + rank + + + svd + + + qr + + + + + Fonctions Utilisées + + La fonction pinv est basée sur la decomposition en valeurs + singulières (fonction Scilab svd). + + + diff --git a/modules/linear_algebra/help/fr_FR/linear/qr.xml b/modules/linear_algebra/help/fr_FR/linear/qr.xml new file mode 100755 index 000000000..3cb813d07 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/linear/qr.xml @@ -0,0 +1,194 @@ + + + + + qr + factorisation QR + + + Séquence d'appel + [Q,R]=qr(X [,"e"]) + [Q,R,E]=qr(X [,"e"]) + [Q,R,rk,E]=qr(X [,tol]) + + + + Paramètres + + + X + + matrice réelle ou complexe + + + + + tol + + nombre réel positif + + + + + Q + + matrice carrée unitaire + + + + + R + + + matrice de même dimensions que X + + + + + E + + matrice de permutation + + + + + rk + + + entier (rang de X) + + + + + + + Description + + [Q,R] = qr(X) renvoie une matrice triangulaire supérieure + R de même + dimensions que X et une matrice carrée othogonale + (unitaire dans le cas complexe) Q telles que + X = Q*R. + + [Q,R] = qr(X,"e") renvoie une decomposition de + taille réduite: si X est une matrice m x + n + + avec m > n alors seulement les + n premières colonnes de Q sont calculées + ainsi que les n premières lignes de + R. + + + Il découle de Q*R = X que la + kième colonne de X peut s'exprimer comme + une combinaison linéaire des k premieres colonnes de + Q (avec les coefficients R(1,k), ..., + R(k,k) + + .Les k premieres colonnes de + Q forment une base orthogonale du sous espace généré + par les Les k premieres colonnes de + X. Si la colonne k de X est + une combinaison linéaire des p premiéres colonnes de + X alors les éléments R(p+1,k), ..., + R(k,k) + + sont nuls. Dans cette situation R est + une matrice trapézoidale supérieure. Si X est de rang + rk alors les lignes R(rk+1,:), R(rk+2,:), + ... + + sont nulles. + + + + [Q,R,E] = qr(X) renvoie une matrice de permutations (de + colonnes) E, + une matrice triangulaire supérieure R dont les + éléments diagonaux sont classés par ordre décroissant et une + matrice unitaire Q telles que X*E = Q*R. + si rk est le rang de X les + rk premiers éléménts diagonaux de R sont + tous non nuls. [Q,R,E] = qr(X,"e") renvoie une decomposition de + taille réduite: si X est une matrice m x + n + + avec m > n alors seulement les + n premières colonnes de Q sont calculées + ainsi que les n premières lignes de + R. + + + [Q,R,rk,E] = qr(X [,tol])renvoie de plus + rk =rang estimé de X. + Plus précisément, + rk est le nombre d'éléments diagonaux de + R supérieurs à tol. La valeur par défaut + de tol est R(1,1)*%eps*max(size(R)) + + + renvoie rk = rang estimé de X. Ici, + rk est le nombre d'éléments diagonaux de R + supérieurs à R(1,1)*%eps*max(size(R). + + + + Exemples + rk first +//diagonal entries of R are non zero : +A=rand(5,2)*rand(2,5); +[Q,R,rk,E] = qr(A,1.d-10); +norm(Q'*A-R) +svd([A,Q(:,1:rk)]) //span(A) =span(Q(:,1:rk)) + ]]> + + + Voir aussi + + + rank + + + svd + + + rowcomp + + + colcomp + + + + + Fonctions Utilisées + La décomposition QR est basée sur les routines Lapack DGEQRF, DGEQPF, + DORGQR pour les matrices réelles et ZGEQRF, ZGEQPF, ZORGQR pour le cas + complexe. + + + diff --git a/modules/linear_algebra/help/fr_FR/markov/CHAPTER b/modules/linear_algebra/help/fr_FR/markov/CHAPTER new file mode 100755 index 000000000..deb78b04a --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/markov/CHAPTER @@ -0,0 +1,2 @@ +title = Matrices de Markov + diff --git a/modules/linear_algebra/help/fr_FR/matrix/CHAPTER b/modules/linear_algebra/help/fr_FR/matrix/CHAPTER new file mode 100755 index 000000000..bb89125cd --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/matrix/CHAPTER @@ -0,0 +1,2 @@ +title = Matrix Analysis + diff --git a/modules/linear_algebra/help/fr_FR/matrix/cond.xml b/modules/linear_algebra/help/fr_FR/matrix/cond.xml new file mode 100755 index 000000000..7c53f27ac --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/matrix/cond.xml @@ -0,0 +1,160 @@ + + + + + cond + conditionnement + + + Séquence d'appel + + c = cond(X) + c = cond(X, p) + + + + Paramètres + + + X + + + matrice réelle ou complexe. Si c = cond(X, p), X doit être une matrice carrée réelle ou complexe. + + + + + p + + scalaire ou chaine de caractères (valeur par défaut p = 2). + + + + c + + scalaire réel. + + + + + + Description + + + c = cond(X) + + + retourne le conditionnement en norme 2.cond(X) est le quotient entre + la plus grande et la plus petite valeur singulière de X. + + + + + c = cond(X, p) + + + retourne le conditionnement en norme p : norm(X, p) * norm(inv(X), p). + Si p est spécifié, p est égal soit à: + + + + + p = 1. cond(X, p) retourne le conditionnement en norme 1. + + + + + p = 2. cond(X, p) retourne le conditionnement en norme 2. + + + + + p = %inf or 'inf'. cond(X, p) retourne le conditionnement en norme infinie. + + + + + p = 'fro'. cond(X, p) retourne le conditionnement en norme de Frobenius. + + + + + + + + + Exemples + + + + + + Voir aussi + + + rcond + + + svd + + + norm + + + + + Historique + + + 5.4.0 + + + Appel de cond(X), où X est une matrice non + carrée, est maintenant gérée. Par exemple : + + + + + + Appel de cond(X, p) permet de calculer le contionnement + en norme p. Par exemple : + + + + + + + + + diff --git a/modules/linear_algebra/help/fr_FR/matrix/det.xml b/modules/linear_algebra/help/fr_FR/matrix/det.xml new file mode 100755 index 000000000..441b723c5 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/matrix/det.xml @@ -0,0 +1,94 @@ + + + + + det + déterminant + + + Séquence d'appel + det(X) + [e,m]=det(X) + + + + Paramètres + + + X + + matrice carrée réelle ou complexe (creuse ou pleine), polynomiale ou rationnelle + + + + + m + + nombre réel ou complexe, mantisse du déterminant en base 10 + + + + + e + + entier, exposant du déterminant en base 10 + + + + + + + Description + + det(X) ( m*10^e ) est le déterminant de la matrice carrée X. + + + Pour les matrices polynomiales det(X) est équivalent à determ(X). + + + Pour les matrices rationnelles det(X) est équivalent à detr(X). + + + + Exemples + + + + Voir aussi + + + detr + + + determ + + + + + Fonctions Utilisées + + Le calcul du determinant est basé sur les routines Lapack : + DGETRF pour les matrices réelles et ZGETRF pour le cas complexe. + + + Concernant le cas des matrices creuses, le calcul du déterminant est effectué + à partir de la décomposition LU de la librairie umfpack. + + + diff --git a/modules/linear_algebra/help/fr_FR/matrix/orth.xml b/modules/linear_algebra/help/fr_FR/matrix/orth.xml new file mode 100755 index 000000000..1c190bc72 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/matrix/orth.xml @@ -0,0 +1,78 @@ + + + + + orth + calcul d'une base orthogonale + + + Séquence d'appel + Q=orth(A) + + + Paramètres + + + A + + matrice réelle ou complexe + + + + + Q + + matrice réelle ou complexe + + + + + + + Description + + Q=orth(A) renvoie Q, une base + orthogonale de l'image de A. Im(Q) + = Im(A) et Q'*Q = I. + + + Le nombre de colonnes de Q est égal au rang de + A, comme déterminé par l'algorithme QR. + + + + Exemples + + + + Voir aussi + + + qr + + + rowcomp + + + colcomp + + + range + + + + diff --git a/modules/linear_algebra/help/fr_FR/matrix/rank.xml b/modules/linear_algebra/help/fr_FR/matrix/rank.xml new file mode 100755 index 000000000..06761fea1 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/matrix/rank.xml @@ -0,0 +1,94 @@ + + + + + rank + rang + + + Séquence d'appel + [i]=rank(X) + [i]=rank(X,tol) + + + + Paramètres + + + X + + matrice réelle ou complexe + + + + + tol + + nombre réel positif + + + + + + + Description + + rank(X) calcule le rang "numérique" de + X c'est à dire le nombre de ses valeurs + singulières supérieures à norm(size(X),'inf') * + norm(X) * %eps + + . + + + rank(X,tol) est le nombre de valeurs singulières de + X supérieures à tol. + + + + Notez que la valeur par défaut de tol est + proportionnelle à norm(X). Par exemple + + + + rank([1.d-80,0;0,1.d-80]) vaut 2 !. + + + + Exemples + + + + Voir aussi + + + svd + + + qr + + + rowcomp + + + colcomp + + + lu + + + + diff --git a/modules/linear_algebra/help/fr_FR/matrix/rcond.xml b/modules/linear_algebra/help/fr_FR/matrix/rcond.xml new file mode 100755 index 000000000..40f871b7f --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/matrix/rcond.xml @@ -0,0 +1,74 @@ + + + + + rcond + estimation de l'inverse du conditionnement + + + Séquence d'appel + rcond(X) + + + Paramètres + + + X + + matrice carrée réelle ou complexe + + + + + + + Description + + rcond(X) est une estimation de l'inverse du conditionnement de X pour la norme l_1. + + + Si X est bien conditionnée, rcond(X) est proche 1. + Sinon, rcond(X) est proche de 0. + + + [r,z]=rcond(X) renvoie rcond(X) dans + r et renvoie aussi z tel que norm(X*z,1) = r*norm(X,1)*norm(z,1) + + + Ainsi, si rcond est très petit z est un vecteur se trouvant dans le noyau de X. + + + + Exemples + + + + Voir aussi + + + svd + + + cond + + + inv + + + + diff --git a/modules/linear_algebra/help/fr_FR/matrix/trace.xml b/modules/linear_algebra/help/fr_FR/matrix/trace.xml new file mode 100755 index 000000000..f37190fa4 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/matrix/trace.xml @@ -0,0 +1,58 @@ + + + + + trace + trace d'une matrice + + + Séquence d'appel + trace(X) + + + Paramètres + + + X + + matrice carrée, réelle, complexe, polynomiale ou rationnelle. + + + + + + + Description + + trace(X) calcule la trace de X. + + + Identique à sum(diag(X)). + + + + Exemples + + + + Voir aussi + + + det + + + + diff --git a/modules/linear_algebra/help/fr_FR/pencil/CHAPTER b/modules/linear_algebra/help/fr_FR/pencil/CHAPTER new file mode 100755 index 000000000..2c9344edc --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/pencil/CHAPTER @@ -0,0 +1,2 @@ +title = Matrice compagnon + diff --git a/modules/linear_algebra/help/fr_FR/pencil/companion.xml b/modules/linear_algebra/help/fr_FR/pencil/companion.xml new file mode 100755 index 000000000..80fc6d410 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/pencil/companion.xml @@ -0,0 +1,78 @@ + + + + + companion + matrice compagnon + + + Séquence d'appel + A=companion(p) + + + Paramètres + + + p + + polynôme ou vecteur de polynômes + + + + + A + + matrice carrée + + + + + + + Description + + Renvoie une matrice A dont le polynôme caractéristique est + p si p est unitaire (le coefficient de plus haut degré est égal à un). Si p n'est pas unitaire + le polynôme caractéristique de A est égal à + p/c où c est le coefficient de plus haut degré de p. + + + Si p est un vecteur de polynômes unitaires, A est bloc-diagonale, + et le polynôme caractéristique du i-ème bloc est égal à p(i). + + + + Exemples + + + + Voir aussi + + + spec + + + poly + + + randpencil + + + + diff --git a/modules/linear_algebra/help/fr_FR/pencil/glever.xml b/modules/linear_algebra/help/fr_FR/pencil/glever.xml new file mode 100755 index 000000000..784429a4f --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/pencil/glever.xml @@ -0,0 +1,123 @@ + + + + + glever + inverse d'un faisceau de matrices + + + Séquence d'appel + [Bfs,Bis,chis]=glever(E,A [,s]) + + + Paramètres + + + E, A + + matrices carrées réelles de même dimensions + + + + + s + + + chaîne de caractères (indéterminée des polynômes, 's' par défaut ) + + + + + Bfs,Bis + + deux matrices polynomiales + + + + + chis + + polynôme + + + + + + + Description + + Calcul de + + + (s*E-A)^-1 + + + par l'algorithme généralisé de Leverrier pour un faisceau de matrices. + + + + chis = polynôme caractéristique (à une constante multiplicative près). + + + Bfs = matrice polynomiale de numérateurs + + + Bis + = matrice polynomiale ( - développement de (s*E-A)^-1 à l'infini). + + + Noter le signe - devant Bis. + + + + + + Attention + + Cette fonction utilise cleanp pour simplifier Bfs,Bis et chis. + + + + Exemples + + + + Voir aussi + + + rowshuff + + + det + + + invr + + + coffg + + + pencan + + + penlaur + + + + diff --git a/modules/linear_algebra/help/fr_FR/pencil/lyap.xml b/modules/linear_algebra/help/fr_FR/pencil/lyap.xml new file mode 100755 index 000000000..68e7a0eca --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/pencil/lyap.xml @@ -0,0 +1,82 @@ + + + + + lyap + Equation de Lyapunov + + + Séquence d'appel + [X]=lyap(A,C,flag) + + + Paramètres + + + A, C + + + matrices réelles, C doit être symétrique + + + + + flag + + chaîne de caractères, 'c' ou 'd' + + + + + + + Description + + X= lyap(A,C,flag) résout l'équation matricielle de + Lyapunov en temps continu ou discret + + + + Une solution unique existe si A n'a pas de valeur propre + sur l'axe imaginaire (flag='c') ou si 1 n'est pas + valeur propre de A (flag='d'). + + + + Exemples + + + + Voir aussi + + + sylv + + + ctr_gram + + + obs_gram + + + + diff --git a/modules/linear_algebra/help/fr_FR/proj.xml b/modules/linear_algebra/help/fr_FR/proj.xml new file mode 100755 index 000000000..5bd02b507 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/proj.xml @@ -0,0 +1,73 @@ + + + + + proj + projection + + + Séquence d'appel + P = proj(X1,X2) + + + Paramètres + + + X1,X2 + + deux matrices réelles avec un nombre identique de colonnes. + + + + + P + + + matrice réelle de projection (P^2=P) + + + + + + + Description + + P est la projection sur X2 parallèlement à X1. + + + + Exemples + + + + Voir aussi + + + projspec + + + orth + + + fullrf + + + + diff --git a/modules/linear_algebra/help/fr_FR/state_space/CHAPTER b/modules/linear_algebra/help/fr_FR/state_space/CHAPTER new file mode 100755 index 000000000..9f51d7351 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/state_space/CHAPTER @@ -0,0 +1 @@ +title = State-Space Matrices diff --git a/modules/linear_algebra/help/fr_FR/state_space/coff.xml b/modules/linear_algebra/help/fr_FR/state_space/coff.xml new file mode 100755 index 000000000..e69a389ef --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/state_space/coff.xml @@ -0,0 +1,99 @@ + + + + + coff + résolvante (méthode des cofacteurs) + + + Séquence d'appel + [N,d]=coff(M [,var]) + + + Paramètres + + + M + + matrice carrée réelle + + + + + var + + chaîne de caractères (indéterminée des polynômes) + + + + + N + + + matrice de polynômes (de même taille que M) + + + + + d + + + polynôme (polynôme caractéristique de M : poly(M,var)) + + + + + + + Description + + coff calcule R=(s*eye()-M)^-1 pour M une matrice réelle. + R est donnée par N/d. + + + N = matrice des numérateurs (polynômes). + + + d = dénominateur commun. + + + var chaîne de caractères (indéterminée des polynômes, 's' par défaut) + + + + Exemples + + + + Voir aussi + + + coffg + + + ss2tf + + + nlev + + + poly + + + + diff --git a/modules/linear_algebra/help/fr_FR/state_space/nlev.xml b/modules/linear_algebra/help/fr_FR/state_space/nlev.xml new file mode 100755 index 000000000..f309b8233 --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/state_space/nlev.xml @@ -0,0 +1,88 @@ + + + + + nlev + Algorithme de Leverrier + + + Séquence d'appel + [num,den]=nlev(A,z [,rmax]) + + + Paramètres + + + A + + matrice réelle carrée + + + + + z + + chaîne de caractères + + + + + rmax + + + paramètre optionnel (voir bdiag) + + + + + + + Description + + [num,den]=nlev(A,z [,rmax]) calcule + (z*eye()-A)^(-1) par bloc diagonalisation de + A suivie de l'algorithme de Leverrier sur chaque + bloc. + + + Cet algorithme est plus efficace que l'algorithme de + Leverrier seul, mais il n'est toujours pas parfait ! + + + + Exemples + + + + Voir aussi + + + coff + + + coffg + + + glever + + + ss2tf + + + + diff --git a/modules/linear_algebra/help/fr_FR/subspaces/CHAPTER b/modules/linear_algebra/help/fr_FR/subspaces/CHAPTER new file mode 100755 index 000000000..90541a88e --- /dev/null +++ b/modules/linear_algebra/help/fr_FR/subspaces/CHAPTER @@ -0,0 +1,2 @@ +title = Sous-espaces + diff --git a/modules/linear_algebra/help/ja_JP/addchapter.sce b/modules/linear_algebra/help/ja_JP/addchapter.sce new file mode 100755 index 000000000..4b62a3ac7 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/addchapter.sce @@ -0,0 +1,11 @@ +// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +// Copyright (C) 2009 - DIGITEO +// +// This file must be used under the terms of the CeCILL. +// This source file is licensed as described in the file COPYING, which +// you should have received as part of this distribution. The terms +// are also available at +// http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt + +add_help_chapter("Linear Algebra",SCI+"/modules/linear_algebra/help/ja_JP",%T); + diff --git a/modules/linear_algebra/help/ja_JP/eigen/CHAPTER b/modules/linear_algebra/help/ja_JP/eigen/CHAPTER new file mode 100755 index 000000000..88f8bc42b --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/eigen/CHAPTER @@ -0,0 +1,2 @@ +title = Eigenvalue and Singular Value + diff --git a/modules/linear_algebra/help/ja_JP/eigen/balanc.xml b/modules/linear_algebra/help/ja_JP/eigen/balanc.xml new file mode 100755 index 000000000..b150c2ea6 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/eigen/balanc.xml @@ -0,0 +1,203 @@ + + + + + + + + + balanc + + 行列ã¾ãŸã¯ãƒšãƒ³ã‚·ãƒ«ã®å¹³è¡¡åŒ– + + + + + + 呼ã³å‡ºã—手順 + + [Ab,X]=balanc(A) + + [Eb,Ab,X,Y]=balanc(E,A) + + + + + + + + 引数 + + + + + + A: + + + + 実数正方行列 + + + + + + + + X: + + + + å¯é€†ãªå®Ÿæ•°æ­£æ–¹è¡Œåˆ— + + + + + + + + E: + + + + + + 実数正方行列 (Aã¨åŒã˜æ¬¡å…ƒ) + + + + + + + + + + Y: + + + + å¯é€†ãªå®Ÿæ•°æ­£æ–¹è¡Œåˆ—. + + + + + + + + + + + + 説明 + + + + 正方行列ã®æ¡ä»¶æ•°ã‚’改善ã™ã‚‹ãŸã‚ã«å¹³è¡¡åŒ–ã—ã¾ã™. + + + + + + [Ab,X] = balanc(A) ã¯, + + 以下ã®ã‚ˆã†ãªç›¸ä¼¼å¤‰æ›Xを見ã¤ã‘ã¾ã™: + + + + + + Ab = inv(X)*A*XãŒè¿‘似的ã«ç­‰ã—ã„ + + 行ノルムãŠã‚ˆã³åˆ—ノルムを有ã™ã‚‹. + + + + + + 行列ペンシルã®å ´åˆ,平衡化ã¯ä¸€èˆ¬åŒ–固有値å•é¡Œã‚’改善ã™ã‚‹ã“ã¨ã«ã‚ˆã‚Š + + è¡Œã‚ã‚Œã¾ã™. + + + + + + [Eb,Ab,X,Y] = balanc(E,A) ã¯, + + Eb=inv(X)*E*Y, Ab=inv(X)*A*Y ã¨ãªã‚‹ã‚ˆã†ãª + + å·¦ãŠã‚ˆã³å³å¤‰æ› + + X ãŠã‚ˆã³ Y ã‚’è¿”ã—ã¾ã™. + + + + + + + + æ³¨æ„ + + + + 平衡化ã¯é–¢æ•°bdiag ãŠã‚ˆã³ spec + + ã§è¡Œã‚ã‚Œã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + bdiag + + + + + + spec + + + + + + schur + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/eigen/bdiag.xml b/modules/linear_algebra/help/ja_JP/eigen/bdiag.xml new file mode 100755 index 000000000..ba3a9117c --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/eigen/bdiag.xml @@ -0,0 +1,189 @@ + + + + + + + + + bdiag + + ブロック対角化, 一般化固有ベクトル + + + + + + 呼ã³å‡ºã—手順 + + [Ab [,X [,bs]]]=bdiag(A [,rmax]) + + + + + + 引数 + + + + + + A + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®æ­£æ–¹è¡Œåˆ— + + + + + + + + rmax + + + + 実数 + + + + + + + + Ab + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®æ­£æ–¹è¡Œåˆ— + + + + + + + + X + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®æ­£å‰‡è¡Œåˆ— + + + + + + + + bs + + + + 整数ベクトル + + + + + + + + + + + + 説明 + + + + + + ã¯,行列Aã®ãƒ–ロック対角化を行ãªã„ã¾ã™. + + bs ã¯ãƒ–ロックã®æ§‹é€ (個々ã®ãƒ–ロックã®å¤§ãã•)を出力ã—ã¾ã™. + + X ã¯åŸºåº•å¤‰æ›ã§ã™. + + ã™ãªã‚ã¡, Ab = inv(X)*A*X ã¯ãƒ–ロック対角ã§ã™. + + + + + + rmax ã¯Xã® + + æ¡ä»¶æ•°ã‚’制御ã—ã¾ã™; + + デフォルト値㯠A ã® l1ノルムã§ã™. + + + + + + (存在ã™ã‚‹å ´åˆ,)対角形å¼ã‚’å¾—ã‚‹ã«ã¯rmaxã« + + 大ããªå€¤ã‚’指定ã—ã¾ã™(例ãˆã°,rmax=1/%eps). + + 一般ã«(ランダムãªå®Ÿæ•°ã® Aã®å ´åˆ) ブロック㯠(1x1) ãŠã‚ˆã³ (2x2) ã§, + + X ã¯å›ºæœ‰å€¤ã®è¡Œåˆ—ã§ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + schur + + + + + + sylv + + + + + + spec + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/eigen/gschur.xml b/modules/linear_algebra/help/ja_JP/eigen/gschur.xml new file mode 100755 index 000000000..2f42eb262 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/eigen/gschur.xml @@ -0,0 +1,180 @@ + + + + + + + + + gschur + + + + 一般化Schur分解. + + ã“ã®é–¢æ•°ã¯å»ƒæ­¢ã•ã‚Œã¾ã—ãŸ. + + + + + + + + 呼ã³å‡ºã—手順 + + [As,Es]=gschur(A,E) + + [As,Es,Q,Z]=gschur(A,E) + + [As,Es,Z,dim] = gschur(A,E,flag) + + [As,Es,Z,dim]= gschur(A,E,extern) + + + + + + + + 説明 + + + + ã“ã®é–¢æ•°ã¯å»ƒæ­¢ã•ã‚Œã¦ãŠã‚Š, schur関数ã«çµ±åˆã•ã‚Œã¦ã„ã¾ã™. + + 多ãã®å ´åˆ, gschur関数ã¯ä»¥å‰ã¨åŒæ§˜ã«å‹•ä½œã—ã¾ã™ãŒ, + + å°†æ¥ã®ãƒªãƒªãƒ¼ã‚¹ã§ã¯å‰Šé™¤ã•ã‚Œã‚‹äºˆå®šã§ã™. + + + + + + 最åˆã®3ã¤ã®æ§‹æ–‡ã¯ä»¥ä¸‹ã®ã‚ˆã†ã«ç½®ãæ›ãˆã‚‹ã“ã¨ãŒã§ãã¾ã™ + + + + + + + + 最後ã®æ§‹æ–‡ã¯ã•ã‚‰ã«è‹¥å¹²ã®èª¿æ•´ãŒå¿…è¦ã§ã™: + + + + + + + + ã‚‚ã—, + + + + + + extern ãŒã€€Scilab関数ã®å ´åˆ, + + Nextern を以下ã®ã‚ˆã†ã«å®šç¾©ã™ã‚‹ã¨, + + æ–°ã—ã„呼ã³å‡ºã—手順ã¯, + + [As,Es,Z,dim]= schur(A,E,Nextern) + + ã¨ãªã‚Šã¾ã™: + + + + + + + + + + + + ã‚‚ã—, + + + + + + extern ã¯,Fortran ã¾ãŸã¯ Cã§è¨˜è¿°ã•ã‚ŒãŸå¤–部関数ã®åå‰ã®å ´åˆ, + + nextern を以下ã®ã‚ˆã†ã«å®šç¾©ã™ã‚‹ã¨ + + æ–°ã—ã„呼ã³å‡ºã—手順ã¯, + + [As,Es,Z,dim]= schur(A,E,'nextern') + + ã®ã‚ˆã†ã«ãªã‚Šã¾ã™: + + + + + + + + + + + + + + + + å‚ç…§ + + + + + + external + + + + + + schur + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/eigen/gspec.xml b/modules/linear_algebra/help/ja_JP/eigen/gspec.xml new file mode 100755 index 000000000..fffb2f624 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/eigen/gspec.xml @@ -0,0 +1,79 @@ + + + + + + + + + gspec + + + + 行列ペンシルã®å›ºæœ‰å€¤. + + ã“ã®é–¢æ•°ã¯å»ƒæ­¢ã•ã‚Œã¾ã—ãŸ. + + + + + + + + 呼ã³å‡ºã—手順 + + [al,be]=gspec(A,E) + + [al,be,Z]=gspec(A,E) + + + + + + + + 説明 + + + + ã“ã®é–¢æ•°ã¯ç¾åœ¨ã§ã¯ spec 関数ã«çµ±åˆã•ã‚Œã¦ã„ã¾ã™. + + 呼ã³å‡ºã—手順ã¯ä»¥ä¸‹ã®ã‚ˆã†ã«ç½®ãæ›ã‚‰ã‚Œã¦ã„ã¾ã™ + + + + + + + + + + å‚ç…§ + + + + + + spec + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/eigen/hess.xml b/modules/linear_algebra/help/ja_JP/eigen/hess.xml new file mode 100755 index 000000000..9ffb1de41 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/eigen/hess.xml @@ -0,0 +1,179 @@ + + + + + + + + + hess + + ãƒ˜ãƒƒã‚»ãƒ³ãƒ™ãƒ«ã‚¯å½¢å¼ + + + + + + 呼ã³å‡ºã—手順 + + H = hess(A) + + [U,H] = hess(A) + + + + + + + + 引数 + + + + + + A + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®æ­£æ–¹è¡Œåˆ— + + + + + + + + H + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®æ­£æ–¹è¡Œåˆ— + + + + + + + + U + + + + 直交ã¾ãŸã¯ãƒ¦ãƒ‹ã‚¿ãƒªæ­£æ–¹è¡Œåˆ— + + + + + + + + + + + + 説明 + + + + [U,H] = hess(A) ã¯, + + A = U*H*U' ãŠã‚ˆã³ U'*U =å˜ä½è¡Œåˆ— ã¨ãªã‚‹ã‚ˆã†ãª + + ユニタリ行列U ãŠã‚ˆã³ãƒ˜ãƒƒã‚»ãƒ³ãƒ™ãƒ«ã‚¯è¡Œåˆ—Hを出力ã—ã¾ã™. + + ã“ã‚Œã«ã‚ˆã‚Š, hess(A) 㯠Hã‚’è¿”ã—ã¾ã™. + + + + 行列ã®ãƒ˜ãƒƒã‚»ãƒ³ãƒ™ãƒ«ã‚¯å½¢å¼ã¯æœ€åˆã®å‰¯å¯¾è§’線以下ã§ã¯ 0ã¨ãªã‚Šã¾ã™. + + ã“ã®è¡Œåˆ—ãŒå¯¾ç§°ã¾ãŸã¯ã‚¨ãƒ«ãƒŸãƒ¼ãƒˆè¡Œåˆ—ã®å ´åˆ, + + å½¢ã¯3é‡å¯¾è§’ã¨ãªã‚Šã¾ã™. + + + + + + + + リファレンス + + + + hess 関数㯠Lapack ルーãƒãƒ³ + + DGEHRD, DORGHR (実数行列ã®å ´åˆ) ãŠã‚ˆã³ ZGEHRD, ZORGHR (複素数行列ã®å ´åˆ)ã«åŸºã¥ã„ã¦ã„ã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + qr + + + + + + contr + + + + + + schur + + + + + + + + + + 使用ã•ã‚Œã‚‹é–¢æ•° + + + + hess 関数ã¯Lapack ルーãƒãƒ³ + + DGEHRD, DORGHR (実数行列ã®å ´åˆ) ãŠã‚ˆã³ ZGEHRD, ZORGHR (複素数行列ã®å ´åˆ)ã« + + 基ã¥ã„ã¦ã„ã¾ã™. + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/eigen/pbig.xml b/modules/linear_algebra/help/ja_JP/eigen/pbig.xml new file mode 100755 index 000000000..feaaa4c4c --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/eigen/pbig.xml @@ -0,0 +1,234 @@ + + + + + + + + + pbig + + 固有投影 + + + + + + 呼ã³å‡ºã—手順 + + [Q,M]=pbig(A,thres,flag) + + + + + + 引数 + + + + + + A + + + + 実数正方行列 + + + + + + + + thres + + + + 実数 + + + + + + + + flag + + + + + + 文字列 ('c' ã¾ãŸã¯ 'd') + + + + + + + + + + Q,M + + + + 実数行列 + + + + + + + + + + + + 説明 + + + + 実部>= thres (flag='c') + + ã¾ãŸã¯ + + 大ãã•>= thres(flag='d') + + ã®å›ºæœ‰å€¤ã‚’有ã™ã‚‹å›ºæœ‰å€¤-部分空間ã¸ã®æŠ•å½±. + + + + + + 投影ã¯Q*Mã«ã‚ˆã‚Šå®šç¾©ã•ã‚Œ,Q + + ã¯åˆ—フルランク, Mã¯è¡Œãƒ•ãƒ«ãƒ©ãƒ³ã‚¯ãŠã‚ˆã³ + + M*Q=eye. + + + + + + flag='c'ã®å ´åˆ, + + M*A*Qã®å›ºæœ‰å€¤ = 実部>= thres + + ã®Aã®å›ºæœ‰å€¤ã¨ãªã‚Šã¾ã™. + + + + + + flag='d'ã®å ´åˆ, + + M*A*Qã®å›ºæœ‰å€¤ = 大ãã•>= thresã® + + Aã®å›ºæœ‰å€¤ã¨ãªã‚Šã¾ã™. + + + + + + flag='c' ã®å ´åˆ,ãã—㦠+ + [Q1,M1] = eye()-Q*Mã® + + フルランク分解 (fullrf)ã®å ´åˆ, + + M1*A*Q1ã®å›ºæœ‰å€¤ = + + 実部 < thresã®Aã®å›ºæœ‰å€¤ã¨ãªã‚Šã¾ã™. + + + + + + flag='d'ã®å ´åˆ,ãã—㦠[Q1,M1] = + + eye()-Q*Mã®ãƒ•ãƒ«ãƒ©ãƒ³ã‚¯åˆ†è§£ (fullrf)ã®å ´åˆ, + + M1*A*Q1ã®å›ºæœ‰å€¤ =大ãã• <thresã® + + Aã®å›ºæœ‰å€¤ã¨ãªã‚Šã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + psmall + + + + + + projspec + + + + + + fullrf + + + + + + schur + + + + + + + + + + 使用ã•ã‚Œã‚‹é–¢æ•° + + + + pbig 㯠+ + ソートã•ã‚ŒãŸ Schur å½¢å¼ã«åŸºã¥ã„ã¦ã„ã¾ã™ + + (Scilab関数 schur). + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/eigen/projspec.xml b/modules/linear_algebra/help/ja_JP/eigen/projspec.xml new file mode 100755 index 000000000..351331f8e --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/eigen/projspec.xml @@ -0,0 +1,165 @@ + + + + + + + + + projspec + + ã‚¹ãƒšã‚¯ãƒˆãƒ«æ¼”ç®—å­ + + + + + + 呼ã³å‡ºã—手順 + + [S,P,D,i]=projspec(A) + + + + + + 引数 + + + + + + A + + + + 正方行列 + + + + + + + + S, P, D + + + + s正方行列 + + + + + + + + i + + + + + + æ•´æ•° (Aã®ã‚¼ãƒ­å›ºæœ‰å€¤ã®æ·»å­—). + + + + + + + + + + + + + + 説明 + + + + Aã®0ã«ãŠã‘るスペクトル特性. + + + + + + S = 0ã«ãŠã‘る縮å°ãƒ¬ã‚¾ãƒ«ãƒ™ãƒ³ãƒˆ + + (S = -Drazin_inverse(A)). + + + + + + P = 0ã«ãŠã‘るスペクトル投影. + + + + + + D = 0ã«ãŠã‘る冪零演算å­. + + + + + + index = 0固有値ã®æ·»å­—. + + + + + + 特異点s=0ã®å‘¨ã‚Šã§ã® + + (s*eye()-A)^(-1) = D^(i-1)/s^i +... + D/s^2 + P/s - S - s*S^2 -... + + ãŒå‡ºåŠ›ã•ã‚Œã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + coff + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/eigen/psmall.xml b/modules/linear_algebra/help/ja_JP/eigen/psmall.xml new file mode 100755 index 000000000..e67b13bdf --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/eigen/psmall.xml @@ -0,0 +1,232 @@ + + + + + + + + + psmall + + スペクトル投影 + + + + + + 呼ã³å‡ºã—手順 + + [Q,M]=psmall(A,thres,flag) + + + + + + 引数 + + + + + + A + + + + 実数ã®æ­£æ–¹è¡Œåˆ— + + + + + + + + thres + + + + 実数 + + + + + + + + flag + + + + + + 文字列 ('c' ã¾ãŸã¯ 'd') + + + + + + + + + + Q,M + + + + 実数行列 + + + + + + + + + + + + 説明 + + + + 実部 < thres (flag='c') + + ã¾ãŸã¯çµ¶å¯¾å€¤ < thres(flag='d') + + ã¨ãªã‚‹å›ºæœ‰å€¤-部分空間ã¸ã®æŠ•å½±. + + + + + + ã“ã®æŠ•å½±ã¯Q*Mã«ã‚ˆã‚Šå®šç¾©ã•ã‚Œã¾ã™. + + ã“ã“ã§, + + Qã¯åˆ—フルランク,Mã¯è¡Œãƒ•ãƒ«ãƒ©ãƒ³ã‚¯, + + ãã—ã¦M*Q=eyeã§ã™. + + + + + + flag='c'ã®å ´åˆ, + + M*A*Qã®å›ºæœ‰å€¤ = + + 実部< thresã®Aã®å›ºæœ‰å€¤. + + + + + + flag='d'ã®å ´åˆ, + + M*A*Qã®å›ºæœ‰å€¤ = + + 大ãã• < thresã®Aã®å›ºæœ‰å€¤. + + + + + + flag='c'ã®å ´åˆ, + + [Q1,M1] = eye()-Q*Mã® + + フルランク分解(fullrf)ã®å ´åˆ, + + M1*A*Q1ã®å›ºæœ‰å€¤ =実部>= + + thresã® + + Aã®å›ºæœ‰å€¤ã¨ãªã‚Šã¾ã™. + + + + + + flag='d'ã®å ´åˆ, + + [Q1,M1] =eye()-Q*M + + ã®ãƒ•ãƒ«ãƒ©ãƒ³ã‚¯åˆ†è§£(fullrf)ã®å ´åˆ, + + M1*A*Q1ã®å›ºæœ‰å€¤ = + + 大ãã•>=thresã® + + Aã®å›ºæœ‰å€¤. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + pbig + + + + + + proj + + + + + + projspec + + + + + + + + + + 使用ã•ã‚Œã‚‹é–¢æ•° + + + + ã“ã®é–¢æ•°ã¯ã‚½ãƒ¼ãƒˆã•ã‚ŒãŸ Schurå½¢å¼(scilab + + 関数 schur)ã«åŸºã¥ã„ã¦ã„ã¾ã™. + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/eigen/schur.xml b/modules/linear_algebra/help/ja_JP/eigen/schur.xml new file mode 100755 index 000000000..be68d6eed --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/eigen/schur.xml @@ -0,0 +1,711 @@ + + + + + + + + + schur + + 行列ãŠã‚ˆã³ãƒšãƒ³ã‚·ãƒ«ã®[ソートã•ã‚ŒãŸ] Schur 分解 + + + + + + 呼ã³å‡ºã—手順 + + [U,T] = schur(A) + + [U,dim [,T] ]=schur(A,flag) + + [U,dim [,T] ]=schur(A,extern1) + + + + [As,Es [,Q,Z]]=schur(A,E) + + [As,Es [,Q],Z,dim] = schur(A,E,flag) + + [Z,dim] = schur(A,E,flag) + + [As,Es [,Q],Z,dim]= schur(A,E,extern2) + + [Z,dim]= schur(A,E,extern2) + + + + + + + + 引数 + + + + + + A + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®æ­£æ–¹è¡Œåˆ—. + + + + + + + + E + + + + + + Aã¨åŒã˜æ¬¡å…ƒã®å®Ÿæ•°ã¾ãŸã¯è¤‡ç´ æ•°ã®æ­£æ–¹è¡Œåˆ—. + + + + + + + + + + flag + + + + + + 文字列 ('c' ã¾ãŸã¯ 'd') + + + + + + + + + + extern1 + + + + an ``external'', 以下ã®å‚ç…§ + + + + + + + + extern2 + + + + an ``external'', 以下ã®å‚ç…§ + + + + + + + + U + + + + 直交ã¾ãŸã¯ãƒ¦ãƒ‹ã‚¿ãƒªæ­£æ–¹è¡Œåˆ— + + + + + + + + Q + + + + 直交ã¾ãŸã¯ãƒ¦ãƒ‹ã‚¿ãƒªæ­£æ–¹è¡Œåˆ— + + + + + + + + Z + + + + o直交ã¾ãŸã¯ãƒ¦ãƒ‹ã‚¿ãƒªæ­£æ–¹è¡Œåˆ— + + + + + + + + T + + + + 上三角ã¾ãŸã¯æº–三角正方行列 + + + + + + + + As + + + + 上三角ã¾ãŸã¯æº–三角正方行列 + + + + + + + + Es + + + + 上三角正方行列 + + + + + + + + dim + + + + æ•´æ•° + + + + + + + + + + + + 説明 + + + + Schur å½¢å¼, 行列ãŠã‚ˆã³ãƒšãƒ³ã‚·ãƒ«ã®ã‚½ãƒ¼ãƒˆã•ã‚ŒãŸ Schur å½¢å¼ + + + + + + + + 行列Schurå½¢å¼ + + + + + + + + 通常ã®Schurå½¢å¼: + + + + + + [U,T] = schur(A) ã¯, + + A = U*T*U' ãŠã‚ˆã³ U'*U = eye(U)ã¨ãªã‚‹ã‚ˆã†ãª + + Schur行列T ãŠã‚ˆã³ãƒ¦ãƒ‹ã‚¿ãƒªè¡Œåˆ— U + + を出力ã—ã¾ã™. + + Schur(A)ã¯,Tã‚’è¿”ã—ã¾ã™. + + A ãŒè¤‡ç´ æ•°ã®å ´åˆ, 複素Schurå½¢å¼ã¯,行列Tã«è¿”ã—ã¾ã™. + + 複素Schurå½¢å¼ã¯,Aã®å›ºæœ‰å€¤ã‚’対角項ã«æœ‰ã™ã‚‹ä¸Šä¸‰è§’行列ã§ã™. + + A ãŒå®Ÿæ•°ã®å ´åˆ, 実数Schurå½¢å¼ãŒè¿”ã•ã‚Œã¾ã™. + + 実数Schurå½¢å¼ã¯,対角項ã«å®Ÿæ•°å›ºæœ‰å€¤ã€è¤‡ç´ æ•°å›ºæœ‰å€¤ã‚’対角項ã®2x2ブロック㫠+ + 有ã—ã¾ã™. + + + + + + + + + + ソートã•ã‚ŒãŸSchurå½¢å¼ + + + + + + [U,dim]=schur(A,'c') ã¯, + + Aã‚’ Schur å½¢å¼ã«å¤‰æ›ã™ã‚‹ + + ユニタリ行列 U ã‚’è¿”ã—ã¾ã™. + + æ›´ã«,Uã®æœ€åˆã®åˆ— dim ã¯, + + 実部ãŒè² ã®å›ºæœ‰å€¤(安定ãª"連続時間"固有値空間) + + ã«é–¢é€£ã™ã‚‹Aã®å›ºæœ‰å€¤ç©ºé–“ + + ã®åŸºåº•ã‚’構æˆã—ã¾ã™. + + + + + + [U,dim]=schur(A,'d') ã¯, + + Aã‚’ Schur å½¢å¼ã«å¤‰æ›ã™ã‚‹ + + ユニタリ行列 U ã‚’è¿”ã—ã¾ã™. + + æ›´ã«,Uã®æœ€åˆã®åˆ— dim ã¯, + + 大ãã•ãŒ1未満ã®å›ºæœ‰å€¤(安定ãª"離散時間"固有値空間) + + ã«é–¢é€£ã™ã‚‹Aã®å›ºæœ‰å€¤ç©ºé–“ + + ã®åŸºåº•ã‚’構æˆã—ã¾ã™. + + + + + + [U,dim]=schur(A,extern1) ã¯, + + Aã‚’ Schur å½¢å¼ã«å¤‰æ›ã™ã‚‹ + + ユニタリ行列Uã‚’è¿”ã—ã¾ã™. + + æ›´ã«,Uã®æœ€åˆã®åˆ— dim ã¯, + + 外部関数 extern1 (詳細㯠external å‚ç…§) + + ã«ã‚ˆã‚Šé¸æŠžã•ã‚ŒãŸå›ºæœ‰å€¤ã«é–¢é€£ã™ã‚‹Aã®å›ºæœ‰å€¤ç©ºé–“ + + ã®åŸºåº•ã‚’構æˆã—ã¾ã™. + + ã“ã® external ã¯Scilab関数ã¾ãŸã¯Cã¾ãŸã¯Fortranプロシージャã«ã‚ˆã‚Š + + 次ã®ã‚ˆã†ã«è¨˜è¿°ã™ã‚‹ã“ã¨ãŒã§ãã¾ã™: + + + + + + + + Scilab関数 + + + + + + extern1㌠+ + Scilab関数ã«ã‚ˆã‚Šè¨˜è¿°ã•ã‚Œã‚‹å ´åˆ, + + 以下ã®å‘¼ã³å‡ºã—手順を有ã™ã‚‹å¿…è¦ãŒã‚ã‚Šã¾ã™: + + s=extern1(Ev), ãŸã ã— Ev ã¯å›ºæœ‰å€¤, + + s ã¯è«–ç†å€¤ã§ã™. + + + + + + + + + + C ã¾ãŸã¯ Fortran プロシージャ + + + + + + extern1 ãŒCã¾ãŸã¯Fortran関数ã«ã‚ˆã‚Š + + 記述ã•ã‚Œã‚‹å ´åˆ,以下ã®å‘¼ã³å‡ºã—手順を有ã™ã‚‹å¿…è¦ãŒã‚ã‚Šã¾ã™: + + int extern1(double *EvR, double *EvI) + + ãŸã ã— EvR ãŠã‚ˆã³ EvI 㯠+ + 固有値ã®å®Ÿéƒ¨ãŠã‚ˆã³è™šéƒ¨ã§ã™. + + trueã¾ãŸã¯ã‚¼ãƒ­ã§ãªã„戻り値ã¯,é¸æŠžã•ã‚ŒãŸå›ºæœ‰å€¤ã‚’æ„味ã—ã¾ã™. + + + + + + + + + + + + + + + + + + + + + + ペンシルSchurå½¢å¼ + + + + + + + + 通常ã®ãƒšãƒ³ã‚·ãƒ«Schurå½¢å¼ + + + + + + [As,Es] = schur(A,E) ã¯, + + 対A, Eã®ä¸€èˆ¬åŒ–Schurå½¢å¼ã§ã‚ã‚‹ + + 準三角行列As行列ãŠã‚ˆã³ä¸‰è§’行列Es + + を出力ã—ã¾ã™. + + + + + + [As,Es,Q,Z] = schur(A,E)ã¯,æ›´ã« + + As=Q'*A*Z ãŠã‚ˆã³ Es=Q'*E*Zã¨ãªã‚‹ã‚ˆã†ãª + + 2ã¤ã®ãƒ¦ãƒ‹ã‚¿ãƒªè¡Œåˆ—Q ãŠã‚ˆã³ Zã‚’è¿”ã—ã¾ã™. + + + + + + + + + + ソートã•ã‚ŒãŸSchurå½¢å¼: + + + + + + [As,Es,Z,dim] = schur(A,E,'c')ã¯, + + ペンシルs*E-Aã®å®Ÿæ•°ä¸€èˆ¬åŒ–Schurå½¢å¼ã‚’è¿”ã—ã¾ã™. + + æ›´ã«, Zã®æœ€åˆã®åˆ— dim ã¯, + + 実部ãŒè² ã®å›ºæœ‰å€¤ (安定ãª"連続時間"一般化固有値空間)ã«é–¢é€£ã™ã‚‹ + + 固有値空間ã®åŸºåº•ã‚’構æˆã—ã¾ã™. + + + + + + [As,Es,Z,dim] = schur(A,E,'d') + + + + + + ã¯,ペンシルs*E-Aã® + + 実数一般化Schurå½¢å¼ã‚’è¿”ã—ã¾ã™. + + æ›´ã«, Zã®æœ€åˆã®åˆ— dim ã¯, + + 大ãã•1未満ã®å›ºæœ‰å€¤ (安定ãª"離散時間"一般化固有値空間)ã«é–¢é€£ã™ã‚‹ + + 固有値空間ã®åŸºåº•ã‚’構æˆã—ã¾ã™. + + + + + + [As,Es,Z,dim] = schur(A,E,extern2) + + + + + + ã¯,ペンシルs*E-Aã®å®Ÿä¸€èˆ¬åŒ–Schurå½¢å¼ã‚’è¿”ã—ã¾ã™. + + æ›´ã«, Zã®æœ€åˆã®åˆ— dim ã¯, + + 関数extern2ã«ã‚ˆã‚ŠæŒ‡å®šã•ã‚ŒãŸè¦å‰‡ã«åŸºã¥ãé¸æŠžã•ã‚ŒãŸ + + ペンシルã®å›ºæœ‰å€¤ã«é–¢ã™ã‚‹å›ºæœ‰å€¤ç©ºé–“ã®åŸºåº•ã‚’構æˆã—ã¾ã™. + + (詳細㯠external å‚ç…§) + + ã“ã® external 㯠Scilab 関数ã¾ãŸã¯Cã¾ãŸã¯Fortranプロシージャ + + ã«ã‚ˆã‚Šæ¬¡ã®ã‚ˆã†ã«è¨˜è¿°ã™ã‚‹ã“ã¨ãŒã§ãã¾ã™: + + + + + + + + Scilab関数 + + + + + + extern2ãŒScilab関数ã«ã‚ˆã‚Šè¨˜è¿°ã•ã‚Œã‚‹å ´åˆ, + + 以下ã®å‘¼ã³å‡ºã—手順を有ã™ã‚‹å¿…è¦ãŒã‚ã‚Šã¾ã™: + + s=extern2(Alpha,Beta), ãŸã ã— Alpha ãŠã‚ˆã³ + + Beta ã¯ä¸€èˆ¬åŒ–固有値ãŠã‚ˆã³è«–ç†å€¤ s + + を定義ã—ã¾ã™. + + + + + + + + + + C ã¾ãŸã¯Fortranプロシージャ + + + + + + if external extern2 ãŒCã¾ãŸã¯Fortran関数ã«ã‚ˆã‚Š + + 記述ã•ã‚Œã‚‹å ´åˆ,以下ã®å‘¼ã³å‡ºã—手順を有ã™ã‚‹å¿…è¦ãŒã‚ã‚Šã¾ã™: + + + + + + int extern2(double *AlphaR, double *AlphaI, double *Beta) + + + + + + : A ãŠã‚ˆã³ E ãŒå®Ÿæ•°ã®å ´åˆ. + + + + + + int extern2(double *AlphaR, double *AlphaI, double *BetaR, double *BetaI) + + + + + + : A ãŠã‚ˆã³ E ãŒè¤‡ç´ æ•°ã®å ´åˆ. + + Alpha, ãŠã‚ˆã³ Beta ã¯ä¸€èˆ¬åŒ–固有値を定義ã—ã¾ã™. + + trueã¾ãŸã¯éžã‚¼ãƒ­ã®æˆ»ã‚Šå€¤ã¯,é¸æŠžã•ã‚ŒãŸä¸€èˆ¬åŒ–固有値をæ„味ã—ã¾ã™. + + + + + + + + + + + + + + + + + + + + + + + + + + å‚考 + + + + 行列Schurå½¢å¼ã®è¨ˆç®—ã¯Lapackルーãƒãƒ³DGEES ãŠã‚ˆã³ ZGEESã«åŸºã¥ã„ã¦ã„ã¾ã™. + + + + + + ペンシルSchurå½¢å¼ã®è¨ˆç®—ã¯Lapackルーãƒãƒ³DGGES ãŠã‚ˆã³ ZGGESã«åŸºã¥ã„ã¦ã„ã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + spec + + + + + + bdiag + + + + + + ricc + + + + + + pbig + + + + + + psmall + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/eigen/spec.xml b/modules/linear_algebra/help/ja_JP/eigen/spec.xml new file mode 100755 index 000000000..ad29f187a --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/eigen/spec.xml @@ -0,0 +1,522 @@ + + + + + + + + + spec + + 行列ã¨ãƒšãƒ³ã‚·ãƒ«ã®å›ºæœ‰å€¤ + + + + + + 呼ã³å‡ºã—手順 + + evals=spec(A) + + [R,diagevals]=spec(A) + + + + evals=spec(A,B) + + [alpha,beta]=spec(A,B) + + [alpha,beta,Z]=spec(A,B) + + [alpha,beta,Q,Z]=spec(A,B) + + + + + + + + 引数 + + + + + + A + + + + 実数ã¾ãŸã¯è¤‡ç´ æ­£æ–¹è¡Œåˆ— + + + + + + + + B + + + + + + Aã¨åŒã˜æ¬¡å…ƒã®å®Ÿæ•°ã¾ãŸã¯è¤‡ç´ æ­£æ–¹è¡Œåˆ— + + + + + + + + + + evals + + + + 実数ã¾ãŸã¯è¤‡ç´ ãƒ™ã‚¯ãƒˆãƒ«, 固有値 + + + + + + + + diagevals + + + + 実数ã¾ãŸã¯(対角項ã«å›ºæœ‰å€¤ã‚’有ã™ã‚‹)複素対角行列 + + + + + + + + alpha + + + + 実数ã¾ãŸã¯è¤‡ç´ ãƒ™ã‚¯ãƒˆãƒ«, al./be ã«ã‚ˆã‚Šå›ºæœ‰å€¤ãŒå¾—られã¾ã™ + + + + + + + + beta + + + + 実数ベクトル, al./be ã«ã‚ˆã‚Šå›ºæœ‰å€¤ãŒå¾—られã¾ã™ + + + + + + + + R + + + + å¯é€†ãªå®Ÿæ•°ã¾ãŸã¯è¤‡ç´ æ­£æ–¹è¡Œåˆ—, 行列å³å›ºæœ‰ãƒ™ã‚¯ãƒˆãƒ«. + + + + + + + + L + + + + å¯é€†ãªå®Ÿæ•°ã¾ãŸã¯è¤‡ç´ æ­£æ–¹è¡Œåˆ—,　ペンシル左固有ベクトル. + + + + + + + + R + + + + å¯é€†ãªå®Ÿæ•°ã¾ãŸã¯è¤‡ç´ æ­£æ–¹è¡Œåˆ—,　ペンシルå³å›ºæœ‰ãƒ™ã‚¯ãƒˆãƒ«. + + + + + + + + + + + + 説明 + + + + + + evals=spec(A) + + + + + + ベクトルevals ã«å›ºæœ‰å€¤ã‚’è¿”ã—ã¾ã™. + + + + + + + + + + [R,diagevals] =spec(A) + + + + + + 対角行列r evals ã«å›ºæœ‰å€¤, + + Rã«å›ºæœ‰ãƒ™ã‚¯ãƒˆãƒ«ã‚’è¿”ã—ã¾ã™. + + + + + + + + + + evals=spec(A,B) + + + + 行列ペンシル A - s B ã®ã‚¹ãƒšã‚¯ãƒˆãƒ«,ã™ãªã‚ã¡, + + 多項å¼è¡Œåˆ— s B - Aã®æ ¹,ã‚’è¿”ã—ã¾ã™. + + + + + + + + + + [alpha,beta] = spec(A,B) + + + + + + 行列ペンシルA- s Bã®ã‚¹ãƒšã‚¯ãƒˆãƒ«, + + ã™ãªã‚ã¡,多項å¼è¡Œåˆ— A - s Bã®æ ¹ã‚’è¿”ã—ã¾ã™. + + 一般化固有値 alpha 㨠beta ã¯è¡Œåˆ— + + A - alpha./beta B ãŒç‰¹ç•°è¡Œåˆ—ã¨ãªã‚‹å€¤ã§ã™. + + 固有値㯠al./be ã«ã‚ˆã‚ŠæŒ‡å®šã•ã‚Œ, + + beta(i) = 0ã®å ´åˆ,i番目ã®å›ºæœ‰å€¤ã¯ç„¡é™å¤§ã¨ãªã‚Šã¾ã™. + + (B = eye(A)ã®å ´åˆ, alpha./beta㯠+ + spec(A)ã¨ãªã‚Šã¾ã™). + + 通常,beta=0や両方ãŒã‚¼ãƒ­ã®å ´åˆã«é–¢ã—ã¦éƒ½åˆãŒè‰¯ã„解釈ãŒå­˜åœ¨ã™ã‚‹ãŸã‚, + + (alpha,beta)ã®çµ„ã¿åˆã‚ã›ã§è¡¨ã•ã‚Œã¾ã™. + + + + + + + + + + [alpha,beta,R] = spec(A,B) + + + + 上記ã«åŠ ãˆã¦ãƒšãƒ³ã‚·ãƒ«ã®ä¸€èˆ¬åŒ–å³å›ºæœ‰ãƒ™ã‚¯ãƒˆãƒ«ã¨ãªã‚‹ + + 行列 Rã‚’è¿”ã—ã¾ã™. + + + + + + + + + + [al,be,L,R] = spec(A,B) + + + + + + 上記ã«åŠ ãˆã¦ãƒšãƒ³ã‚·ãƒ«ã®ä¸€èˆ¬åŒ–å³ãŠã‚ˆã³å·¦å›ºæœ‰ãƒ™ã‚¯ãƒˆãƒ«ã§ã‚る行列 + + L ãŠã‚ˆã³Rã‚’è¿”ã—ã¾ã™. + + + + + + + + + + [al,be,Z] = spec(A,E) + + + + + + 一般化å³å›ºæœ‰ãƒ™ã‚¯ãƒˆãƒ«ã§ã‚る行列 Z ã‚’è¿”ã—ã¾ã™. + + + + + + + + + + [al,be,Q,Z] = spec(A,E) + + + + + + 一般化å³ãŠã‚ˆã³å·¦å›ºæœ‰ãƒ™ã‚¯ãƒˆãƒ«ã§ã‚る行列 Q + + ãŠã‚ˆã³ Zã‚’è¿”ã—ã¾ã™. + + + + + + + + + + 大ããªå®Œå…¨ / 疎行列ã®å ´åˆ, Arnoldi モジュールを使用ã™ã‚‹ã“ã¨ãŒã§ãã¾ã™. + + + + + + å‚ç…§ + + 行列ã®å›ºæœ‰å€¤è¨ˆç®—㯠Lapack ルーãƒãƒ³ã«åŸºã¥ã„ã¦ã„ã¾ã™ + + + + + + 行列ãŒå¯¾ç§°ã§ãªã„å ´åˆ, DGEEV ãŠã‚ˆã³ ZGEEV. + + + + + + 行列ãŒå¯¾ç§°ã®å ´åˆ, DSYEV ãŠã‚ˆã³ ZHEEV. + + + + + + 複素対象行列ã¯è¤‡ç´ å…±å½¹ã®éžå¯¾è§’é …ã¨å®Ÿæ•°ã®å¯¾è§’項を有ã—ã¾ã™. + + ペンシル固有値計算㯠Lapack ルーãƒãƒ³ + + DGGEV ãŠã‚ˆã³ ZGGEVã«åŸºã¥ã„ã¦ã„ã¾ã™. + + + + + + + + 実数ãŠã‚ˆã³è¤‡ç´ è¡Œåˆ— + + + + 例ãˆã° evals ã‚„ R ã®ã‚ˆã†ãªå‡ºåŠ›å¤‰æ•°ã®åž‹ã¯å…¥åŠ›è¡Œåˆ— A ãŠã‚ˆã³ B ã®åž‹ã¨ + + åŒã˜ã§ã‚ã‚‹å¿…è¦ã¯ãªã„ã“ã¨ã«æ³¨æ„ã—ã¦ãã ã•ã„. + + 以下ã®ãƒ‘ラグラフã§ã¯ã€è¡Œåˆ— A ã®å›ºæœ‰å€¤ãŠã‚ˆã³å›ºæœ‰ãƒ™ã‚¯ãƒˆãƒ«ã‚’ + + 計算ã™ã‚‹éš›ã®å‡ºåŠ›å¤‰æ•°ã®åž‹ã‚’解æžã—ã¾ã™. + + + + + + + + 実数 A 行列 + + + + + + 対称 + + 固有値ã¨å›ºæœ‰ãƒ™ã‚¯ãƒˆãƒ«ã¯å®Ÿæ•°. + + + + + + éžå¯¾ç§° + + 固有値ã¨å›ºæœ‰ãƒ™ã‚¯ãƒˆãƒ«ã¯è¤‡ç´ æ•°. + + + + + + + + + + 複素 A 行列 + + + + + + 対称 + + 固有値ã¯å®Ÿæ•°ã ãŒå›ºæœ‰ãƒ™ã‚¯ãƒˆãƒ«ã¯è¤‡ç´ æ•°. + + + + + + éžå¯¾ç§° + + 固有値,固有ベクトルã¯è¤‡ç´ æ•°. + + + + + + + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + poly + + + + + + det + + + + + + schur + + + + + + bdiag + + + + + + colcomp + + + + + + dsaupd + + + + + + dnaupd + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/eigen/sva.xml b/modules/linear_algebra/help/ja_JP/eigen/sva.xml new file mode 100755 index 000000000..64e9f801f --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/eigen/sva.xml @@ -0,0 +1,155 @@ + + + + + + + + + sva + + 特異値近似 + + + + + + 呼ã³å‡ºã—手順 + + [U,s,V]=sva(A,k) + + [U,s,V]=sva(A,tol) + + + + + + + + 引数 + + + + + + A + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ— + + + + + + + + k + + + + æ•´æ•° + + + + + + + + tol + + + + éžè² ã®å®Ÿæ•° + + + + + + + + + + + + 説明 + + + + 特異値近似. + + + + + + kã‚’>=1ã®æ•´æ•°ã¨ã™ã‚‹ã¨ã, + + [U,S,V]=sva(A,k) ã¯, + + rank(B)=kã¨ã—㦠+ + B=U*S*V'ãŒAã®æœ€è‰¯ã®L2è¿‘ä¼¼ã¨ãªã‚‹ + + よã†ãª + + U,S ãŠã‚ˆã³Vã‚’è¿”ã—ã¾ã™. + + + + + + 実数tolを指定ã—ãŸ[U,S,V]=sva(A,tol)ã¯, + + A-Bã®L2ノルムã§ã‚ã‚‹B=U*S*V'ã® + + 最大値ãŒtolã¨ãªã‚‹ã‚ˆã†ãª + + U,S ãŠã‚ˆã³ V ã‚’è¿”ã—ã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + svd + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/eigen/svd.xml b/modules/linear_algebra/help/ja_JP/eigen/svd.xml new file mode 100755 index 000000000..d5e536a04 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/eigen/svd.xml @@ -0,0 +1,252 @@ + + + + + + + + + svd + + 特異値分解 + + + + + + 呼ã³å‡ºã—手順 + + s=svd(X) + + [U,S,V]=svd(X) + + [U,S,V]=svd(X,0) (obsolete) + + [U,S,V]=svd(X,"e") + + [U,S,V,rk]=svd(X [,tol]) + + + + + + + + 引数 + + + + + + X + + + + 実数ã¾ãŸã¯è¤‡ç´ è¡Œåˆ— + + + + + + + + s + + + + 実数ベクトル (特異値) + + + + + + + + S + + + + 実数対角行列 (特異値) + + + + + + + + U,V + + + + 直交ã¾ãŸã¯ãƒ¦ãƒ‹ã‚¿ãƒªæ­£æ–¹è¡Œåˆ—(特異値). + + + + + + + + tol + + + + 実数 + + + + + + + + + + + + 説明 + + + + [U,S,V] = svd(X) 㯠+ + X ã¨åŒæ¬¡å…ƒã§ + + é™é †ã«éžè² ã®å¯¾è§’è¦ç´ ã‚’有ã™ã‚‹ + + 対角行列 SãŠã‚ˆã³ + + X = U*S*V'ã¨ãªã‚‹ + + ユニタリ行列 U 㨠V + + を出力ã—ã¾ã™. + + + + + + [U,S,V] = svd(X,0) 㯠+ + "エコノミーサイズ"分解を出力ã—ã¾ã™. + + X ãŒmè¡Œn列 (m > n)ã®å ´åˆ, + + U ã®æœ€åˆã®n列ã®ã¿ãŒè¨ˆç®—ã•ã‚Œ, + + S㯠nè¡Œn列ã¨ãªã‚Šã¾ã™. + + + + + + s= svd(X) 㯠+ + 特異値をå«ã‚€ãƒ™ã‚¯ãƒˆãƒ«sã‚’è¿”ã—ã¾ã™. + + + + + + [U,S,V,rk]=svd(X,tol) 㯠+ + rkã«åŠ ãˆã¦, + + X ã®æ•°å€¤ãƒ©ãƒ³ã‚¯,ã™ãªã‚ã¡ + + tolより大ããªç‰¹ç•°å€¤ã®æ•°ã‚’出力ã—ã¾ã™. + + + + + + tolã®ãƒ‡ãƒ•ã‚©ãƒ«ãƒˆå€¤ã¯ + + rankã¨åŒã˜ã§ã™. + + + + + + + + 例 + + + + + + + + å‚考 + + + + + + rank + + + + + + qr + + + + + + colcomp + + + + + + rowcomp + + + + + + sva + + + + + + spec + + + + + + + + + + 使用ã•ã‚Œã‚‹é–¢æ•° + + + + svd 分解ã¯Lapackã®ãƒ«ãƒ¼ãƒãƒ³ DGESVD (実数行列ã®å ´åˆ)ãŠã‚ˆã³ + + ZGESVD (複素数ã®å ´åˆ)ã«åŸºã¥ã„ã¦ã„ã‚‹. + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/factorization/CHAPTER b/modules/linear_algebra/help/ja_JP/factorization/CHAPTER new file mode 100755 index 000000000..e6daeb8eb --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/factorization/CHAPTER @@ -0,0 +1,2 @@ +title = Factorization + diff --git a/modules/linear_algebra/help/ja_JP/factorization/givens.xml b/modules/linear_algebra/help/ja_JP/factorization/givens.xml new file mode 100755 index 000000000..7eae55a1e --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/factorization/givens.xml @@ -0,0 +1,162 @@ + + + + + + + + + givens + + ã‚®ãƒ–ãƒ³ã‚¹å¤‰æ› + + + + + + 呼ã³å‡ºã—手順 + + U=givens(xy) + + U=givens(x,y) + + [U,c]=givens(xy) + + [U,c]=givens(x,y) + + + + + + + + 引数 + + + + + + x,y + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•° + + + + + + + + xy + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®è¦ç´ æ•°2ã®åˆ—ベクトル + + + + + + + + U + + + + 2x2 ユニタリ行列 + + + + + + + + c + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®è¦ç´ æ•°2ã®åˆ—ベクトル + + + + + + + + + + + + 説明xy = [x;y]ã¨ã—㦠+ + + + U= givens(x, y) ã¾ãŸã¯ U = givens(xy) + + ã¯,次ã®ã‚ˆã†ãª2x2 ã® + + ユニタリ行列 U ã‚’è¿”ã—ã¾ã™: + + + + + + U*xy=[r;0]=c. + + + + + + + + givens(x,y) ãŠã‚ˆã³ givens([x;y]) ã¯ç­‰ä¾¡ã§ã‚ã‚‹ã“ã¨ã« + + 注æ„ã—ã¦ãã ã•ã„. + + + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + qr + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/factorization/householder.xml b/modules/linear_algebra/help/ja_JP/factorization/householder.xml new file mode 100755 index 000000000..9643f73ee --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/factorization/householder.xml @@ -0,0 +1,140 @@ + + + + + + + + + householder + + ãƒã‚¦ã‚¹ãƒ›ãƒ«ãƒ€ãƒ¼ç›´äº¤é¡æ˜ è¡Œåˆ— + + + + + + 呼ã³å‡ºã—手順 + + u=householder(v [,w]) + + + + + + 引数 + + + + + + v + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®åˆ—ベクトル + + + + + + + + w + + + + + + vã¨åŒã˜å¤§ãã•ã®å®Ÿæ•°ã¾ãŸã¯è¤‡ç´ æ•°ã®åˆ—ベクトル. + + デフォルト値ã¯eye(v) + + + + + + + + + + u + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®åˆ—ベクトル + + + + + + + + + + + + 説明 + + + + åŒã˜å¤§ãã•ã®åˆ—ベクトル + + v, w を指定ã™ã‚‹ã¨, + + householder(v,w) ã¯, + + (eye()-2*u*u')*vãŒwã«æ¯”例ã™ã‚‹ã‚ˆã†ãª + + ユニタリ列ベクトルuã‚’è¿”ã—ã¾ã™. + + (eye()-2*u*u') ã¯ãƒã‚¦ã‚¹ãƒ›ãƒ«ãƒ€ãƒ¼ç›´äº¤é¡æ˜ è¡Œåˆ—ã§ã™. + + + + + + w ã®ãƒ‡ãƒ•ã‚©ãƒ«ãƒˆå€¤ã¯ eye(v)ã§ã™. + + ã“ã®å ´åˆ,ベクトル (eye()-2*u*u')*v ã¯ãƒ™ã‚¯ãƒˆãƒ« + + eye(v)*norm(v)ã§ã™. + + + + + + + + å‚ç…§ + + + + + + qr + + + + + + givens + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/factorization/sqroot.xml b/modules/linear_algebra/help/ja_JP/factorization/sqroot.xml new file mode 100755 index 000000000..0498d89f5 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/factorization/sqroot.xml @@ -0,0 +1,105 @@ + + + + + + + + + sqroot + + W*W' エルミート分解 + + + + + + 呼ã³å‡ºã—手順 + + sqroot(X) + + + + + + 引数 + + + + + + X + + + + 対称éžè² å®šå®Ÿã¾ãŸã¯è¤‡ç´ è¡Œåˆ— + + + + + + + + + + + + 説明 + + + + X=W*W' ã¨ãªã‚‹ã‚ˆã†ãªWã‚’è¿”ã—ã¾ã™(SVDを使用). + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + chol + + + + + + svd + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/kernel/CHAPTER b/modules/linear_algebra/help/ja_JP/kernel/CHAPTER new file mode 100755 index 000000000..be67920e1 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/kernel/CHAPTER @@ -0,0 +1,2 @@ +title = Kernel + diff --git a/modules/linear_algebra/help/ja_JP/kernel/colcomp.xml b/modules/linear_algebra/help/ja_JP/kernel/colcomp.xml new file mode 100755 index 000000000..31a6a6ba2 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/kernel/colcomp.xml @@ -0,0 +1,206 @@ + + + + + + + + + colcomp + + 列圧縮,カーãƒãƒ«,ヌル空間 + + + + + + 呼ã³å‡ºã—手順 + + [W,rk]=colcomp(A [,flag] [,tol]) + + + + + + 引数 + + + + + + A + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ— + + + + + + + + flag + + + + 文字列 + + + + + + + + tol + + + + 実数 + + + + + + + + W + + + + 正方正則行列 (基底変æ›) + + + + + + + + rk + + + + + + æ•´æ•° (Aã®ãƒ©ãƒ³ã‚¯) + + + + + + + + + + + + + + 説明 + + + + Aã®åˆ—圧縮: Ac = A*W 㯠+ + 列圧縮,ã™ãªã‚ã¡ Ac=[0,Af] ã¨ãªã‚Šã¾ã™. + + + + + + ãŸã ã—, Af ã¯ãƒ•ãƒ«åˆ—ランクを有ã—ã¾ã™: + + rank(Af) = rank(A) = rk. + + + + + + flag ãŠã‚ˆã³ tol 㯠+ + オプションã®ãƒ‘ラメータ: flag = 'qr' + + ã¾ãŸã¯ 'svd' (デフォルト㯠+ + 'svd')ã§ã™. + + + + + + tol = 許容誤差パラメータ (デフォルト値㯠+ + %epsã®ã‚ªãƒ¼ãƒ€ãƒ¼). + + + + + + Wã®æœ€åˆã®ma-rk列ã¯, + + size(A)=(na,ma)ã¨ã™ã‚‹ã¨ã, + + Aã®ã‚«ãƒ¼ãƒãƒ«ã«åºƒãŒã‚Šã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + rowcomp + + + + + + fullrf + + + + + + fullrfk + + + + + + kernel + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/kernel/fullrf.xml b/modules/linear_algebra/help/ja_JP/kernel/fullrf.xml new file mode 100755 index 000000000..6085443bb --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/kernel/fullrf.xml @@ -0,0 +1,198 @@ + + + + + + + + + fullrf + + フルランク分解 + + + + + + 呼ã³å‡ºã—手順 + + [Q,M,rk]=fullrf(A,[tol]) + + + + + + 引数 + + + + + + A + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ— + + + + + + + + tol + + + + 実数 (ランク定義時ã®é–¾å€¤) + + + + + + + + Q,M + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ— + + + + + + + + rk + + + + + + æ•´æ•° (Aã®ãƒ©ãƒ³ã‚¯) + + + + + + + + + + + + + + 説明 + + + + フルランク分解 : fullrf ã¯, + + A = Q*Mã¨ãªã‚‹ã‚ˆã†ãª + + Q ãŠã‚ˆã³ Mã‚’è¿”ã—ã¾ã™. + + ãŸã ã—, + + range(Q)=range(A) ãŠã‚ˆã³ + + ker(M)=ker(A), + + Q フル列ランク , M フル行ランク, + + rk = rank(A) = #columns(Q) = #rows(M)ã§ã™. + + + + + + tol ã¯ã‚ªãƒ—ションã®å®Ÿæ•°ãƒ‘ラメータã§ã™ + + (デフォルト値㯠sqrt(%eps)ã§ã™). + + Aã®ãƒ©ãƒ³ã‚¯rk㯠+ + norm(A)*tolより大ã㪠+ + 特異値ã®æ•°ã¨ã—ã¦å®šç¾©ã•ã‚Œã¾ã™. + + + + + + AãŒå¯¾ç§°ã®å ´åˆ, + + fullrf 㯠M=Q'ã‚’è¿”ã—ã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + svd + + + + + + qr + + + + + + fullrfk + + + + + + rowcomp + + + + + + colcomp + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/kernel/fullrfk.xml b/modules/linear_algebra/help/ja_JP/kernel/fullrfk.xml new file mode 100755 index 000000000..8ce763a2c --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/kernel/fullrfk.xml @@ -0,0 +1,143 @@ + + + + + + + + + fullrfk + + A^kã®ãƒ•ãƒ«ãƒ©ãƒ³ã‚¯åˆ†è§£ + + + + + + 呼ã³å‡ºã—手順 + + [Bk,Ck]=fullrfk(A,k) + + + + + + 引数 + + + + + + A + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ— + + + + + + + + k + + + + æ•´æ•° + + + + + + + + Bk,Ck + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ— + + + + + + + + + + + + 説明 + + + + ã“ã®é–¢æ•°ã¯,A^kã®ãƒ•ãƒ«ãƒ©ãƒ³ã‚¯åˆ†è§£, + + ã™ãªã‚ã¡, Bk*Ck=A^k を計算ã—ã¾ã™. + + ãŸã ã—, Bk ã¯åˆ—フルランク, + + Ckã¯è¡Œãƒ•ãƒ«ãƒ©ãƒ³ã‚¯ã§ã™. + + range(Bk)=range(A^k) + + ãŠã‚ˆã³ ker(Ck)=ker(A^k)ã¨ãªã‚Šã¾ã™. + + + + + + k=1ã®å ´åˆ, fullrfk 㯠+ + fullrfã¨ç­‰ä¾¡ã«ãªã‚Šã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + fullrf + + + + + + range + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/kernel/im_inv.xml b/modules/linear_algebra/help/ja_JP/kernel/im_inv.xml new file mode 100755 index 000000000..de6c340d7 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/kernel/im_inv.xml @@ -0,0 +1,202 @@ + + + + + + + + + im_inv + + åŽŸåƒ + + + + + + 呼ã³å‡ºã—手順 + + [X,dim]=im_inv(A,B [,tol]) + + [X,dim,Y]=im_inv(A,B, [,tol]) + + + + + + + + 引数 + + + + + + A,B + + + + åŒã˜åˆ—ã®æ•°ã‚’有ã™ã‚‹å®Ÿæ•°ã¾ãŸã¯è¤‡ç´ æ•°è¡Œåˆ— + + + + + + + + X + + + + + + 次数ãŒAã®åˆ—ã®æ•°ã«ç­‰ã—ã„直交ã¾ãŸã¯ãƒ¦ãƒ‹ã‚¿ãƒªæ­£æ–¹è¡Œåˆ— + + + + + + + + + + dim + + + + æ•´æ•° (部分空間ã®æ¬¡å…ƒ) + + + + + + + + Y + + + + + + 次数ãŒAãŠã‚ˆã³Bã®è¡Œã®æ•°ã«ç­‰ã—ã„直交行列. + + + + + + + + + + + + + + 説明 + + + + [X,dim]=im_inv(A,B) 㯠(A^-1)(B), + + ã™ãªã‚ã¡, Aã¸ã®åƒãŒ range(B) ã« + + ã‚るベクトルを計算ã—ã¾ã™. + + + + + + Xã®æœ€åˆã®åˆ— dim 㯠+ + (A^-1)(B)ã«åºƒãŒã£ã¦ã„ã¾ã™. + + + + + + tol ã¯éƒ¨åˆ†ç©ºé–“ã®å–ã‚Šè¾¼ã¿ã‚’確èªã™ã‚‹ãŸã‚ã« + + 閾値ãŒä½¿ç”¨ã•ã‚Œã¦ãŠã‚Š, + + ãã®ãƒ‡ãƒ•ã‚©ãƒ«ãƒˆå€¤ã¯ tol = 100*%eps ã§ã™. + + Y ãŒè¿”ã•ã‚Œã‚‹æ™‚, + + [Y*A*X,Y*B] ã¯ä»¥ä¸‹ã®ã‚ˆã†ã«åˆ†å‰²ã•ã‚Œã¾ã™: + + [A11,A12;0,A22],[B1;0] + + + + + + ãŸã ã—, B1ã¯è¡Œãƒ•ãƒ«ãƒ©ãƒ³ã‚¯ ( + + rank(B)ã«ç­‰ã—ã„) ãã—㦠A22 㯠+ + 列フルランク㧠dim 列ã¨ãªã‚Šã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + rowcomp + + + + + + spaninter + + + + + + spanplus + + + + + + linsolve + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/kernel/kernel.xml b/modules/linear_algebra/help/ja_JP/kernel/kernel.xml new file mode 100755 index 000000000..f46881733 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/kernel/kernel.xml @@ -0,0 +1,173 @@ + + + + + + + + + kernel + + カーãƒãƒ«, ヌル空間 + + + + + + 呼ã³å‡ºã—手順 + + W=kernel(A [,tol,[,flag]) + + + + + + 引数 + + + + + + A + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®ãƒ•ãƒ«è¡Œåˆ—ã¾ãŸã¯å®Ÿæ•°ç–Žè¡Œåˆ— + + + + + + + + flag + + + + + + 文字列 'svd' (デフォルト) ã¾ãŸã¯ 'qr' + + + + + + + + + + tol + + + + 実数 + + + + + + + + W + + + + 列フルランク行列 + + + + + + + + + + + + 説明 + + + + W=kernel(A) ã¯Aã®ã‚«ãƒ¼ãƒãƒ« (ヌル空間)ã‚’è¿”ã—ã¾ã™. + + A ãŒåˆ—フルランクã®å ´åˆ, 空ã®è¡Œåˆ— [] ãŒè¿”ã•ã‚Œã¾ã™. + + + + + + flag ãŠã‚ˆã³ tol 㯠+ + オプションã®ãƒ‘ラメータã§ã™: flag = 'qr' + + ã¾ãŸã¯ 'svd' (デフォルト㯠'svd'). + + + + + + tol = 許容誤差パラメータ (デフォルト値㯠%eps ã®ã‚ªãƒ¼ãƒ€). + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + colcomp + + + + + + fullrf + + + + + + fullrfk + + + + + + linsolve + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/kernel/range.xml b/modules/linear_algebra/help/ja_JP/kernel/range.xml new file mode 100755 index 000000000..30905174c --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/kernel/range.xml @@ -0,0 +1,173 @@ + + + + + + + + + range + + A^kã®ç¯„囲 + + + + + + 呼ã³å‡ºã—手順 + + [X,dim]=range(A,k) + + + + + + 引数 + + + + + + A + + + + 実数正方行列 + + + + + + + + k + + + + æ•´æ•° + + + + + + + + X + + + + 直交実数行列 + + + + + + + + dim + + + + æ•´æ•° (部分空間ã®æ¬¡å…ƒ) + + + + + + + + + + + + 説明 + + + + 範囲 A^kを計算ã—ã¾ã™ ; X ã® + + 最åˆã® dim è¡Œã¯, A^kã®ç¯„囲ã«åºƒãŒã‚Šã¾ã™. + + Xã®æœ€å¾Œã®è¡Œã¯, + + ã“ã®ç›´äº¤ç›¸è£œãªç¯„囲ã«åºƒãŒã‚Šã¾ã™. + + X*X' ã¯å˜ä½è¡Œåˆ—ã§ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + fullrfk + + + + + + rowcomp + + + + + + + + + + 使用ã•ã‚Œã‚‹é–¢æ•° + + + + range 関数ã¯, + + svd分解を使用ã™ã‚‹ + + rowcomp 関数 + + ã«åŸºã¥ã„ã¦ã„ã¾ã™. + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/kernel/rowcomp.xml b/modules/linear_algebra/help/ja_JP/kernel/rowcomp.xml new file mode 100755 index 000000000..dc29889fb --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/kernel/rowcomp.xml @@ -0,0 +1,233 @@ + + + + + + + + + rowcomp + + 行圧縮, 範囲 + + + + + + 呼ã³å‡ºã—手順 + + [W,rk]=rowcomp(A [,flag [,tol]]) + + + + + + 引数 + + + + + + A + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ— + + + + + + + + flag + + + + オプションã®æ–‡å­—列, 指定å¯èƒ½ãªå€¤ã¯ + + 'svd' ã¾ãŸã¯ 'qr'ã§ã™. + + デフォルト値 'svd'ã¯ã§ã™. + + + + + + + + + + tol + + + + オプションã®éžè² ã®å®Ÿæ•°. デフォルト値㯠+ + sqrt(%eps)*norm(A,1). + + + + + + + + + + W + + + + 正方正則行列 (基底ã®å¤‰æ›´) + + + + + + + + rk + + + + + + æ•´æ•° (Aã®ãƒ©ãƒ³ã‚¯) + + + + + + + + + + + + + + 説明 + + + + Aã®è¡Œåœ§ç¸®. Ac = W*A ã¯è¡Œåœ§ç¸®ã•ã‚ŒãŸè¡Œåˆ—ã§ã™: ã™ãªã‚ã¡, + + Afを行フルランクã¨ã—㦠+ + Ac=[Af;0] ã¨ãªã‚Šã¾ã™. + + + + + + flag ãŠã‚ˆã³ tol ã¯ã‚ªãƒ—ションã®ãƒ‘ラメータã§ã™: flag='qr' + + ã¾ãŸã¯ 'svd' (デフォルト 'svd'). + + + + + + tol ã¯è¨±å®¹èª¤å·®ãƒ‘ラメータã§ã™. + + + + + + W'ã®æœ€åˆã®rk 列ã«ã¯, + + Aã®ç¯„囲ãŒåºƒãŒã‚Šã¾ã™. + + + + + + Wã®æœ€åˆã®(上å´ã®)rk è¡Œã«ã¯, + + Aã®è¡Œç¯„囲ãŒåºƒãŒã‚Šã¾ã™. + + + + + + éžã‚¼ãƒ­ãƒ™ã‚¯ãƒˆãƒ« x ã¯, + + W*xãŒAcã«åŸºã¥ã行圧縮ã•ã‚ŒãŸå ´åˆ, + + ã™ãªã‚ã¡,ãã®æœ€å¾Œã®è¦ç´ ã®ãƒŽãƒ«ãƒ ãŒæœ€åˆã®è¦ç´ ã«å¯¾ã—ã¦å°ã•ã„å ´åˆã«é™ã‚Š, + + range(A)ã«å±žã—ã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + colcomp + + + + + + fullrf + + + + + + fullrfk + + + + + + + + + + 使用ã•ã‚Œã‚‹é–¢æ•° + + + + rowcomp 関数ã¯svd ã¾ãŸã¯ + + qr 分解d. + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/linear/CHAPTER b/modules/linear_algebra/help/ja_JP/linear/CHAPTER new file mode 100755 index 000000000..7d9d9cf49 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/linear/CHAPTER @@ -0,0 +1,2 @@ +title = Linear Equations + diff --git a/modules/linear_algebra/help/ja_JP/linear/aff2ab.xml b/modules/linear_algebra/help/ja_JP/linear/aff2ab.xml new file mode 100755 index 000000000..f15783cf4 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/linear/aff2ab.xml @@ -0,0 +1,258 @@ + + + + + + + + + aff2ab + + ç·šå½¢ (アフィン)関数を A,b ã«å¤‰æ› + + + + + + 呼ã³å‡ºã—手順 + + [A,b]=aff2ab(afunction,dimX,D [,flag]) + + + + + + 引数 + + + + + + afunction + + + + + + scilab 関数 Y =fct(X,D) + + ãŸã ã—, X, D, Y ã¯è¡Œåˆ—ã®ãƒªã‚¹ãƒˆ + + + + + + + + + + dimX + + + + + + p x 2 整数行列 (p 㯠+ + Xã®è¡Œåˆ—ã®æ•°) + + + + + + + + + + D + + + + + + 実数行列ã®list (ã¾ãŸã¯ä»»æ„ã®æœ‰åŠ¹ãªScilab オブジェクト). + + + + + + + + + + flag + + + + + + オプションã®ãƒ‘ラメータ (flag='f' + + ã¾ãŸã¯ flag='sp') + + + + + + + + + + A + + + + 実数行列 + + + + + + + + b + + + + + + Aã¨åŒã˜è¡Œæ¬¡å…ƒã‚’有ã™ã‚‹å®Ÿæ•°ãƒ™ã‚¯ãƒˆãƒ« + + + + + + + + + + + + + + 説明 + + + + aff2abã¯,アフィン関数ã®(正準形å¼ã®)行列表ç¾ã‚’è¿”ã—ã¾ã™. + + + + + + afunction ã¯ä»¥ä¸‹ã®è¦å®šã®æ§‹æ–‡ã‚’有ã™ã‚‹é–¢æ•°ã§ã™: + + Y=afunction(X,D) + + ãŸã ã—, X=list(X1,X2,...,Xp) 㯠+ + p 個ã®å®Ÿæ•°è¡Œåˆ—ã®ãƒªã‚¹ãƒˆ, Y=list(Y1,...,Yq) 㯠+ + Xiã«ç·šå½¢ã«ä¾å­˜ã™ã‚‹q 個ã®å®Ÿæ•°è¡Œåˆ—ã®ãƒªã‚¹ãƒˆã§ã™. + + (オプションã®) 入力 D ã¯, X ã®é–¢æ•°ã¨ã—㦠+ + Yを計算ã™ã‚‹ãŸã‚ã«å¿…è¦ãªãƒ‘ラメータを有ã—ã¦ã„ã¾ã™. + + + + + + dimX 㯠p x 2 行列ã§ã™: dimX(i)=[nri,nci] + + ã¯è¡Œåˆ—Xiã®è¡Œã¨åˆ—ã®å®Ÿéš›ã®æ•°ã§ã™. + + ã“れらã®æ¬¡å…ƒã¯,çµæžœã®è¡Œåˆ—Aã®åˆ—ã®æ¬¡å…ƒã§ã‚ã‚‹ + + na を以下ã®ã‚ˆã†ã«å®šç¾©ã—ã¾ã™: + + na=nr1*nc1 +...+ nrp*ncp. + + + + + + オプションã®ãƒ‘ラメータ flag='sp' ãŒæŒ‡å®šã•ã‚ŒãŸå ´åˆ, + + çµæžœã®è¡Œåˆ— Aã¯ç–Žè¡Œåˆ—ã¨ã—ã¦è¿”ã•ã‚Œã¾ã™. + + + + + + ã“ã®é–¢æ•°ã¯,未知変数ãŒè¡Œåˆ—ã§ã‚るよã†ãª + + 線形方程å¼ã®ã‚·ã‚¹ãƒ†ãƒ ã‚’解ããŸã‚ã«æœ‰ç”¨ã§ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + linsolve + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/linear/chol.xml b/modules/linear_algebra/help/ja_JP/linear/chol.xml new file mode 100755 index 000000000..3838ddb87 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/linear/chol.xml @@ -0,0 +1,149 @@ + + + + + + + + + chol + + コレスキー分解 + + + + + + 呼ã³å‡ºã—手順 + + [R]=chol(X) + + + + + + 引数 + + + + + + X + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®æ­£å®šå¯¾ç§°è¡Œåˆ—. + + + + + + + + + + + + 説明 + + + + X ãŒæ­£å®šã®å ´åˆ, R = chol(X) ã¯, + + R'*R = Xã¨ãªã‚‹ã‚ˆã†ãª + + 上三角行列Rを出力ã—ã¾ã™. + + + + + + chol(X) ã¯Xã®å¯¾è§’é … + + ã¨ä¸Šä¸‰è§’部ã®ã¿ã‚’使用ã—ã¾ã™. + + 下三角部ã¯ä¸Šä¸‰è§’部ã®è»¢ç½®(複素共役)ã¨ã¿ãªã•ã‚Œã¾ã™. + + + + + + + + å‚考文献 + + + + コレスキー分解ã¯Lapackルーãƒãƒ³ DPOTRF (実数行列ã®å ´åˆ)ãŠã‚ˆã³ ZPOTRF (複素行列ã®å ´åˆ) + + ã«åŸºã¥ãã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + spchol + + + + + + qr + + + + + + svd + + + + + + bdiag + + + + + + fullrf + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/linear/inv.xml b/modules/linear_algebra/help/ja_JP/linear/inv.xml new file mode 100755 index 000000000..64e289d34 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/linear/inv.xml @@ -0,0 +1,195 @@ + + + + + + + + + inv + + 逆行列 + + + + + + 呼ã³å‡ºã—手順 + + inv(X) + + + + + + 引数 + + + + + + X + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®æ­£æ–¹è¡Œåˆ—, 多項å¼è¡Œåˆ—ãŠã‚ˆã³ + + ä¼é”関数ã¾ãŸã¯çŠ¶æ…‹ç©ºé–“表ç¾ã®æœ‰ç†è¡Œåˆ—. + + + + + + + + + + + + + + 説明 + + + + inv(X) ã¯,正方行列 Xã®é€†è¡Œåˆ— + + ã¨ãªã‚Šã¾ã™. + + X ã®ã‚¹ã‚±ãƒ¼ãƒªãƒ³ã‚°ãŒ + + 悪ã„å ´åˆã‚„特異行列ã«è¿‘ã„å ´åˆã«ã¯è­¦å‘Šã‚’出力ã—ã¾ã™. + + + + + + 多項å¼è¡Œåˆ—ã¾ãŸã¯ä¼é”関数表ç¾ã®æœ‰ç†è¡Œåˆ—ã®å ´åˆ, + + inv(X) 㯠invr(X)ã«ç­‰ã—ããªã‚Šã¾ã™. + + + + + + 状態空間表ç¾ã®ç·šå½¢ã‚·ã‚¹ãƒ†ãƒ (syslin リスト)ã®å ´åˆ, + + invr(X) 㯠invsyslin(X)ã«ç­‰ã—ããªã‚Šã¾ã™. + + + + + + + + å‚ç…§ + + + + 数値行列用ã®inv 関数㯠Lapack ルーãƒãƒ³ + + DGETRF, DGETRI (実数行列ã®å ´åˆ)ãŠã‚ˆã³ ZGETRF, ZGETRI + + (複素数ã®å ´åˆ)ã«åŸºã¥ã„ã¦ã„ã¾ã™. + + 多項å¼ãŠã‚ˆã³æœ‰ç†è¡Œåˆ—ã«é–¢ã™ã‚‹ inv 㯠+ + Scilab関数invrã«åŸºã¥ã„ã¦ã„ã¾ã™. + + + + + + + + 例 + + + + + + + + å‚考 + + + + + + slash + + + + + + backslash + + + + + + pinv + + + + + + qr + + + + + + lufact + + + + + + lusolve + + + + + + invr + + + + + + coff + + + + + + coffg + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/linear/linsolve.xml b/modules/linear_algebra/help/ja_JP/linear/linsolve.xml new file mode 100755 index 000000000..f34d07ead --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/linear/linsolve.xml @@ -0,0 +1,211 @@ + + + + + + + + + linsolve + + 線形方程å¼ã‚½ãƒ«ãƒ + + + + + + 呼ã³å‡ºã—手順 + + [x0,kerA]=linsolve(A,b [,x0]) + + + + + + 引数 + + + + + + A + + + + + + a na x ma 実数行列 (疎行列ã®å ´åˆã‚‚ã‚ã‚Š) + + + + + + + + + + b + + + + + + na x 1ベクトル (Aã®è¡Œã¨åŒã˜æ¬¡å…ƒ) + + + + + + + + + + x0 + + + + 実数ベクトル + + + + + + + + kerA + + + + + + ma x k 実数行列 + + + + + + + + + + + + + + 説明 + + + + linsolve ã¯, + + A*x+b=0ã®è§£ã‚’å…¨ã¦è¨ˆç®—ã—ã¾ã™. + + + + + + x0 ã¯ç‰¹è§£ (存在ã™ã‚‹å ´åˆ), + + kerA ã¯Aã®ãƒŒãƒ«ç©ºé–“ã§ã™. + + ä»»æ„ã®wã«ã¤ã„ã¦x=x0+kerA*wã¯, + + A*x+b=0を満ãŸã—ã¾ã™. + + + + + + 互æ›æ€§ã®ã‚ã‚‹ x0 ãŒã‚¨ãƒ³ãƒˆãƒªã«æŒ‡å®šã•ã‚ŒãŸå ´åˆ, + + x0ãŒè¿”ã•ã‚Œã¾ã™. + + ãã†ã§ãªã„å ´åˆ,x0ã¨äº’æ›æ€§ã®ã‚ã‚‹ã‚‚ã®(存在ã™ã‚‹å ´åˆ)ãŒè¿”ã•ã‚Œã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + inv + + + + + + pinv + + + + + + colcomp + + + + + + im_inv + + + + + + umfpack + + + + + + backslash + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/linear/lsq.xml b/modules/linear_algebra/help/ja_JP/linear/lsq.xml new file mode 100755 index 000000000..51c55183d --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/linear/lsq.xml @@ -0,0 +1,192 @@ + + + + + + + + + lsq + + 線形最å°äºŒä¹—å•é¡Œ. + + + + + + 呼ã³å‡ºã—手順 + + X=lsq(A,B [,tol]) + + + + + + 引数 + + + + + + A + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã® (m x n) 行列 + + + + + + + + B + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã® (m x p) 行列 + + + + + + + + tol + + + + Aã®å®ŸåŠ¹ãƒ©ãƒ³ã‚¯ã‚’定義ã™ã‚‹ãŸã‚ã«ä½¿ç”¨ã•ã‚Œã‚‹æ­£ã®ã‚¹ã‚«ãƒ©ãƒ¼ + + (Aã®ãƒ”ボットæ“作付ãQR分解ã«ãŠã‘る最å‰éƒ¨ã«ã‚る部分三角行列R11ã®æ¬¡æ•°ã¨ã—㦠+ + 定義ã•ã‚Œ,æ¡ä»¶æ•°ã®æŽ¨å®šå€¤ã¯<= 1/tolã¨ãªã‚Šã¾ã™. + + tolã®ãƒ‡ãƒ•ã‚©ãƒ«ãƒˆå€¤ã¯ sqrt(%eps)ã«è¨­å®šã•ã‚Œã¾ã™ ) + + + + + + + + + + X + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã® (n x p) 行列 + + + + + + + + + + + + 説明 + + + + X=lsq(A,B) ã¯æ–¹ç¨‹å¼ A*X=Bã® + + 最å°äºŒä¹—解ã®æœ€å°ãƒŽãƒ«ãƒ ã‚’計算ã—ã¾ã™. + + 一方, X=A \ B㯠+ + å„列ã«æœ€å¤§rank(A)個ã®éžã‚¼ãƒ­è¦ç´ ã‚’有ã™ã‚‹æœ€å°äºŒä¹—解を計算ã—ã¾ã™. + + + + + + + + å‚考文献 + + + + lsq 関数ã¯LApack 関数 DGELSY (実行列ã®å ´åˆ)ãŠã‚ˆã³ + + ZGELSY (複素行列ã®å ´åˆ)ã«åŸºã¥ã„ã¦ã„ã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + backslash + + + + + + inv + + + + + + pinv + + + + + + rank + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/linear/lu.xml b/modules/linear_algebra/help/ja_JP/linear/lu.xml new file mode 100755 index 000000000..6ce19d0e3 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/linear/lu.xml @@ -0,0 +1,299 @@ + + + + + + + + + lu + + ピボットé¸æŠžä»˜ãã®LU 分解 + + + + + + 呼ã³å‡ºã—手順 + + [L,U]= lu(A) + + [L,U,E]= lu(A) + + + + + + + + 引数 + + + + + + A + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ— (m x n). + + + + + + + + L + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ— (m x min(m,n)). + + + + + + + + U + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ— (min(m,n) x n ). + + + + + + + + E + + + + a (n x n) ç½®æ›è¡Œåˆ—. + + + + + + + + + + + + 説明 + + + + [L,U]= lu(A) ã¯, + + Uを上三角行列, + + Lを何らã‹ã®ç‰¹åˆ¥ãªæ§‹é€ ã‚’æŒãŸãªã„一般的ãªè¡Œåˆ—ã¨ã—ã¦, + + A = L*U ã¨ãªã‚‹ã‚ˆã†ãª + + 2ã¤ã®è¡Œåˆ— L ãŠã‚ˆã³ + + U を出力ã—ã¾ã™. + + 実際ã¯,行列Aã¯E*A=B*U + + ã®ã‚ˆã†ã«åˆ†è§£ã•ã‚Œã¾ã™. + + ãŸã ã—, 行列Bã¯ä¸‹ä¸‰è§’行列, + + 行列Lã¯L=E'*Bã‹ã‚‰è¨ˆç®—ã•ã‚Œã¾ã™. + + + + + + A ãŒãƒ©ãƒ³ã‚¯ kを有ã—ã¦ã„ã‚‹å ´åˆ, + + Uã®è¡Œ k+1 ã‹ã‚‰ + + n ã¾ã§ã¯ 0　ã¨ãªã‚Šã¾ã™. + + + + + + [L,U,E]= lu(A) ã¯, + + 上三角行列UãŠã‚ˆã³ + + ç½®æ›è¡Œåˆ—ã‚’ Eã¨ã—ãŸä¸‹ä¸‰è§’行列 E*L, + + ã«ã‚ˆã‚ŠE*A = L*Uã¨ãªã‚‹ã‚ˆã†ãª + + 3ã¤ã®è¡Œåˆ— L, U ãŠã‚ˆã³ + + Eを出力ã—ã¾ã™. + + + + + + A ãŒå®Ÿæ•°è¡Œåˆ—ã®å ´åˆ, + + 関数lufact ãŠã‚ˆã³ lugetã‚’ + + 用ã„ã‚‹ã“ã¨ã«ã‚ˆã‚Š, + + ç½®æ›è¡Œåˆ—ã‚’å¾—ã‚‹ã“ã¨ãŒã§ãã¾ã™. + + AãŒãƒ•ãƒ«ãƒ©ãƒ³ã‚¯ã§ãªã„å ´åˆ,行列 L + + ã®åˆ—圧縮も得るã“ã¨ãŒã§ãã‚‹. + + + + + + + + 例 #1 + + + + 以下ã®ä¾‹ã§ã¯,大ãã•4ã®ãƒ’ルãƒãƒ¼ãƒˆè¡Œåˆ—を作æˆã—, + + A=LU ã¨åˆ†è§£ã—ã¾ã™. + + 行列 L ã¯ä¸‹ä¸‰è§’行列ã§ã¯ãªã„ã“ã¨ã«æ³¨æ„ã—ã¦ãã ã•ã„. + + 下三角行列Lã‚’å–å¾—ã™ã‚‹ã«ã¯, + + 出力引数 E ã‚’ Scilab ã«æŒ‡å®šã™ã‚‹å¿…è¦ãŒã‚ã‚Šã¾ã™. + + + + + + + + + + 例 #2 + + + + 以下ã®ä¾‹ã§ã¯,大ãã•4ã®ãƒ’ルãƒãƒ¼ãƒˆè¡Œåˆ—を作æˆã—, + + EA=LU ã¨åˆ†è§£ã—ã¾ã™. + + 行列 L ã¯ä¸‹ä¸‰è§’行列ã§ã‚ã‚‹ã“ã¨ã«æ³¨æ„ã—ã¦ãã ã•ã„. + + + + + + + + + + 例 #3 + + + + 以下ã®ä¾‹ã§ã¯, lufact ãŠã‚ˆã³ luget 関数を使用ã™ã‚‹ + + 方法を示ã—ã¦ã„ã¾ã™. + + + + + + + + + + å‚ç…§ + + + + + + lufact + + + + + + luget + + + + + + lusolve + + + + + + qr + + + + + + svd + + + + + + + + + + 使用ã™ã‚‹é–¢æ•° + + + + lu 分解 Lapack ルーãƒãƒ³ DGETRF (実数行列ã®å ´åˆ) + + ãŠã‚ˆã³ ZGETRF (複素数ã®å ´åˆ) ã«åŸºã¥ã„ã¦ã„ã¾ã™. + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/linear/pinv.xml b/modules/linear_algebra/help/ja_JP/linear/pinv.xml new file mode 100755 index 000000000..1b59ae16c --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/linear/pinv.xml @@ -0,0 +1,159 @@ + + + + + + + + + pinv + + 擬似逆行列 + + + + + + 呼ã³å‡ºã—手順 + + pinv(A,[tol]) + + + + + + 引数 + + + + + + A + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ— + + + + + + + + tol + + + + 実数 + + + + + + + + + + + + 説明 + + + + X= pinv(A) ã¯, + + A'ã¨åŒã˜æ¬¡å…ƒã®ä»¥ä¸‹ã®ã‚ˆã†ãª + + 行列Xを出力ã—ã¾ã™: + + + + + + A*X*A = A, X*A*X = X ãã—㦠+ + A*X ãŠã‚ˆã³ X*A + + ã¯å…±ã«ã‚¨ãƒ«ãƒŸãƒ¼ãƒˆè¡Œåˆ—ã§ã™. + + + + + + 計算ã¯ç‰¹ç•°å€¤åˆ†è§£ã«åŸºã¥ã„ã¦ãŠã‚Š, + + 許容値よりもå°ã•ã„特異値㯠0 ã¨ã—ã¦æ‰±ã‚ã‚Œã¾ã™: + + ã“ã®è¨±å®¹èª¤å·®ã¯ X=pinv(A,tol) + + ã§ã‚¢ã‚¯ã‚»ã‚¹ã•ã‚Œã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + rank + + + + + + svd + + + + + + qr + + + + + + + + + + 使用ã•ã‚Œã‚‹é–¢æ•° + + + + pinv 関数ã¯ç‰¹ç•°å€¤åˆ†è§£ã«åŸºã¥ã„ã¦ã„ã¾ã™ + + (Scilab関数 svd). + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/linear/qr.xml b/modules/linear_algebra/help/ja_JP/linear/qr.xml new file mode 100755 index 000000000..d1b1acdad --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/linear/qr.xml @@ -0,0 +1,378 @@ + + + + + + + + + qr + + QR 分解 + + + + + + 呼ã³å‡ºã—手順 + + [Q,R]=qr(X [,"e"]) + + [Q,R,E]=qr(X [,"e"]) + + [Q,R,rk,E]=qr(X [,tol]) + + + + + + + + 引数 + + + + + + X + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ— + + + + + + + + tol + + + + éžè² ã®å®Ÿæ•° + + + + + + + + Q + + + + 正方直交ã¾ãŸã¯ãƒ¦ãƒ‹ã‚¿ãƒªè¡Œåˆ— + + + + + + + + R + + + + + + Xã¨åŒã˜æ¬¡å…ƒã®è¡Œåˆ— + + + + + + + + + + E + + + + ç½®æ›è¡Œåˆ— + + + + + + + + rk + + + + + + æ•´æ•° (Xã®QRランク) + + + + + + + + + + + + + + 説明 + + + + + + [Q,R] = qr(X) + + + + + + X= Q*Rã¨ãªã‚‹ã‚ˆã†ãª + + Xã¨åŒã˜æ¬¡å…ƒã® + + 上三角行列RãŠã‚ˆã³ç›´äº¤(複素数ã®å ´åˆã¯ãƒ¦ãƒ‹ã‚¿ãƒª)行列 + + Qを出力ã—ã¾ã™. + + [Q,R] = qr(X,"e")ã¯æ¬¡ã«ã‚ˆã†ã« + + "エコノミーサイズ"ã§å‡ºåŠ›ã—ã¾ã™: + + X ㌠mè¡Œn列 (m > n)ã®å ´åˆ, + + Qã®æœ€åˆã®n列ã®ã¿ãŒ + + Rã®æœ€åˆã®nè¡Œã¨åŒæ™‚ã«è¨ˆç®—ã•ã‚Œã¾ã™. + + + + + + Q*R = X ã‹ã‚‰, + + 行列 Xã®k番目ã®åˆ—ã¯, + + (ä¿‚æ•° R(1,k), ..., R(k,k) を用ã„ã¦) + + Qã®æœ€åˆã®k列ã®ç·šå½¢çµåˆã§è¡¨ã•ã‚Œã¾ã™. + + Qã®æœ€åˆã®k列ã¯,Xã®æœ€åˆã®k列 + + ã«åºƒãŒã‚‹éƒ¨åˆ†ç©ºé–“ã®ç›´äº¤åŸºåº•ã‚’作æˆã—ã¾ã™. + + Xã®åˆ—k(ã™ãªã‚ã¡, X(:,k) ) + + ãŒXã®æœ€åˆã®p列ã®ç·šå½¢çµåˆã®å ´åˆ, + + エントリR(p+1,k), ..., R(k,k)㯠0 ã¨ãªã‚Šã¾ã™. + + ã“ã®å ´åˆ,Rã¯ä¸Šå°å½¢ã¨ãªã‚Šã¾ã™. + + X ãŒãƒ©ãƒ³ã‚¯rkを有ã™ã‚‹å ´åˆ, + + è¡Œ R(rk+1,:), R(rk+2,:), ... 㯠0 ã¨ãªã‚Šã¾ã™. + + + + + + + + + + [Q,R,E] = qr(X) + + + + + + X*E = Q*Rã¨ãªã‚‹ã‚ˆã†ãª + + (列)ç½®æ›è¡Œåˆ—E, + + é™é †ã®å¯¾è§’è¦ç´ ã‚’有ã™ã‚‹ä¸Šä¸‰è§’行列 R, + + 直交(ã¾ãŸã¯ãƒ¦ãƒ‹ã‚¿ãƒª)Q + + を出力ã—ã¾ã™. + + rkãŒXã®ãƒ©ãƒ³ã‚¯ã®å ´åˆ, + + Rã®ä¸»å¯¾è§’é …ã«æ²¿ã£ãŸ + + 最åˆã®rk個ã®ã‚¨ãƒ³ãƒˆãƒª, + + ã™ãªã‚ã¡,R(1,1), R(2,2), ..., R(rk,rk)㯠+ + å…¨ã¦0以外ã¨ãªã‚Šã¾ã™. + + [Q,R,E] = qr(X,"e") 㯠+ + "エコノミーサイズ"ã§å‡ºåŠ›ã—ã¾ã™: + + X ㌠mè¡Œn列 (m > n)ã®å ´åˆ, + + Qã®æœ€åˆã®n列ã®ã¿ãŒ + + Rã®æœ€åˆã®nè¡Œã¨åŒæ™‚ã«è¨ˆç®—ã•ã‚Œã¾ã™. + + + + + + + + + + [Q,R,rk,E] = qr(X ,tol) + + + + + + rk = Xã®ãƒ©ãƒ³ã‚¯ã®æŽ¨å®šå€¤ + + ã‚’è¿”ã—ã¾ã™. + + ã™ãªã‚ã¡, rkã¯, + + 指定ã—ãŸé–¾å€¤tolより大ã㪠+ + Rã®å¯¾è§’è¦ç´ ã®æ•°ã¨ãªã‚Šã¾ã™. + + + + + + + + + + [Q,R,rk,E] = qr(X) + + + + + + rk = Xã®ãƒ©ãƒ³ã‚¯ã®æŽ¨å®šå€¤ + + ã‚’è¿”ã—ã¾ã™. + + ã™ãªã‚ã¡,rk 㯠+ + tol=R(1,1)*%eps*max(size(R))より大ã㪠+ + Rã®å¯¾è§’è¦ç´ ã®æ•°ã¨ãªã‚Šã¾ã™. + + Rã®æ¡ä»¶æ•°ã‚’用ã„ã‚‹ + + ランク計算型ã®QR分解ã«ã¤ã„ã¦ã¯,rankqrã‚’ + + å‚ç…§ã—ã¦ãã ã•ã„. + + + + + + + + + + + + + + 例 + + rk first +//diagonal entries of R are non zero : +A=rand(5,2)*rand(2,5); +[Q,R,rk,E] = qr(A,1.d-10); +norm(Q'*A-R) +svd([A,Q(:,1:rk)]) //span(A) =span(Q(:,1:rk)) + ]]> + + + + + + å‚ç…§ + + + + + + rankqr + + + + + + rank + + + + + + svd + + + + + + rowcomp + + + + + + colcomp + + + + + + + + + + 使用ã™ã‚‹é–¢æ•° + + + + qr 分解ã¯Lapack ルーãƒãƒ³ DGEQRF, DGEQPF, + + DORGQR (実数行列)ãŠã‚ˆã³ ZGEQRF, ZGEQPF, ZORGQR (複素数ã®å ´åˆ) + + ã«åŸºã¥ã„ã¦ã„ã¾ã™. + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/linear/rankqr.xml b/modules/linear_algebra/help/ja_JP/linear/rankqr.xml new file mode 100755 index 000000000..b6013b2fa --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/linear/rankqr.xml @@ -0,0 +1,290 @@ + + + + + + + + + rankqr + + QR分解ã«åŸºã¥ã階数 + + + + + + 呼ã³å‡ºã—手順 + + [Q,R,JPVT,RANK,SVAL]=rankqr(A, [RCOND,JPVT]) + + + + + + 引数 + + + + + + A + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ— + + + + + + + + RCOND + + + + + + Aã®å®ŸåŠ¹éšŽæ•°(ランク)を定義ã™ã‚‹ãŸã‚ã«ä½¿ç”¨ã•ã‚Œã‚‹å®Ÿæ•°ã§ã™. + + ã“ã®éšŽæ•°ã¯, + + Aã®ãƒ”ボットé¸æŠžä»˜ãã®QR分解ã®ä¸­ã® + + 最大ã®å…ˆé ­ã®éƒ¨åˆ†ä¸‰è§’行列R11ã®æ¬¡æ•°ã¨ã—ã¦å®šç¾©ã•ã‚Œã¾ã™. + + ãã®æŽ¨å®šã•ã‚ŒãŸæ¡ä»¶æ•°ã¯ < 1/RCOND ã¨ãªã‚Šã¾ã™. + + + + + + + + + + JPVT + + + + + + エントリã®æ•´æ•°ãƒ™ã‚¯ãƒˆãƒ«, JPVT(i) ㌠0ã§ãªã„å ´åˆ, + + Aã®i列目㯠+ + APã®å…ˆé ­ã¨äº¤æ›ã•ã‚Œ, + + ãれ以外ã®å ´åˆ,iã¯è‡ªç”±ãªåˆ—ã¨ãªã‚Šã¾ã™. + + 処ç†çµ‚了時ã«JPVT(i) = kã®å ´åˆ, + + A*Pã®i列目ã¯, + + Aã®k列目ã¨ãªã£ã¦ã„ã¾ã™. + + + + + + + + + + RANK + + + + + + Aã®å®ŸåŠ¹ãƒ©ãƒ³ã‚¯,ã™ãªã‚ã¡, + + 部分行列R11ã®æ¬¡æ•°. + + ã“ã‚Œã¯,Aã®å®Œå…¨ãªç›´äº¤åˆ†è§£ã«ãŠã‘ã‚‹ + + 部分行列T1ã®æ¬¡æ•°ã¨åŒã˜ã§ã™. + + + + + + + + + + SVAL + + + + + + 3ã¤ã®è¦ç´ ã‚’有ã™ã‚‹å®Ÿæ•°ãƒ™ã‚¯ãƒˆãƒ«;三角分解Rã® + + 特異値ã®æŽ¨å®šå€¤. + + + + + + SVAL(1) ã¯, + + R(1:RANK,1:RANK)ã®æœ€å¤§ç‰¹ç•°å€¤ã§ã™; + + + + + + SVAL(2) ã¯, + + R(1:RANK,1:RANK)ã®æœ€å°ç‰¹ç•°å€¤ã§ã™; + + + + + + SVAL(3) ã¯, + + RANK < MIN(M,N)ã®å ´åˆ, + + R(1:RANK+1,1:RANK+1), + + ãã†ã§ãªã„å ´åˆ, R(1:RANK,1:RANK)ã®æœ€å°ç‰¹ç•°å€¤ã§ã™. + + + + + + + + + + + + + + 説明 + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®Mè¡ŒN列一般行列Aã®(オプションã§) + + ランク出力を伴ãªã†QR分解を計算ã—ã¾ã™. + + ランクãŒä¸å®Œå…¨ã«ãªã‚‹å¯èƒ½æ€§ãŒã‚ã‚Š,実効ランクをæ¡ä»¶æ•°ã®ã‚¤ãƒ³ã‚¯ãƒªãƒ¡ãƒ³ã‚¿ãƒ«æŽ¨å®šã«ã‚ˆã‚Š + + 推定ã—ã¾ã™. + + + + + + ã“ã®ãƒ«ãƒ¼ãƒãƒ³ã¯åˆ—ピボットé¸æŠžä»˜ãã®QR分解を使用ã—ã¾ã™: + + + + + + + + R11ã¯,æ¡ä»¶æ•°ã®æŽ¨å®šå€¤ãŒ1/RCOND未満ã¨ãªã‚‹ + + 最大ã®éƒ¨åˆ†è¡Œåˆ—ã¨ã—ã¦å®šç¾©ã•ã‚Œã¾ã™. + + R11, RANKã®æ¬¡æ•°ã¯, + + Aã®å®ŸåŠ¹éšŽæ•°ã§ã™. + + + + + + 三角分解ãŒéšŽæ•°å‡ºåŠ›ã‚’ä¼´ãªã†å ´åˆ (ã“ã‚Œã¯å…ˆé ­ã®åˆ—ãŒå¥å…¨(well-conditioned)ãªå ´åˆã§ã™), + + SVAL(1)ã¯Aã®æœ€å¤§ç‰¹ç•°å€¤ã® + + 推定値ã¨ãªã‚Š,SVAL(2) ãŠã‚ˆã³ + + SVAL(3)ã¯,ãã‚Œãžã‚Œ Aã® + + RANK番目ãŠã‚ˆã³(RANK+1)番目㮠+ + 特異値ã®æŽ¨å®šå€¤ã¨ãªã‚Šã¾ã™. + + + + + + ã“れらã®å€¤ã‚’評価ã™ã‚‹ã“ã¨ã«ã‚ˆã‚Š,é¸æŠžã—ãŸRCONDã® + + 値ã«ã‚ˆã‚ŠéšŽæ•°ãŒè‰¯å¥½ã«å®šç¾©ã•ã‚Œã‚‹ã“ã¨ã‚’確èªã™ã‚‹ã“ã¨ãŒã§ãã¾ã™. + + 比 SVAL(1)/SVAL(2) ã¯, + + R(1:RANK,1:RANK)ã®æ¡ä»¶æ•°ã®æŽ¨å®šå€¤ã§ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + qr + + + + + + rank + + + + + + + + + + 使用ã•ã‚Œã‚‹é–¢æ•° + + + + Slicot library routines MB03OD, ZB03OD. + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/markov/CHAPTER b/modules/linear_algebra/help/ja_JP/markov/CHAPTER new file mode 100755 index 000000000..c29eb913c --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/markov/CHAPTER @@ -0,0 +1,2 @@ +title = Markov Matrices + diff --git a/modules/linear_algebra/help/ja_JP/markov/classmarkov.xml b/modules/linear_algebra/help/ja_JP/markov/classmarkov.xml new file mode 100755 index 000000000..48055e45c --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/markov/classmarkov.xml @@ -0,0 +1,176 @@ + + + + + + + + + classmarkov + + マルコフ行列ã®å†å¸°çš„ã‹ã¤ä¸€æ™‚çš„ãªã‚¯ãƒ©ã‚¹ + + + + + + 呼ã³å‡ºã—手順 + + [perm,rec,tr,indsRec,indsT]=classmarkov(M) + + + + + + 引数 + + + + + + M + + + + 実数 N x N マルコフ行列. å„è¡Œã®ã‚¨ãƒ³ãƒˆãƒªã®åˆè¨ˆã‚’ + + 1ã«åŠ ãˆãŸã‚‚ã® + + + + + + + + + + perm + + + + 整数交æ›ãƒ™ã‚¯ãƒˆãƒ«. + + + + + + + + rec, tr + + + + 整数ベクトル, 数値 (å„å†å¸°çš„クラスã«ãŠã‘る状態é‡ã®æ•°, + + 一時的ãªçŠ¶æ…‹é‡ã®æ•°). + + + + + + + + + + indsRec,indsT + + + + 整数ベクトル. (å†å¸°çš„ãŠã‚ˆã³ä¸€æ™‚çš„ãªçŠ¶æ…‹é‡ã®æ·»å­—). + + + + + + + + + + + + 説明 + + + + 以下ã®ã‚ˆã†ãªç½®æ›ãƒ™ã‚¯ãƒˆãƒ«permã‚’è¿”ã—ã¾ã™ + + + + + + + + å„ Mii ã¯rec(i) i=1,..,r次㮠+ + マルコフ行列ã§ã™. + + Qã¯,tr次ã®ã‚µãƒ–マルコフ行列ã§ã™. + + 1 ã‹ã‚‰ sum(rec)ã®çŠ¶æ…‹é‡ã¯å†å¸°çš„ã§, + + r+1ã‹ã‚‰nã¯ä¸€æ™‚çš„ãªçŠ¶æ…‹é‡ã§ã™. + + perm=[indsRec,indsT]ã¨ãªã‚Šã¾ã™. + + ãŸã ã—, indsRec ã¯å¤§ãã• sum(rec)ã®ãƒ™ã‚¯ãƒˆãƒ«, + + indsT ã¯å¤§ãã• trã®ãƒ™ã‚¯ãƒˆãƒ«ã§ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + genmarkov + + + + + + eigenmarkov + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/markov/eigenmarkov.xml b/modules/linear_algebra/help/ja_JP/markov/eigenmarkov.xml new file mode 100755 index 000000000..11762fb51 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/markov/eigenmarkov.xml @@ -0,0 +1,150 @@ + + + + + + + + + eigenmarkov + + æ­£è¦åŒ–ã•ã‚ŒãŸå·¦ãŠã‚ˆã³å³ãƒžãƒ«ã‚³ãƒ•å›ºæœ‰ãƒ™ã‚¯ãƒˆãƒ« + + + + + + 呼ã³å‡ºã—手順 + + [M,Q]=eigenmarkov(P) + + + + + + 引数 + + + + + + P + + + + 実数 N x N マルコフ行列. 1ã«åŠ ãˆã‚‹å„è¡Œã®ã‚¨ãƒ³ãƒˆãƒªã®åˆè¨ˆ. + + + + + + + + M + + + + N個ã®åˆ—を有ã™ã‚‹å®Ÿæ•°è¡Œåˆ—. + + + + + + + + Q + + + + N個ã®è¡Œã‚’有ã™ã‚‹å®Ÿæ•°è¡Œåˆ—. + + + + + + + + + + + + 説明 + + + + マルコフ推移行列 P ã®å›ºæœ‰å€¤ 1 ã«é–¢é€£ã™ã‚‹ + + æ­£è¦åŒ–ã•ã‚ŒãŸå·¦ãŠã‚ˆã³å³å›ºæœ‰ãƒ™ã‚¯ãƒˆãƒ«ã‚’è¿”ã—ã¾ã™. + + ã“ã®å›ºæœ‰å€¤ã®å¤šé‡åº¦ãŒ m　ã§, P㌠N x N ã®å ´åˆ, + + M 㯠m x N 行列㧠Q 㯠N x m 行列ã¨ãªã‚Šã¾ã™. + + M(k,:) ã¯k番目ã®ã‚¨ãƒ«ã‚´ãƒ¼ãƒ‰é›†åˆ(å†å¸°çš„クラス)ã«é–¢é€£ã™ã‚‹ + + 確率分布ベクトルã§ã™. + + M(k,x) ã¯ã€€x ㌠k番目ã®å†å¸°çš„クラスã«ãªã„å ´åˆã«ã¯ + + 0ã¨ãªã‚Šã¾ã™. + + Q(x,k) ã¯x ã‹ã‚‰å§‹ã¾ã‚‹ k 番目ã®å†å¸°çš„クラスã«æœ€çµ‚çš„ã«ã‚る確率ã§ã™. + + 大ããªkã«é–¢ã—ã¦P^k ㌠+ + åŽæŸã™ã‚‹å ´åˆ(1以外ã«å˜ä½å††ä¸Šã«å›ºæœ‰å€¤ãŒãªã„), + + 極é™ã¯Q*Mã¨ãªã‚Šã¾ã™(固有投影). + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + genmarkov + + + + + + classmarkov + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/markov/genmarkov.xml b/modules/linear_algebra/help/ja_JP/markov/genmarkov.xml new file mode 100755 index 000000000..287e19b83 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/markov/genmarkov.xml @@ -0,0 +1,161 @@ + + + + + + + + + genmarkov + + + + å†å¸°çš„ãŠã‚ˆã³ä¸€æ™‚çš„ãªã‚¯ãƒ©ã‚¹ã‚’有ã™ã‚‹ãƒ©ãƒ³ãƒ€ãƒ ãªãƒžãƒ«ã‚³ãƒ•è¡Œåˆ—を生æˆã™ã‚‹ + + + + + + + + 呼ã³å‡ºã—手順 + + M=genmarkov(rec,tr) + + M=genmarkov(rec,tr,flag) + + + + + + + + 引数 + + + + + + rec + + + + 整数行ベクトル (次元ã¯å†å¸°çš„クラスã®æ•°). + + + + + + + + tr + + + + æ•´æ•° (一時的ãªçŠ¶æ…‹é‡ã®æ•°) + + + + + + + + M + + + + 実数ã®ãƒžãƒ«ã‚³ãƒ•è¡Œåˆ—. + + 1ã«è¿½åŠ ã™ã‚‹å„è¡Œã®ã‚¨ãƒ³ãƒˆãƒªã®åˆè¨ˆ. + + + + + + + + + + flag + + + + + + 文字列 'perm'. 指定ã—ãŸå ´åˆ, + + 状態é‡ã®ãƒ©ãƒ³ãƒ€ãƒ ãªç½®æ›ãŒè¡Œã‚ã‚Œã¾ã™. + + + + + + + + + + + + + + 説明 + + + + ãã‚Œãžã‚Œrec(1),...rec($)個ã®ã‚¨ãƒ³ãƒˆãƒªã‚’有ã™ã‚‹ + + size(rec,1)個ã®å†å¸°çš„ãªã‚¯ãƒ©ã‚¹ã¨tr個ã®ä¸€æ™‚çš„ãªçŠ¶æ…‹é‡ã‚’有ã™ã‚‹ + + ランダムãªãƒžãƒ«ã‚³ãƒ•æŽ¨ç§»ç¢ºçŽ‡è¡Œåˆ—ã‚’Mã«è¿”ã—ã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + classmarkov + + + + + + eigenmarkov + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/matrix/CHAPTER b/modules/linear_algebra/help/ja_JP/matrix/CHAPTER new file mode 100755 index 000000000..bb89125cd --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/matrix/CHAPTER @@ -0,0 +1,2 @@ +title = Matrix Analysis + diff --git a/modules/linear_algebra/help/ja_JP/matrix/cond.xml b/modules/linear_algebra/help/ja_JP/matrix/cond.xml new file mode 100755 index 000000000..874d2fd98 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/matrix/cond.xml @@ -0,0 +1,292 @@ + + + + + + + + + cond + + æ¡ä»¶æ•° + + + + + + 呼ã³å‡ºã—手順 + + + + c = cond(X) + + c = cond(X, p) + + + + + + + + 引数 + + + + + + X + + + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ—. c = cond(X, p)ã®å ´åˆ, Xã¯å®Ÿæ•°ã¾ãŸã¯è¤‡ç´ æ•°ã® + + 正方行列ã¨ã™ã‚‹å¿…è¦ãŒã‚ã‚Šã¾ã™. + + + + + + + + + + c + + + + 実数スカラー, æ¡ä»¶æ•°. + + + + + + + + + + + + 説明 + + + + + + c = cond(X) + + + + + + ã¯,2-ノルムã®æ¡ä»¶æ•°ã‚’è¿”ã—ã¾ã™. + + cond(X)ã¯Xã® + + 最大特異値ã¨æœ€å°ç‰¹ç•°å€¤ã®æ¯”ã§ã™. + + + + + + + + + + c = cond(X, p) + + + + + + ã¯,p-ノルムã®æ¡ä»¶æ•°ã‚’è¿”ã—ã¾ã™ : + + norm(X, p) * norm(inv(X), p). + + p ãŒæŒ‡å®šã•ã‚ŒãŸå ´åˆ, + + p ã¯ä»¥ä¸‹ã«ç­‰ã—ããªã‚Šã¾ã™ : + + + + + + + + + + p = 1. cond(X, p) ã¯,1-ノルムã®æ¡ä»¶æ•°ã‚’è¿”ã—ã¾ã™. + + + + + + + + + + p = 2. cond(X, p) ã¯,1-ノルムã®æ¡ä»¶æ•°ã‚’è¿”ã—ã¾ã™. + + + + + + + + + + p = %inf ã¾ãŸã¯ 'inf'. cond(X, p) + + ã¯,ç„¡é™å¤§ãƒŽãƒ«ãƒ ã®æ¡ä»¶æ•°ã‚’è¿”ã—ã¾ã™. + + + + + + + + + + p = 'fro'. cond(X, p) + + ã¯ãƒ•ãƒ­ãƒ™ãƒ‹ã‚¦ã‚¹ãƒŽãƒ«ãƒ ã®æ¡ä»¶æ•°ã‚’è¿”ã—ã¾ã™. + + + + + + + + + + + + + + + + + + 例 + + + + + + + + + + + + å‚考 + + + + + + rcond + + + + + + svd + + + + + + norm + + + + + + + + + + 履歴 + + + + + + 5.4.0 + + + + + + éžæ­£æ–¹è¡Œåˆ—ã®Xを指定ã—ã¦, + + cond(X)をコールã—ã¦ã‚‚管ç†ã•ã‚Œã‚‹ã‚ˆã†ã«ãªã‚Šã¾ã—ãŸ. + + 例ãˆã°: + + + + + + + + + + + + cond(X, p)をコールã™ã‚‹ã“ã¨ã§ã€ + + p-ノルムæ¡ä»¶æ•°ã‚’計算ã§ãるよã†ã«ãªã‚Šã¾ã—ãŸ. + + 例ãˆã°: + + + + + + + + + + + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/matrix/det.xml b/modules/linear_algebra/help/ja_JP/matrix/det.xml new file mode 100755 index 000000000..c7ea8f509 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/matrix/det.xml @@ -0,0 +1,212 @@ + + + + + + + + + det + + è¡Œåˆ—å¼ + + + + + + 呼出ã—手順 + + det(X) + + [e,m]=det(X) + + + + + + + + 引数 + + + + + + X + + + + 実数ã¾ãŸã¯è¤‡ç´ æ­£æ–¹è¡Œåˆ—, 多項å¼ã¾ãŸã¯æœ‰ç†è¡Œåˆ—. + + + + + + + + m + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°, 行列å¼ã® 10 を基底ã¨ã™ã‚‹ä»®æ•° + + + + + + + + e + + + + æ•´æ•°, 行列å¼ã® 10 を基底ã¨ã™ã‚‹æŒ‡æ•° + + + + + + + + + + + + 説明 + + + + det(X) (m*10^e)ã¯, + + 正方行列Xã®è¡Œåˆ—å¼ã§ã™. + + + + + + 多項å¼è¡Œåˆ—ã®å ´åˆ,det(X) 㯠+ + determ(X)ã¨ç­‰ã—ããªã‚Šã¾ã™. + + + + + + 有ç†æ•°è¡Œåˆ—ã®å ´åˆ, det(X) 㯠+ + detr(X)ã¨ç­‰ã—ããªã‚Šã¾ã™. + + + + + + + + det ãŠã‚ˆã³ detr 関数㯠+ + åŒã˜ã‚¢ãƒ«ã‚´ãƒªã‚ºãƒ ã‚’使用ã—ã¾ã›ã‚“. + + 有ç†æ•°é–¢æ•°ã®å ´åˆ, det(X) 㯠+ + determ関数ã«åŸºã¥ã + + %r_det(X) ã§ã‚ªãƒ¼ãƒãƒ¼ãƒ­ãƒ¼ãƒ‰ã•ã‚Œã¾ã™. + + detr() ã¯, Leverrier法を使用ã—ã¾ã™. + + + + + + 時々, + + det ãŠã‚ˆã³ detr 関数㯠+ + 有ç†æ•°é–¢æ•°ã¨ç•°ãªã‚‹å€¤ã‚’è¿”ã™å¯èƒ½æ€§ãŒã‚ã‚Šã¾ã™. + + ã“ã®ã‚ˆã†ãªå ´åˆ,åŒã˜çµæžœã‚’å¾—ã‚‹ãŸã‚ã«, + + 有ç†æ•°ã¯simp_mode(%f)を使用ã™ã‚‹ã“ã¨ã«ã‚ˆã‚Š + + 有ç†æ•°ã‚’ç°¡å˜åŒ–ã™ã‚‹ãƒ¢ãƒ¼ãƒ‰ã‚’無効ã«ã™ã‚‹å¿…è¦ãŒã‚ã‚Šã¾ã™. + + + + + + + + + + å‚ç…§ + + + + det ã®è¨ˆç®—㯠Lapack ルーãƒãƒ³ DGETRF (実数行列ã®å ´åˆ) ãŠã‚ˆã³ + + ZGETRF (複素数ã®å ´åˆ)ã«åŸºã¥ã„ã¦ã„ã¾ã™. + + + + + + 疎行列ã®å ´åˆ, 行列å¼ã¯ umfpack ライブラリã®LU分解ã«ã‚ˆã‚Šå¾—られã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + detr + + + + + + determ + + + + + + simp_mode + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/matrix/orth.xml b/modules/linear_algebra/help/ja_JP/matrix/orth.xml new file mode 100755 index 000000000..c66862e31 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/matrix/orth.xml @@ -0,0 +1,144 @@ + + + + + + + + + orth + + 直交基底 + + + + + + 呼ã³å‡ºã—手順 + + Q=orth(A) + + + + + + 引数 + + + + + + A + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ— + + + + + + + + Q + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ— + + + + + + + + + + + + 説明 + + + + Q=orth(A) ã¯, + + Aã®ç¯„囲ã§ã®ç›´äº¤åŸºåº•ã§ã‚ã‚‹ + + Qã‚’è¿”ã—ã¾ã™. + + Range(Q) = + + Range(A) ãŠã‚ˆã³ Q'*Q=eye. + + + + + + Qã®åˆ—ã®æ•°ã¯, + + QRアルゴリズムã§å®šç¾©ã•ã‚ŒãŸ + + Aã®ãƒ©ãƒ³ã‚¯ã§ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + qr + + + + + + rowcomp + + + + + + colcomp + + + + + + range + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/matrix/rank.xml b/modules/linear_algebra/help/ja_JP/matrix/rank.xml new file mode 100755 index 000000000..cdbd5aee6 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/matrix/rank.xml @@ -0,0 +1,157 @@ + + + + + + + + + rank + + 階数 + + + + + + 呼ã³å‡ºã—手順 + + [i]=rank(X) + + [i]=rank(X,tol) + + + + + + + + 引数 + + + + + + X + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ— + + + + + + + + tol + + + + éžè² å®Ÿæ•° + + + + + + + + + + + + 説明 + + + + rank(X) ã¯, Xã®æ•°å€¤çš„ãªéšŽæ•°(ランク), + + ã™ãªã‚ã¡, norm(size(X),'inf') * norm(X) * %eps より大ã㪠+ + X ã®ç‰¹ç•°å€¤ã®æ•°ã§ã™. + + + + + + rank(X,tol) ã¯,tol + + より大ããªXã®ç‰¹ç•°å€¤ã®æ•°ã§ã™. + + + + + + tol ã®ãƒ‡ãƒ•ã‚©ãƒ«ãƒˆå€¤ã¯ + + norm(X)ã«æ¯”例ã™ã‚‹ã“ã¨ã«æ³¨æ„ã—ã¦ãã ã•ã„. + + çµæžœã¨ã—ã¦,rank([1.d-80,0;0,1.d-80]) 㯠2 ã«ãªã‚Šã¾ã™!. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + svd + + + + + + qr + + + + + + rowcomp + + + + + + colcomp + + + + + + lu + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/matrix/rcond.xml b/modules/linear_algebra/help/ja_JP/matrix/rcond.xml new file mode 100755 index 000000000..e12e728b5 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/matrix/rcond.xml @@ -0,0 +1,145 @@ + + + + + + + + + rcond + + æ¡ä»¶æ•°ã®é€†æ•° + + + + + + 呼ã³å‡ºã—手順 + + rcond(X) + + + + + + 引数 + + + + + + X + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®æ­£æ–¹è¡Œåˆ— + + + + + + + + + + + + 説明 + + + + rcond(X) ã¯,1-ノルムã«ãŠã‘ã‚‹ + + Xã®æ¡ä»¶ã®é€†æ•°ã®æŽ¨å®šå€¤ã§ã™. + + + + + + XãŒå¥å…¨ãªå ´åˆ, + + rcond(X) 㯠1 ã«è¿‘ããªã‚Šã¾ã™. + + ãã†ã§ãªã„å ´åˆ, rcond(X) 㯠0ã«è¿‘ããªã‚Šã¾ã™. + + + + + + Aã®1-ノルムを Lapack/DLANGEã§è¨ˆç®—, ãã®LU分解をLapack/DGETRFã§è¨ˆç®—, + + 最後ã«æ¡ä»¶ã‚’Lapack/DGECONã§æŽ¨å®šã—ã¾ã™. + + + + + + + + 例 + + + + + + rcondã«ã‚ˆã‚‹1-ノルム逆æ¡ä»¶æ•°ã®æŽ¨å®šã¯, + + condã«ã‚ˆã‚‹2-ノルムæ¡ä»¶æ•°ã®è¨ˆç®—よりã¯ã‚‹ã‹ã«é«˜é€Ÿã§ã™. + + トレードオフã¨ã—ã¦,rcond ã¯è‹¥å¹²ä¿¡é ¼æ€§ãŒä½Žä¸‹ã™ã‚‹å¯èƒ½æ€§ãŒã‚ã‚Šã¾ã™. + + + + + + + + + + å‚ç…§ + + + + + + svd + + + + + + cond + + + + + + inv + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/matrix/rref.xml b/modules/linear_algebra/help/ja_JP/matrix/rref.xml new file mode 100755 index 000000000..cf47e7b89 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/matrix/rref.xml @@ -0,0 +1,127 @@ + + + + + + + + + rref + + LU分解ã«ã‚ˆã‚Šè¡Œã‚¨ã‚·ãƒ¥ãƒ­ãƒ³å½¢å¼ã®è¡Œåˆ—を計算 + + + + + + 呼ã³å‡ºã—手順 + + R=rref(A) + + + + + + 引数 + + + + + + A + + + + スカラーã®ã‚¨ãƒ³ãƒˆãƒªã‚’有ã™ã‚‹m x n 行列 + + + + + + + + R + + + + Aã®è¡Œã‚¨ã‚·ãƒ¥ãƒ­ãƒ³å½¢å¼ã®m x n行列 + + + + + + + + + + + + 説明 + + + + rref ã¯,å·¦LU分解ã«ã‚ˆã‚ŠæŒ‡å®šã—ãŸè¡Œåˆ— + + ã®è¡Œã‚¨ã‚·ãƒ¥ãƒ­ãƒ³å½¢å¼ã‚’計算ã—ã¾ã™. + + X=rref([A,eye(m,m)])をコールã™ã‚‹éš›ã«ä½¿ç”¨ã—㟠+ + 変æ›ã ã‘ãŒå¿…è¦ãªå ´åˆ,行エシュロン形å¼R㯠+ + X(:,1:n)ã¨ãªã‚Šã¾ã™. + + 左変æ›Lã¯,L*A=Rã¨ãªã‚‹ã‚ˆã†ãª + + X(:,n+1:n+m) ã«ã‚ˆã‚Šå¾—ã‚‹ã“ã¨ãŒã§ãã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + lu + + + + + + qr + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/matrix/trace.xml b/modules/linear_algebra/help/ja_JP/matrix/trace.xml new file mode 100755 index 000000000..c3b6fd4ba --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/matrix/trace.xml @@ -0,0 +1,103 @@ + + + + + + + + + trace + + トレース + + + + + + 呼ã³å‡ºã—手順 + + trace(X) + + + + + + 引数 + + + + + + X + + + + 実数ã¾ãŸã¯è¤‡ç´ æ•°ã®æ­£æ–¹è¡Œåˆ—, 多項å¼ã¾ãŸã¯æœ‰ç†è¡Œåˆ—. + + + + + + + + + + + + 説明 + + + + trace(X)ã¯,行列Xã® + + トレースã§ã™. + + + + + + sum(diag(X))ã¨åŒã˜ã§ã™. + + + + + + + + 例 + + + + + + + + å‚考 + + + + + + det + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/pencil/CHAPTER b/modules/linear_algebra/help/ja_JP/pencil/CHAPTER new file mode 100755 index 000000000..86d1da116 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/pencil/CHAPTER @@ -0,0 +1,2 @@ +title = Matrix Pencil + diff --git a/modules/linear_algebra/help/ja_JP/pencil/companion.xml b/modules/linear_algebra/help/ja_JP/pencil/companion.xml new file mode 100755 index 000000000..93a9adcec --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/pencil/companion.xml @@ -0,0 +1,150 @@ + + + + + + + + + companion + + コンパニオン行列 + + + + + + 呼ã³å‡ºã—手順 + + A=companion(p) + + + + + + 引数 + + + + + + p + + + + 多項å¼ã¾ãŸã¯å¤šé …å¼ã®ãƒ™ã‚¯ãƒˆãƒ« + + + + + + + + A + + + + 正方行列 + + + + + + + + + + + + 説明 + + + + 特性多項å¼ã‚’有ã™ã‚‹ + + 行列Aã‚’è¿”ã—ã¾ã™. + + pãŒãƒ¢ãƒ‹ãƒƒã‚¯ãªå ´åˆ,特性多項å¼ã¯ + + pã«ç­‰ã—ããªã‚Šã¾ã™. + + pãŒãƒ¢ãƒ‹ãƒƒã‚¯ã§ãªã„å ´åˆ, + + Aã®ç‰¹æ€§æ–¹ç¨‹å¼ã¯ + + p/cã«ç­‰ã—ããªã‚Šã¾ã™. + + ãŸã ã—,cã¯p + + ã®æœ€å¤§æ¬¡æ•°ã®ä¿‚æ•°ã§ã™. + + + + + + p ãŒãƒ¢ãƒ‹ãƒƒã‚¯ãªå¤šé …å¼ã®ãƒ™ã‚¯ãƒˆãƒ«ã®å ´åˆ, + + A ã¯ãƒ–ロック対角ã¨ãªã‚Š, + + i番目ã®ç‰¹æ€§å¤šé …å¼ã¯ + + p(i)ã¨ãªã‚Šã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + spec + + + + + + poly + + + + + + randpencil + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/pencil/ereduc.xml b/modules/linear_algebra/help/ja_JP/pencil/ereduc.xml new file mode 100755 index 000000000..b1d93ff64 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/pencil/ereduc.xml @@ -0,0 +1,223 @@ + + + + + + + + + ereduc + + QZ変æ›ã«ã‚ˆã‚Šåˆ—階段型行列を計算 + + + + + + 呼ã³å‡ºã—手順 + + [E,Q,Z [,stair [,rk]]]=ereduc(X,tol) + + + + + + 引数 + + + + + + X + + + + 実数エントリを有ã™ã‚‹m x n 行列. + + + + + + + + tol + + + + 実数ã®æ­£ã®ã‚¹ã‚«ãƒ©ãƒ¼. + + + + + + + + E + + + + 列階段型行列 + + + + + + + + Q + + + + m x m ユニタリ行列 + + + + + + + + Z + + + + n x n ユニタリ行列 + + + + + + + + stair + + + + 添字ベクトル, + + + + + + * + + + + + + 境界è¦ç´ E(i,j)ãŒç«¯ç‚¹ã®å ´åˆ, + + ISTAIR(i) = + j. + + + + + + + + + + * + + + + + + 境界è¦ç´ E(i,j)ãŒç«¯ç‚¹ã§ãªã„å ´åˆ, + + ISTAIR(i) = - j. + + + + + + + + + + + + (i=1,...,M) + + + + + + + + + + rk + + + + æ•´æ•°, 行列ã®ãƒ©ãƒ³ã‚¯ã®æŽ¨å®šå€¤ + + + + + + + + + + + + 説明 + + + + mx n行列X (正則ã§ã‚ã‚‹ + + å¿…è¦ã¯ãªã„)を指定ã™ã‚‹ã¨, + + 関数ereducã¯,列階段型(å°å½¢)ã® + + ユニタリ変æ›è¡Œåˆ—E=Q*X*Z + + を計算ã—ã¾ã™. + + æ›´ã«è¡Œåˆ—Xã®ãƒ©ãƒ³ã‚¯ãŒå®šç¾©ã•ã‚Œã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + fstair + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/pencil/fstair.xml b/modules/linear_algebra/help/ja_JP/pencil/fstair.xml new file mode 100755 index 000000000..7f82fe30d --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/pencil/fstair.xml @@ -0,0 +1,354 @@ + + + + + + + + + fstair + + QZ変æ›ã«ã‚ˆã‚Šåˆ—階段型ペンシルを計算ã™ã‚‹ + + + + + + 呼ã³å‡ºã—手順 + + [AE,EE,QE,ZE,blcks,muk,nuk,muk0,nuk0,mnei]=fstair(A,E,Q,Z,stair,rk,tol) + + + + + + 引数 + + + + + + A + + + + 実数エントリを有ã™ã‚‹m x n行列. + + + + + + + + tol + + + + 実数ã®æ­£ã®ã‚¹ã‚«ãƒ©ãƒ¼. + + + + + + + + E + + + + 列階段型行列 + + + + + + + + Q + + + + m x m ユニタリ行列 + + + + + + + + Z + + + + n x n ユニタリ行列 + + + + + + + + stair + + + + 添字ベクトル (ereducå‚ç…§) + + + + + + + + rk + + + + æ•´æ•°, 行列ランクã®æŽ¨å®šå€¤ + + + + + + + + AE + + + + 実数エントリを有ã™ã‚‹m x n行列. + + + + + + + + EE + + + + 列階段型行列 + + + + + + + + QE + + + + m x m ユニタリ行列 + + + + + + + + ZE + + + + n x n ユニタリ行列 + + + + + + + + nblcks + + + + + + 行列Aã§æ¤œå‡ºã•ã‚ŒãŸ + + フル行ランクを有ã™ã‚‹ã‚µãƒ–行列ã®æ•°(>= 0). + + + + + + + + + + muk: + + + + 次元 (n) ã®æ•´æ•°é…列. + + ペンシルsE(eps)-A(eps)ã«ãŠã„㦠+ + 列フルランクを有ã™ã‚‹ã‚µãƒ–行列ã®åˆ—次元 mu(k) (k=1,...,nblcks) ã‚’å«ã¿ã¾ã™. + + + + + + + + + + nuk: + + + + + + 次元 (m+1) ã®æ•´æ•°é…列. + + ペンシルsE(eps)-A(eps)ã«ãŠã„㦠+ + 行フルランクを有ã™ã‚‹ã‚µãƒ–行列ã®è¡Œæ¬¡å…ƒ nu(k) (k=1,...,nblcks) + + ã‚’å«ã¿ã¾ã™. + + + + + + + + + + muk0: + + + + + + 次元 (n) ã®æ•´æ•°é…列. + + ペンシルsE(eps,inf)-A(eps,inf)ã«ãŠã„㦠+ + 列フルランクを有ã™ã‚‹ã‚µãƒ–行列ã®åˆ—次元 mu(k) (k=1,...,nblcks) ã‚’å«ã¿ã¾ã™. + + + + + + + + + + nuk: + + + + + + 次元 (m+1) ã®æ•´æ•°é…列. + + ペンシルsE(eps,inf)-A(eps,inf)ã«ãŠã„㦠+ + 行フルランクを有ã™ã‚‹ã‚µãƒ–行列ã®è¡Œæ¬¡å…ƒ nu(k) (k=1,...,nblcks) + + ã‚’å«ã¿ã¾ã™. + + + + + + + + + + mnei: + + + + 次元 (4) ã®æ•´æ•°é…列. + + mnei(1) = sE(eps)-A(eps)ã®è¡Œã®æ¬¡å…ƒ + + + + + + + + + + + + + + 説明 + + + + 行列 E を列階段形ã¨ã—ã¦, + + ペンシル sE-A を指定ã™ã‚‹ã¨, + + 関数fstair㯠+ + ユーザã®æŒ‡å®šã«åŸºã¥ã + + ユニタリ変æ›ã•ã‚ŒãŸãƒšãƒ³ã‚·ãƒ«QE(sEE-AE)ZEã‚’ + + 計算ã—ã¾ã™. + + ã“ã®ãƒšãƒ³ã‚·ãƒ«ã¯, ã»ã¼ãƒšãƒ³ã‚·ãƒ«sE-A + + ã®ä¸€èˆ¬åŒ–Schurå½¢å¼ã§ã™. + + ã“ã®é–¢æ•°ã¯,指定ã—ãŸãƒšãƒ³ã‚·ãƒ«ã® + + クロãƒãƒƒã‚«ãƒ¼æ§‹é€ ã®éƒ¨åˆ†ã‚‚出力ã—ã¾ã™. + + + + + + Q,Z ã¯ãƒ¦ãƒ‹ã‚¿ãƒªè¡Œåˆ—ã§, + + ペンシルを計算ã™ã‚‹éš›ã«ä½¿ç”¨ã•ã‚Œã¾ã™. + + ãŸã ã—, E ã¯åˆ—階段形ã§ã™ (ereducå‚ç…§) + + + + + + + + å‚ç…§ + + + + + + quaskro + + + + + + ereduc + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/pencil/glever.xml b/modules/linear_algebra/help/ja_JP/pencil/glever.xml new file mode 100755 index 000000000..b51eac00e --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/pencil/glever.xml @@ -0,0 +1,220 @@ + + + + + + + + + glever + + 行列ペンシルã®é€† + + + + + + 呼ã³å‡ºã—手順 + + [Bfs,Bis,chis]=glever(E,A [,s]) + + + + + + 引数 + + + + + + E, A + + + + åŒã˜æ¬¡å…ƒã®æ­£æ–¹å®Ÿè¡Œåˆ— + + + + + + + + s + + + + + + 文字列 (デフォルト値 's') + + + + + + + + + + Bfs,Bis + + + + 多項å¼è¡Œåˆ— + + + + + + + + chis + + + + å¤šé …å¼ + + + + + + + + + + + + 説明 + + 一般化ã—ãŸLeverrierã®ã‚¢ãƒ«ã‚´ãƒªã‚ºãƒ ã«ã‚ˆã‚Šã‚る行列ペンシルã«ã¤ã„㦠+ + + + + + (s*E-A)^-1 + + + + + + を計算ã—ã¾ã™. + + + + + + + + chis = ç‰¹æ€§å¤šé …å¼ (乗法定数ã¾ã§). + + + + + + Bfs = 分å­ã®ç‰¹æ€§å¤šé …å¼è¡Œåˆ—. + + + + + + Bis + + = 多項å¼è¡Œåˆ— ( - (s*E-A)^-1 ã®ç„¡é™å¤§ã¾ã§ã®ç´šæ•°å±•é–‹). + + + + + + Bisã®å‰ã« - 符å·ãŒã‚ã‚‹ã“ã¨ã«æ³¨æ„ã—ã¦ãã ã•ã„. + + + + + + + + æ³¨æ„ + + + + ã“ã®é–¢æ•°ã¯,Bfs,Bis ãŠã‚ˆã³ chisã‚’ + + ç°¡å˜åŒ–ã™ã‚‹ãŸã‚ã« cleanpを使用ã—ã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + rowshuff + + + + + + det + + + + + + invr + + + + + + coffg + + + + + + pencan + + + + + + penlaur + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/pencil/kroneck.xml b/modules/linear_algebra/help/ja_JP/pencil/kroneck.xml new file mode 100755 index 000000000..73cd277ef --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/pencil/kroneck.xml @@ -0,0 +1,281 @@ + + + + + + + + + kroneck + + 行列ペンシルã®ã‚¯ãƒ­ãƒãƒƒã‚«ãƒ¼å½¢å¼ + + + + + + 呼ã³å‡ºã—手順 + + [Q,Z,Qd,Zd,numbeps,numbeta]=kroneck(F) + + [Q,Z,Qd,Zd,numbeps,numbeta]=kroneck(E,A) + + + + + + + + 引数 + + + + + + F + + + + + + 実数行列ペンシル F=s*E-A + + + + + + + + + + E,A + + + + åŒã˜æ¬¡å…ƒã®å®Ÿæ•°è¡Œåˆ— + + + + + + + + Q,Z + + + + 正方直交行列 + + + + + + + + Qd,Zd + + + + 整数ベクトル + + + + + + + + numbeps,numeta + + + + 整数ベクトル + + + + + + + + + + + + 説明 + + + + 行列ペンシルã®ã‚¯ãƒ­ãƒãƒƒã‚«ãƒ¼å½¢å¼: kroneck ã¯, + + ペンシルF=s*E -Aを以下ã®ã‚ˆã†ãªä¸Šä¸‰è§’å½¢å¼ã«å¤‰æ›ã™ã‚‹ + + 2ã¤ã®ç›´äº¤è¡Œåˆ—Q, Zを計算ã—ã¾ã™: + + + + + + + + 4個ã®ãƒ–ロックã®æ¬¡å…ƒã¯ä»¥ä¸‹ã®ã‚ˆã†ã«æŒ‡å®šã•ã‚Œã¾ã™: + + + + + + eps=Qd(1) x Zd(1), inf=Qd(2) x Zd(2), + + f = Qd(3) x Zd(3), eta=Qd(4)xZd(4) + + + + + + infブロックã«ã¯ãƒšãƒ³ã‚·ãƒ«ã®ç„¡é™å¤§ãƒ¢ãƒ¼ãƒ‰ãŒå«ã¾ã‚Œã¾ã™. + + + + + + f ブロックã«ã¯ãƒšãƒ³ã‚·ãƒ«ã®æœ‰é™ãƒ¢ãƒ¼ãƒ‰ãŒå«ã¾ã‚Œã¾ã™. + + + + + + イプシロンã¨etaブロックã®æ§‹é€ ã¯ä»¥ä¸‹ã®ã‚ˆã†ã«æŒ‡å®šã•ã‚Œã¾ã™: + + + + + + numbeps(1) = 大ãã• 0 x 1ã®epsブロックã®ç•ªå· + + + + + + numbeps(2) = 大ãã• 1 x 2ã®epsブロックã®ç•ªå· + + + + + + numbeps(3) = 大ãã• 2 x 3ã®epsブロックã®ç•ªå· etc... + + + + + + numbeta(1) = 大ãã• 1 x 0ã®etaブロックã®ç•ªå· + + + + + + numbeta(2) = 大ãã• 2 x 1ã®etaブロックã®ç•ªå· + + + + + + numbeta(3) = 大ãã• 3 x 2ã®etaブロックã®ç•ªå· etc... + + + + + + ã“ã®ã‚³ãƒ¼ãƒ‰ã¯T. Beelen (Slicot-WGS group)ã«ã‚ˆã‚‹ã‚‚ã®ã§ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + gschur + + + + + + gspec + + + + + + systmat + + + + + + pencan + + + + + + randpencil + + + + + + trzeros + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/pencil/lyap.xml b/modules/linear_algebra/help/ja_JP/pencil/lyap.xml new file mode 100755 index 000000000..206c3f209 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/pencil/lyap.xml @@ -0,0 +1,143 @@ + + + + + + + + + lyap + + ãƒªã‚¢ãƒ—ãƒŽãƒ•æ–¹ç¨‹å¼ + + + + + + 呼ã³å‡ºã—手順 + + [X]=lyap(A,C,'c') + + [X]=lyap(A,C,'d') + + + + + + + + 引数 + + + + + + A, C + + + + + + 実数正方行列, C ã¯å¯¾ç§°ã§ã‚ã‚‹ã“ã¨ãŒå¿…è¦ + + + + + + + + + + + + + + 説明 + + + + X= lyap(A,C,flag) ã¯é€£ç¶šæ™‚é–“ã¾ãŸã¯é›¢æ•£æ™‚間行列 + + リアプノフ方程å¼ã‚’解ãã¾ã™: + + + + + + + + + + Aã®å›ºæœ‰å€¤ãŒ-Aã® + + 固有値ã§ãªã„å ´åˆ(flag='c') + + ã¾ãŸã¯Aã®å›ºæœ‰å€¤åˆ†ã®1ã®å ´åˆ + + (flag='d')ã«ã®ã¿ãƒ¦ãƒ‹ãƒ¼ã‚¯ãª + + 解ãŒå¾—られるã“ã¨ã«æ³¨æ„ã—ã¦ãã ã•ã„. + + + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + sylv + + + + + + ctr_gram + + + + + + obs_gram + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/pencil/pencan.xml b/modules/linear_algebra/help/ja_JP/pencil/pencan.xml new file mode 100755 index 000000000..842d32110 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/pencil/pencan.xml @@ -0,0 +1,191 @@ + + + + + + + + + pencan + + 行列ペンシルã®æ­£æº–å½¢ + + + + + + 呼ã³å‡ºã—手順 + + [Q,M,i1]=pencan(Fs) + + [Q,M,i1]=pencan(E,A) + + + + + + + + 引数 + + + + + + Fs + + + + + + 標準ペンシル s*E-A + + + + + + + + + + E,A + + + + 実数正方行列 + + + + + + + + Q,M + + + + 正則実数行列 + + + + + + + + i1 + + + + æ•´æ•° + + + + + + + + + + + + 説明 + + + + 標準ペンシル Fs=s*E-Aを指定ã™ã‚‹ã¨, + + pencan ã¯, + + M*(s*E-A)*QãŒ"正準"å½¢ã¨ãªã‚‹ã‚ˆã†ãª + + 行列Q ãŠã‚ˆã³Mã‚’è¿”ã—ã¾ã™. + + + + + + M*E*Q ã¯ãƒ–ロック行列ã§ã™. + + + + + + + + ãŸã ã—,Nã¯å†ªé›¶è¡Œåˆ—N, + + i1 = 行列Iã®å¤§ãã•ã§ã™. + + + + + + M*A*Q ã¯ä»¥ä¸‹ã®ã‚ˆã†ãªãƒ–ロック行列ã§ã™: + + + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + glever + + + + + + penlaur + + + + + + rowshuff + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/pencil/penlaur.xml b/modules/linear_algebra/help/ja_JP/pencil/penlaur.xml new file mode 100755 index 000000000..33556c0e8 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/pencil/penlaur.xml @@ -0,0 +1,224 @@ + + + + + + + + + penlaur + + 行列ペンシルã®ãƒ­ãƒ¼ãƒ©ãƒ³ä¿‚æ•° + + + + + + 呼ã³å‡ºã—手順 + + [Si,Pi,Di,order]=penlaur(Fs) + + [Si,Pi,Di,order]=penlaur(E,A) + + + + + + + + 引数 + + + + + + Fs + + + + + + 標準ペンシル s*E-A + + + + + + + + + + E, A + + + + 実数正方行列 + + + + + + + + Si,Pi,Di + + + + 実数正方行列 + + + + + + + + order + + + + æ•´æ•° + + + + + + + + + + + + 説明 + + + + penlaur ã¯,ç„¡é™å¤§ã«ãŠã‘ã‚‹ + + (s*E-A)^-1ã®ç¬¬ä¸€ãƒ­ãƒ¼ãƒ©ãƒ³ä¿‚æ•°ã‚’ + + 計算ã—ã¾ã™. + + + + + + s = ç„¡é™å¤§ ã«ãŠã„ã¦, + + (s*E-A)^-1 = ... + Si/s - Pi - s*Di + ... + + + + + + order = 特異点ã®æ¬¡æ•° (order=index-1). + + + + + + 行列ペンシル Fs=s*E-A ã¯å¯é€†ã§ã‚ã‚‹å¿…è¦ãŒã‚ã‚Šã¾ã™. + + + + + + 添字0ã®ãƒšãƒ³ã‚·ãƒ«ã®å ´åˆ, + + Pi, Di,... 㯠0,ãŠã‚ˆã³ Si=inv(E) + + ã¨ãªã‚Šã¾ã™. + + + + + + 添字1ã®ãƒšãƒ³ã‚·ãƒ«(order=0)ã®å ´åˆ, + + Di =0 ã¨ãªã‚Šã¾ã™. + + + + + + より大ããªæ·»å­—ã®ãƒšãƒ³ã‚·ãƒ«ã®å ´åˆ, + + é … -s^2 Di(2), -s^3 Di(3),... ã¯æ¬¡ã®ã‚ˆã†ã«æŒ‡å®šã•ã‚Œã¾ã™: + + + + + + Di(2)=Di*A*Di, Di(3)=Di*A*Di*A*Di (最大 Di(order)). + + + + + + + + æ³¨æ„ + + + + 実験的ãªãƒãƒ¼ã‚¸ãƒ§ãƒ³: so*E-Aã®æ¡ä»¶æ•°ãŒæ‚ªã„å ´åˆã« + + å•é¡Œã‚’発生ã—ã¾ã™ + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + glever + + + + + + pencan + + + + + + rowshuff + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/pencil/quaskro.xml b/modules/linear_algebra/help/ja_JP/pencil/quaskro.xml new file mode 100755 index 000000000..cee463831 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/pencil/quaskro.xml @@ -0,0 +1,249 @@ + + + + + + + + + quaskro + + 準クロãƒãƒƒã‚«ãƒ¼å½¢å¼ + + + + + + 呼ã³å‡ºã—手順 + + [Q,Z,Qd,Zd,numbeps,numbeta]=quaskro(F) + + [Q,Z,Qd,Zd,numbeps,numbeta]=quaskro(E,A) + + [Q,Z,Qd,Zd,numbeps,numbeta]=quaskro(F,tol) + + [Q,Z,Qd,Zd,numbeps,numbeta]=quaskro(E,A,tol) + + + + + + + + 引数 + + + + + + F + + + + + + 実数行列ペンシル F=s*E-A (s=poly(0,'s')) + + + + + + + + + + E,A + + + + åŒã˜æ¬¡å…ƒã®å®Ÿæ•°è¡Œåˆ— + + + + + + + + tol + + + + 実数 (許容誤差,デフォルト値=1.d-10) + + + + + + + + Q,Z + + + + 正方直交行列 + + + + + + + + Qd,Zd + + + + 整数ベクトル + + + + + + + + numbeps + + + + 整数ベクトル + + + + + + + + + + + + 説明 + + + + 行列ペンシルã®æº–クロãƒãƒƒã‚«ãƒ¼å½¢å¼: + + quaskroã¯,ペンシルF=s*E -Aを上三角行列形å¼ã«å¤‰æ›ã™ã‚‹ + + 直交行列 Q, Zを計算ã—ã¾ã™: + + + + + + + + ブロックã®æ¬¡å…ƒã¯æ¬¡ã®ã‚ˆã†ã«æŒ‡å®šã•ã‚Œã¾ã™: + + + + + + eps=Qd(1) x Zd(1), inf=Qd(2) x Zd(2), + + r = Qd(3) x Zd(3) + + + + + + inf ブロックã«ã¯, ペンシルã®ç„¡é™å¤§ãƒ¢ãƒ¼ãƒ‰ãŒå«ã¾ã‚Œã¾ã™. + + + + + + f ブロックã«ã¯, ペンシルã®æœ‰é™ãƒ¢ãƒ¼ãƒ‰ãŒå«ã¾ã‚Œã¾ã™. + + + + + + epsilonブロックã®æ§‹é€ ã¯æ¬¡ã®ã‚ˆã†ã«æŒ‡å®šã•ã‚Œã¾ã™: + + + + + + numbeps(1) = 大ãã• 0 x 1ã®epsブロックã®æ•° + + + + + + numbeps(2) = 大ãã• 1 x 2ã®epsブロックã®æ•° + + + + + + numbeps(3) = 大ãã• 2 x 3ã®epsブロックã®æ•° etc... + + + + + + 完全ãª(4ブロックã®)クロãƒãƒƒã‚«ãƒ¼å½¢å¼ã¯, + + (pertransposed)ペンシルsE(r)-A(r)を指定ã—㦠+ + quaskroをコールã™ã‚‹ + + 関数kroneckã«ã‚ˆã‚ŠæŒ‡å®šã•ã‚Œã¾ã™. + + + + + + ã“ã®ã‚³ãƒ¼ãƒ‰ T. Beelenã«ã‚ˆã‚‹ã‚‚ã®ã§ã™. + + + + + + + + å‚ç…§ + + + + + + kroneck + + + + + + gschur + + + + + + gspec + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/pencil/randpencil.xml b/modules/linear_algebra/help/ja_JP/pencil/randpencil.xml new file mode 100755 index 000000000..e83731339 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/pencil/randpencil.xml @@ -0,0 +1,207 @@ + + + + + + + + + randpencil + + ランダムãªãƒšãƒ³ã‚·ãƒ« + + + + + + 呼ã³å‡ºã—手順 + + F=randpencil(eps,infi,fin,eta) + + + + + + 引数 + + + + + + eps + + + + 整数ベクトル + + + + + + + + infi + + + + 整数ベクトル + + + + + + + + fin + + + + 実数ベクトル, ã¾ãŸã¯ãƒ¢ãƒ‹ãƒƒã‚¯ãªå¤šé …å¼, ã¾ãŸã¯ãƒ¢ãƒ‹ãƒƒã‚¯ãªå¤šé …å¼ã®ãƒ™ã‚¯ãƒˆãƒ« + + + + + + + + eta + + + + 整数ベクトル + + + + + + + + F + + + + + + 実数行列ペンシル F=s*E-A (s=poly(0,'s')) + + + + + + + + + + + + + + 説明 + + + + ユーティリティ関数. + + F=randpencil(eps,infi,fin,eta) ã¯,指定ã—ãŸã‚¯ãƒ­ãƒãƒƒã‚«ãƒ¼æ§‹é€ ã‚’有ã™ã‚‹ + + ランダムãªãƒšãƒ³ã‚·ãƒ« Fã‚’è¿”ã—ã¾ã™. + + 構造ã¯ä»¥ä¸‹ã®ã‚ˆã†ã«æŒ‡å®šã•ã‚Œã¾ã™: + + eps=[eps1,...,epsk]: epsilon ブロックã®æ§‹é€  (大ãã• eps1 x(eps1+1),....) + + fin=[l1,...,ln] 有é™ã®å›ºæœ‰å€¤ã®é›†åˆ (実数ã¨ä»®å®š)( []ã®å ´åˆã‚‚ã‚ã‚Šã¾ã™) + + infi=[k1,...,kp] ç„¡é™å¤§ã«ãŠã‘ã‚‹J-ブロックã®å¤§ãã• + + ki>=1 (J ブロックãŒãªã„å ´åˆ: infi=[] ). + + eta=[eta1,...,etap]: ofeta ブロック構造 (大ãã• (eta1+1)x eta1,...) + + + + + + epsi >=0ã§ã‚ã‚‹å¿…è¦ãŒã‚ã‚Š, + + etai >=0ã§ã‚ã‚‹å¿…è¦ãŒã‚ã‚Š, infi + + >=1ã§ã‚ã‚‹å¿…è¦ãŒã‚ã‚Šã¾ã™. + + + + + + fin ㌠(モニックãª) 多項å¼ã®å ´åˆ, + + 有é™ã®ãƒ–ロックã¯finã®æ ¹ã‚’固有値ã¨ã—ã¦è¨±å®¹ã—ã¾ã™. + + + + + + fin ãŒå¤šé …å¼ã®ãƒ™ã‚¯ãƒˆãƒ«ã®å ´åˆ, + + ã“れらã¯Fã®æœ‰é™å˜å› å­,ã™ãªã‚ã¡,p(i)ã®æ ¹ã¯ + + Fã®æœ‰é™ãªå›ºæœ‰å€¤ã¨ãªã‚Šã¾ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + kroneck + + + + + + pencan + + + + + + penlaur + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/pencil/rowshuff.xml b/modules/linear_algebra/help/ja_JP/pencil/rowshuff.xml new file mode 100755 index 000000000..3f6b66c49 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/pencil/rowshuff.xml @@ -0,0 +1,196 @@ + + + + + + + + + rowshuff + + シャッフルアルゴリズム + + + + + + 呼ã³å‡ºã—手順 + + [Ws,Fs1]=rowshuff(Fs, [alfa]) + + + + + + 引数 + + + + + + Fs + + + + + + 正方実数ペンシル Fs = s*E-A + + + + + + + + + + Ws + + + + 多項å¼è¡Œåˆ— + + + + + + + + Fs1 + + + + + + 正方実数ペンシル F1s = s*E1 -A1, + + ãŸã ã— E1 ã¯æ­£å‰‡ + + + + + + + + + + alfa + + + + + + 実数 (alfa = 0 ãŒãƒ‡ãƒ•ã‚©ãƒ«ãƒˆå€¤) + + + + + + + + + + + + + + 説明 + + + + シャッフルアルゴリズム: ペンシル Fs=s*E-Aを指定ã™ã‚‹ã¨, + + 以下ã®ã‚ˆã†ãª(正方多項å¼è¡Œåˆ—) Ws=W(s) ã‚’è¿”ã—ã¾ã™: + + + + + + Fs1 = s*E1-A1 = W(s)*(s*E-A) ã¯, + + 正則ãªãƒšãƒ³ã‚·ãƒ«è¡Œåˆ— E1 ã§ã™. + + + + + + ã“ã‚Œã¯,ペンシルFs = s*E-AãŒæ­£å‰‡ (ã™ãªã‚ã¡,å¯é€†)ã®å ´åˆã«é™ã‚Šå¯èƒ½ã§ã™. + + Ws ã®æ¬¡æ•°ã¯ãƒšãƒ³ã‚·ãƒ«ã®æ·»å­—ã«ç­‰ã—ããªã‚Šã¾ã™. + + + + + + Fsã®ç„¡é™å¤§ã«ã‚る極ã¯alfaã«é…ç½®ã•ã‚Œ, + + Wsã®ã‚¼ãƒ­ã¯alfaã«é…ç½®ã•ã‚Œã¾ã™. + + + + + + (s*E-A)^-1 = (s*E1-A1)^-1 * W(s) = (W(s)*(s*E-A))^-1 *W(s) + + ã¨ãªã‚‹ã“ã¨ã«æ³¨æ„ã—ã¦ãã ã•ã„. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + pencan + + + + + + glever + + + + + + penlaur + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/pencil/sylv.xml b/modules/linear_algebra/help/ja_JP/pencil/sylv.xml new file mode 100755 index 000000000..185456e74 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/pencil/sylv.xml @@ -0,0 +1,132 @@ + + + + + + + + + sylv + + シルベスタ方程å¼. + + + + + + 呼ã³å‡ºã—手順 + + sylv(A, B, C, flag) + + + + + + 引数 + + + + + + A,B,C + + + + é©å½“ãªæ¬¡å…ƒã®å®Ÿæ•°è¡Œåˆ—. + + + + + + + + flag + + + + + + 文字列 ('c' ã¾ãŸã¯ 'd') + + + + + + + + + + + + + + 説明 + + + + X= sylv(A,B,C,'c') ã¯ä»¥ä¸‹ã® + + "連続時間" シルベスタ方程å¼ã®è§£Xを計算ã—ã¾ã™, + + + + + + + + X=sylv(A,B,C,'d') ã¯ä»¥ä¸‹ã® + + "離散時間" シルベスタ方程å¼ã®è§£Xを計算ã—ã¾ã™, + + + + + + + + + + 例 + + + + + + + + å‚考 + + + + + + lyap + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/proj.xml b/modules/linear_algebra/help/ja_JP/proj.xml new file mode 100755 index 000000000..6346f8afd --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/proj.xml @@ -0,0 +1,129 @@ + + + + + + + + + proj + + 投影 + + + + + + 呼ã³å‡ºã—手順 + + P = proj(X1,X2) + + + + + + 引数 + + + + + + X1,X2 + + + + åŒã˜åˆ—ã®æ•°ã‚’有ã™ã‚‹å®Ÿæ•°ã®è¡Œåˆ— + + + + + + + + P + + + + + + 実数ã®æŠ•å½±è¡Œåˆ— (P^2=P) + + + + + + + + + + + + + + 説明 + + + + P ã¯X1ã«å¹³è¡Œãª + + X2ã¸ã®æŠ•å½±ã§ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + projspec + + + + + + orth + + + + + + fullrf + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/state_space/CHAPTER b/modules/linear_algebra/help/ja_JP/state_space/CHAPTER new file mode 100755 index 000000000..a0b62cdee --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/state_space/CHAPTER @@ -0,0 +1,2 @@ +title = State-Space Matrices + diff --git a/modules/linear_algebra/help/ja_JP/state_space/coff.xml b/modules/linear_algebra/help/ja_JP/state_space/coff.xml new file mode 100755 index 000000000..ebf5b06cc --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/state_space/coff.xml @@ -0,0 +1,183 @@ + + + + + + + + + coff + + レゾルベント (余因å­æ³•) + + + + + + 呼ã³å‡ºã—手順 + + [N,d]=coff(M [,var]) + + + + + + 引数 + + + + + + M + + + + 正方実数行列 + + + + + + + + var + + + + 文字列 + + + + + + + + N + + + + + + 多項å¼è¡Œåˆ— (Mã¨åŒã˜å¤§ãã•) + + + + + + + + + + d + + + + + + å¤šé …å¼ (ç‰¹æ€§å¤šé …å¼ poly(A,'s')) + + + + + + + + + + + + + + 説明 + + + + coff ã¯, + + 実数行列 Mã«ã¤ã„㦠R=(s*eye()-M)^-1 + + を計算ã—ã¾ã™. + + R 㯠N/dã§æŒ‡å®šã•ã‚Œã¾ã™. + + + + + + N = 多項å¼è¡Œåˆ—ã®åˆ†å­. + + + + + + d = 共通分æ¯. + + + + + + var 文字列 (çœç•¥æ™‚ã¯'s') + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + coffg + + + + + + ss2tf + + + + + + nlev + + + + + + poly + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/state_space/nlev.xml b/modules/linear_algebra/help/ja_JP/state_space/nlev.xml new file mode 100755 index 000000000..996d6e566 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/state_space/nlev.xml @@ -0,0 +1,161 @@ + + + + + + + + + nlev + + Leverrierã®ã‚¢ãƒ«ã‚´ãƒªã‚ºãƒ  + + + + + + 呼ã³å‡ºã—手順 + + [num,den]=nlev(A,z [,rmax]) + + + + + + 引数 + + + + + + A + + + + 実数正方行列 + + + + + + + + z + + + + 文字列 + + + + + + + + rmax + + + + + + オプションã®ãƒ‘ラメータ (bdiagå‚ç…§) + + + + + + + + + + + + + + 説明 + + + + [num,den]=nlev(A,z [,rmax]) ã¯, + + (z*eye()-A)^(-1)を計算ã—ã¾ã™. + + + + + + 計算ã®éš›ã«ã¯,Aã®ãƒ–ロック対角化ã®å¾Œã« + + Leverrierアルゴリズムをå„ブロックã«é©ç”¨ã—ã¾ã™. + + + + + + ã“ã®ã‚¢ãƒ«ã‚´ãƒªã‚ºãƒ ã¯é€šå¸¸ã® Leverrierアルゴリズム + + より優れã¦ã„ã¾ã™ãŒ,ã¾ã å®Œå…¨ã§ã¯ã‚ã‚Šã¾ã›ã‚“! + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + coff + + + + + + coffg + + + + + + glever + + + + + + ss2tf + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/subspaces/CHAPTER b/modules/linear_algebra/help/ja_JP/subspaces/CHAPTER new file mode 100755 index 000000000..d87d9ca5e --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/subspaces/CHAPTER @@ -0,0 +1,3 @@ +title = Subspaces + + diff --git a/modules/linear_algebra/help/ja_JP/subspaces/spaninter.xml b/modules/linear_algebra/help/ja_JP/subspaces/spaninter.xml new file mode 100755 index 000000000..0ec8c806c --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/subspaces/spaninter.xml @@ -0,0 +1,169 @@ + + + + + + + + + spaninter + + 共通部分空間 + + + + + + 呼ã³å‡ºã—手順 + + [X,dim]=spaninter(A,B [,tol]) + + + + + + 引数 + + + + + + A, B + + + + åŒæ•°ã®è¡Œã‚’有ã™ã‚‹å®Ÿæ•°ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ— + + + + + + + + X + + + + 直交ã¾ãŸã¯ãƒ¦ãƒ‹ã‚¿ãƒªæ­£æ–¹è¡Œåˆ— + + + + + + + + dim + + + + + + æ•´æ•°, range(A) ãŠã‚ˆã³ + + range(B)é–“ã®å…±é€šéƒ¨åˆ†ç©ºé–“ã®æ¬¡å…ƒ + + + + + + + + + + + + + + 説明 + + + + range(A) ãŠã‚ˆã³ range(B)ã® + + 共通部分を計算ã—ã¾ã™. + + + + + + Xã®æœ€åˆã®dim列ã¯,ã“ã®å…±é€šéƒ¨åˆ†ã«å±•é–‹ã—ã¾ã™. + + ã™ãªã‚ã¡,X(:,1:dim)ã¯, + + range(A) 㨠range(B)ã®é–“ã®ç›´äº¤åŸºåº•ã§ã™. + + + + + + Xã®åŸºåº•ã§ã¯, + + A ãŠã‚ˆã³ BãŒãã‚Œãžã‚Œæ¬¡ã®ã‚ˆã†ã« + + 表ã•ã‚Œã¾ã™: + + + + + + X'*A ãŠã‚ˆã³ X'*B. + + + + + + tol ã¯é–¾å€¤ã§ã™ (sqrt(%eps) ãŒãƒ‡ãƒ•ã‚©ãƒ«ãƒˆå€¤ã§ã™). + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + spanplus + + + + + + spantwo + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/subspaces/spanplus.xml b/modules/linear_algebra/help/ja_JP/subspaces/spanplus.xml new file mode 100755 index 000000000..b8717b88e --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/subspaces/spanplus.xml @@ -0,0 +1,184 @@ + + + + + + + + + spanplus + + 部分空間ã®åˆè¨ˆ + + + + + + 呼ã³å‡ºã—手順 + + [X,dim,dima]=spanplus(A,B[,tol]) + + + + + + 引数 + + + + + + A, B + + + + åŒæ•°ã®è¡Œã‚’有ã™ã‚‹å®Ÿæ•°ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ— + + + + + + + + X + + + + 直交ã¾ãŸã¯ãƒ¦ãƒ‹ã‚¿ãƒªæ­£æ–¹è¡Œåˆ— + + + + + + + + dim, dima + + + + æ•´æ•°, 部分空間ã®æ¬¡å…ƒ + + + + + + + + tol + + + + éžè² ã®å®Ÿæ•° + + + + + + + + + + + + 説明 + + + + 以下ã®æ§˜ãªåŸºåº•Xを計算ã—ã¾ã™: + + + + + + Xã®æœ€åˆã®dima列㌠+ + range(A)ã«å±•é–‹ã—, + + ã“れ以é™ã®(dim-dima)列ãŒ, + + A+B ã®Aã«é–¢ã™ã‚‹åŸºåº•ã‚’構æˆ. + + + + + + Xã®æœ€åˆã®dim列ã¯, + + A+Bã®åŸºåº•ã‚’構æˆã—ã¾ã™. + + + + + + [A,B]ã«é–¢ã™ã‚‹ä»¥ä¸‹ã®æ­£æº–å½¢å¼ãŒå®šç¾©ã•ã‚Œã¾ã™: + + + + + + + + tol ã¯ã‚ªãƒ—ションã®å¼•æ•°ã§ã™(関数ã®ã‚³ãƒ¼ãƒ‰ã‚’å‚ç…§). + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + spaninter + + + + + + im_inv + + + + + + spantwo + + + + + + + + + diff --git a/modules/linear_algebra/help/ja_JP/subspaces/spantwo.xml b/modules/linear_algebra/help/ja_JP/subspaces/spantwo.xml new file mode 100755 index 000000000..cca713c63 --- /dev/null +++ b/modules/linear_algebra/help/ja_JP/subspaces/spantwo.xml @@ -0,0 +1,202 @@ + + + + + + + + + spantwo + + 部分空間ã®åˆè¨ˆãŠã‚ˆã³å…±é€šéƒ¨åˆ† + + + + + + 呼ã³å‡ºã—手順 + + [Xp,dima,dimb,dim]=spantwo(A,B, [tol]) + + + + + + 引数 + + + + + + A, B + + + + åŒæ•°ã®è¡Œã‚’有ã™ã‚‹å®Ÿæ•°ã¾ãŸã¯è¤‡ç´ æ•°ã®è¡Œåˆ— + + + + + + + + Xp + + + + 正方正則行列 + + + + + + + + dima, dimb, dim + + + + æ•´æ•°, 部分空間ã®æ¬¡å…ƒ + + + + + + + + tol + + + + éžè² ã®å®Ÿæ•° + + + + + + + + + + + + 説明 + + + + åŒã˜è¡Œæ•°ã‚’有ã™ã‚‹è¡Œåˆ— A ãŠã‚ˆã³ B を指定ã™ã‚‹ã¨, + + 以下ã®ã‚ˆã†ãªæ­£æ–¹è¡Œåˆ—Xp (正則ã ãŒç›´äº¤ã§ã‚ã‚‹å¿…è¦ã¯ã‚ã‚Šã¾ã›ã‚“) + + ã‚’è¿”ã—ã¾ã™: + + + + + + + + inv(Xp)ã®æœ€åˆã®dima列㯠+ + range(A)ã«å±•é–‹ã•ã‚Œã¾ã™. + + + + + + inv(Xp)ã® + + 列 dim-dimb+1 ã‹ã‚‰ dima 㯠+ + range(A) 㨠range(B)ã®å…±é€šéƒ¨åˆ†ã«å±•é–‹ã•ã‚Œã¾ã™. + + + + + + inv(Xp)ã®æœ€åˆã®dim列ã¯, + + range(A)+range(B)ã«å±•é–‹ã•ã‚Œã¾ã™. + + + + + + inv(Xp)ã®åˆ—dim-dimb+1ã‹ã‚‰dim + + ã¯range(B)ã«å±•é–‹ã•ã‚Œã¾ã™. + + + + + + 行列 [A1;A2] ã¯è¡Œãƒ•ãƒ«ãƒ©ãƒ³ã‚¯(=rank(A))ã§ã™. + + 行列[B2;B3]ã¯è¡Œãƒ•ãƒ«ãƒ©ãƒ³ã‚¯(=rank(B))ã§ã™. + + 行列[A2,B2]ã¯è¡Œãƒ•ãƒ«ãƒ©ãƒ³ã‚¯(=rank(A inter B))ã§ã™. + + è¡Œ[A1,0;A2,B2;0,B3] ã¯è¡Œãƒ•ãƒ«ãƒ©ãƒ³ã‚¯(=rank(A+B))ã§ã™. + + + + + + + + 例 + + + + + + + + å‚ç…§ + + + + + + spanplus + + + + + + spaninter + + + + + + + + + diff --git a/modules/linear_algebra/help/pt_BR/addchapter.sce b/modules/linear_algebra/help/pt_BR/addchapter.sce new file mode 100755 index 000000000..99f821fe3 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/addchapter.sce @@ -0,0 +1,11 @@ +// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +// Copyright (C) 2009 - DIGITEO +// +// This file must be used under the terms of the CeCILL. +// This source file is licensed as described in the file COPYING, which +// you should have received as part of this distribution. The terms +// are also available at +// http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt + +add_help_chapter("Linear Algebra",SCI+"/modules/linear_algebra/help/pt_BR",%T); + diff --git a/modules/linear_algebra/help/pt_BR/eigen/CHAPTER b/modules/linear_algebra/help/pt_BR/eigen/CHAPTER new file mode 100755 index 000000000..88f8bc42b --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/eigen/CHAPTER @@ -0,0 +1,2 @@ +title = Eigenvalue and Singular Value + diff --git a/modules/linear_algebra/help/pt_BR/eigen/balanc.xml b/modules/linear_algebra/help/pt_BR/eigen/balanc.xml new file mode 100755 index 000000000..670af0f56 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/eigen/balanc.xml @@ -0,0 +1,109 @@ + + + + + balanc + balanceamento de uma matriz ou de um feixe de + matrizes + + + + Seqüência de Chamamento + [Ab,X]=balanc(A) + [Eb,Ab,X,Y]=balanc(E,A) + + + + Parâmetros + + + A: + + matriz quadrada de reais + + + + X: + + matriz quadrada de reais invertível + + + + E: + + matriz quadrada de reais (com mesma dimensão que + A) + + + + + Y: + + uma matriz quadrada de reais invertível + + + + + + Descrição + Balanceia uma matriz quadrada para melhorar seu número de + condicionamento. + + + [Ab,X] = balanc(A) acha uma transformação de + similaridade X tal que + + + Ab = inv(X)*A*X tem aproximadamente normas iguais + de linha e de coluna. + + Para feixes de matrizes, o balancemento é feito para melhorar o + problema do autovalor generalizado. + + + [Eb,Ab,X,Y] = balanc(E,A) retorna transformações + esquerda e direita X e Y tais que + Eb=inv(X)*E*Y, Ab=inv(X)*A*Y + + + + Observação + + O balanceamento é feito nas funções bdiag e + spec. + + + + Exemplos + + + + Ver Também + + + bdiag + + + spec + + + schur + + + + diff --git a/modules/linear_algebra/help/pt_BR/eigen/bdiag.xml b/modules/linear_algebra/help/pt_BR/eigen/bdiag.xml new file mode 100755 index 000000000..94f586ba1 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/eigen/bdiag.xml @@ -0,0 +1,108 @@ + + + + + bdiag + diagonalização em blocos, autovetores + generalizados + + + + Seqüência de Chamamento + [Ab [,X [,bs]]]=bdiag(A [,rmax]) + + + Parâmetros + + + A + + matriz quadrada de reais ou complexos + + + + rmax + + número real + + + + Ab + + matriz quadrada de reais ou complexos + + + + X + + matriz de reais ou complexos não-singular + + + + bs + + vetor de inteiros + + + + + + Descrição + + + realiza a diagonalização em blocos da matriz A. + bs fornece a estrutura dos blocos (respectivos tamanhos dos blocos). + X é a matriz mudança de base i.e Ab = + inv(X)*A*X + + édiagonal em blocos. + + + rmax controla o condicionamento de + X; o valor padrão é a norma L1 de + A. + + Para encontrar a forma diagonal (se existir) escolha um valor + suficientemente grande para rmax + (rmax=1/%eps , por exemplo). Genericamente, (para uma + matriz A de reais aleatória) os blocos são (1x1) e (2x2) e + X é a matriz de autovetores. + + + + Exemplos + + + + Ver Também + + + schur + + + sylv + + + spec + + + + diff --git a/modules/linear_algebra/help/pt_BR/eigen/gschur.xml b/modules/linear_algebra/help/pt_BR/eigen/gschur.xml new file mode 100755 index 000000000..2b3515dad --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/eigen/gschur.xml @@ -0,0 +1,97 @@ + + + + + gschur + forma de Schur generalizada (função obsoleta) + + + Seqüência de Chamamento + [As,Es]=gschur(A,E) + [As,Es,Q,Z]=gschur(A,E) + [As,Es,Z,dim] = gschur(A,E,flag) + [As,Es,Z,dim]= gschur(A,E,extern) + + + + Descrição + Esta função é obsoleta e agora está inclusa na função + schur function. Na maior parte dos casos, a função + gschur irá ainda trabalhar como antes, mas será + removida em um lançamento futuro do Scilab. + + As três sintaxes podem ser substituídas por + + A última sintaxe requer algumas adaptações a mais: + + + if + + é uma função do Scilab, a nova seqüência de chamamento deve + ser [As,Es,Z,dim]= schur(A,E,Nextern) com Nextern + definido como segue: + + + + + + if + + é o nome de uma função externa codificada em FORTRAN ou C, a + nova seqüência de chamamento deve ser [As,Es,Z,dim]= + schur(A,E,'nextern') + + com nextern definido como + segue: + + + + + + + + Ver Também + + + external + + + schur + + + + diff --git a/modules/linear_algebra/help/pt_BR/eigen/gspec.xml b/modules/linear_algebra/help/pt_BR/eigen/gspec.xml new file mode 100755 index 000000000..ae0f5b92f --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/eigen/gspec.xml @@ -0,0 +1,45 @@ + + + + + gspec + autovalores de feixe de matrizes (função + obsoleta) + + + + Seqüência de Chamamento + [al,be]=gspec(A,E) + [al,be,Z]=gspec(A,E) + + + + Descrição + + Esta função está agora inclusa na função spec . A + seqüência de chamamento deve ser substituida por + + + + + Ver Também + + + spec + + + + diff --git a/modules/linear_algebra/help/pt_BR/eigen/hess.xml b/modules/linear_algebra/help/pt_BR/eigen/hess.xml new file mode 100755 index 000000000..b9daf0150 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/eigen/hess.xml @@ -0,0 +1,91 @@ + + + + + hess + forma de Hessenberg + + + Seqüência de Chamamento + H = hess(A) + [U,H] = hess(A) + + + + Parâmetros + + + A + + matriz quadrada de reais ou complexos + + + + H + + matriz quadrada de reais ou complexos + + + + U + + matriz quadrada ortogonal ou unitária + + + + + + Descrição + + [U,H] = hess(A) produz uma matriz unitária + U e uma matriz de Hessenberg H tais + que A = U*H*U' e U'*U = Identidade. + Por si só, hess(A) retorna H. + + A forma de Hessenberg de uma matriz é zero abaixo da primeira + subdiagonal. Se a matriz é simetrica ou Hermitiana, a forma é + tridiagonal. + + + + Referências + A função hess é baseada nas rotinas Lapack DGEHRD, DORGHR para + matrizes de e ZGEHRD, ZORGHR para matrizes de complexos. + + + + Exemplos + + + + Ver Também + + + qr + + + contr + + + schur + + + + + + <para/> + </refsection> +</refentry> diff --git a/modules/linear_algebra/help/pt_BR/eigen/pbig.xml b/modules/linear_algebra/help/pt_BR/eigen/pbig.xml new file mode 100755 index 000000000..15b493fac --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/eigen/pbig.xml @@ -0,0 +1,125 @@ +<?xml version="1.0" encoding="ISO-8859-1"?> +<!-- + * Scilab ( http://www.scilab.org/ ) - This file is part of Scilab + * Copyright (C) 2008 - INRIA + * + * This file must be used under the terms of the CeCILL. + * This source file is licensed as described in the file COPYING, which + * you should have received as part of this distribution. The terms + * are also available at + * http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt + * + --> +<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ns3="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org" xml:id="pbig" xml:lang="en"> + <refnamediv> + <refname>pbig</refname> + <refpurpose>autoprojeção</refpurpose> + </refnamediv> + <refsynopsisdiv> + <title> Seqüência de Chamamento + [Q,M]=pbig(A,thres,flag) + + + Parâmetros + + + A + + matriz quadrada de reais + + + + thres + + número real + + + + flag + + + string ('c' ou + 'd') + + + + + Q,M + + matrizes de reais + + + + + + Descrição + Projeção sobre um auto-subespaço associado aos autovalores de parte + real>= thres (flag='c') ou com + magnitude >= thres + (flag='d'). + + + A projeção é definida por Q*M, + Q tem posto-coluna cheio, M tem + posto-linha cheio e M*Q=eye. + + + Se flag='c', os autovalores de + M*A*Q = autovalores de A com parte + real >= thres. + + + Se flag='d', os autovalores de + M*A*Q = autovalores de A com + magnitude >= thres. + + + Se flag='c' e se [Q1,M1] = + fatoração em posto cheio (fullrf) de + eye()-Q*M então os autovalores de + M1*A*Q1 = autovalores de A com parte + real < thres. + + + Se flag='d' e se [Q1,M1] + =fatoração em posto cheio (fullrf) de + eye()-Q*M então os autovalores de + M1*A*Q1 = autovalores de A com + magnitude < thres. + + + + Exemplos + + + + Ver Também + + + psmall + + + projspec + + + fullrf + + + schur + + + + + Função Usada + + pbig é baseada na forma ordenada de Schur (função + do Scilab schur). + + + diff --git a/modules/linear_algebra/help/pt_BR/eigen/projspec.xml b/modules/linear_algebra/help/pt_BR/eigen/projspec.xml new file mode 100755 index 000000000..a24fac61f --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/eigen/projspec.xml @@ -0,0 +1,93 @@ + + + + + projspec + operadores espectrais + + + Seqüência de Chamamento + [S,P,D,i]=projspec(A) + + + Parâmetros + + + A + + matriz quadrada + + + + S, P, D + + matrizes quadradas + + + + i + + inteiro (índice do autovalor zero de + A). + + + + + + + Descrição + + Características espectrais de A em 0. + + + S = resolvente reduzido em 0 + (S = -Inverso_de_Drazin(A)). + + + P = projeção espectral em 0. + + + D = operador nilpotente em 0. + + + index = índice do autovalor 0. + + + Tem-se (s*eye()-A)^(-1) = D^(i-1)/s^i +... + D/s^2 + P/s - + S - s*S^2 -... + + ao redor da singularidade s=0. + + + + Exemplos + + + + Ver Também + + + coff + + + + diff --git a/modules/linear_algebra/help/pt_BR/eigen/psmall.xml b/modules/linear_algebra/help/pt_BR/eigen/psmall.xml new file mode 100755 index 000000000..e187285bc --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/eigen/psmall.xml @@ -0,0 +1,120 @@ + + + + + psmall + projeção espectral + + + Calling Sequence + [Q,M]=psmall(A,thres,flag) + + + Parameters + + + A + + matriz quadrada de reais + + + + thres + + número real + + + + flag + + + string ('c' ou + 'd') + + + + + Q,M + + matrizes de reais + + + + + + Description + Projeção sobre auto-subespaço associado com autovalores com parte + real < thres (flag='c') ou com + módulo < thres (flag='d'). + + + A projeção é definda por Q*M, + Q é de posto-coluna cheio, M é de + posto-linha cheio e M*Q=eye. + + + Se flag='c', os autovalores de + M*A*Q = autovalores de A com parte + real < thres. + + + Se flag='d', os autovalores de + M*A*Q = autovalores de A com + magnitude < thres. + + + Se flag='c' e se [Q1,M1] = + fatoração em posto cheio (fullrf) de + eye()-Q*Mentão os autovalores de + M1*A*Q1 = autovalores de A com parte + real >= thres. + + + Se flag='d' e se [Q1,M1] = + fatoração em posto cheio (fullrf) de + eye()-Q*M então os autovalores de + M1*A*Q1 = autovalores de A com + magnitude >= thres. + + + + Examples + + + + See Also + + + pbig + + + proj + + + projspec + + + + + Used Functions + Esta função é baseada na forma de Schur ordenada (Função do + Scilab schur). + + + diff --git a/modules/linear_algebra/help/pt_BR/eigen/schur.xml b/modules/linear_algebra/help/pt_BR/eigen/schur.xml new file mode 100755 index 000000000..f89dd7791 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/eigen/schur.xml @@ -0,0 +1,411 @@ + + + + + schur + decomposição (ordenada) de Schur de matrizes e + feixes + + + + Seqüência de Chamamento + [U,T] = schur(A) + [U,dim [,T] ]=schur(A,flag) + [U,dim [,T] ]=schur(A,extern1) + + [As,Es [,Q,Z]]=schur(A,E) + [As,Es [,Q],Z,dim] = schur(A,E,flag) + [Z,dim] = schur(A,E,flag) + [As,Es [,Q],Z,dim]= schur(A,E,extern2) + [Z,dim]= schur(A,E,extern2) + + + + Parâmetros + + + A + + matriz quadrada de reais ou complexos + + + + E + + matriz quadrada de reais ou complexos com as mesmas dimensões + de A. + + + + + flag + + + string ('c' + ou'd') + + + + + extern1 + + uma ``external'' (função externa), veja abaixo + + + + extern2 + + uma ``external'', veja abaixo + + + + U + + matriz quadrada ortogonal ou unitária + + + + Q + + matriz quadrada ortogonal ou unitária + + + + Z + + matriz quadrada ortogonal ou unitária + + + + T + + matriz quadrada triangular superior ou quasi-triangular + + + + + As + + matriz quadrada triangular superior ou quasi-triangular + + + + + Es + + matriz quadrada triangular superior + + + + dim + + inteiro + + + + + + Descrição + Formas de Schur, formas ordenadas de Schur de matrizes e feixes + + + + FORMA DE SCHUR MATRICIAL + + + + Forma de Schur usual: + + + [U,T] = schur(A) produz uma matriz de + Schur T e uma matriz unitária + U tais que A = U*T*U' e + U'*U = eye(U). Por si mesmo, + schur(A) retorna T. Se + A é de complexos, a Forma de Schur Complexa + é retornada na matriz T. A Forma de Schur + Complexa é triangular superior com os autovalores de + A na diagonal. Se A é de + reais, a Forma de Schur Real é retornada. A Forma de Schur + Real tem autovalores reais na diagonal e os autovalores + complexos em blocos 2-por-2 na diagonal. + + + + + Formas de Schur ordenadas + + + [U,dim]=schur(A,'c') rretorna uma + matriz unitária U que transforma + A em uma forma de Schur. Ainda, as + primeiras dim colunas de U formam uma base + para o autoespaço de A associado aos + autovalores com partes reais negativas (autoespaço de "tempo + contínuo" estável). + + + [U,dim]=schur(A,'d') retorna uma + matriz unitária U que transforma + A em uma forma de Schur. Ainda, as + primeiras dim colunas de + U geram uma base do autoespaço de + A associado aos autovalores de magnitude + menor que 1 (autoespaço de "tempo discreto" estável). + + + [U,dim]=schur(A,extern1) retorna uma + matriz unitária U que transforma + A em uma forma de Schur. Ainda, as + dim primeiras colunas de + U geram uma base para o autoespaço de + A associado aos autovalores que são + selecionados pela "external" extern1 (veja + "external" para detalhes). Esta "external" pode ser descrita + por uma função do Scilab ou por um "procedure" de C ou + FORTRAN: + + + + Uma Função do Scilab + + + Se extern1 é descrita por uma + função do Scilab, deve ter a seguinte seqüência de + chamamento: s=extern1(Ev), onde + Ev é um autovalor e + s um booleano. + + + + + Um "Procedure" C ou FORTRAN + + + Se extern1 é descrita por uma + função de C ou FORTRAN, deve ter a seguinte seqüência de + chamamento: int extern1(double *EvR, double + *EvI) + + onde EvR e + EvI são partes real e complexa de + autovalor. Valor verdadeiro ou diferente de zero + retornado significa autovalor selecionado. + + + + + + + + + + + FORMAS DE SCHUR DE FEIXES + + + + Forma de Schur de Feixe Usual + + + [As,Es] = schur(A,E) produz uma + matriz As quasi-triangular e uma matriz + triangular Es que são a forma generalizada + de Schur do par A, E. + + + [As,Es,Q,Z] = schur(A,E) retorna, + ainda, duas matrizes unitárias Q e + Z tais que As=Q'*A*Z e + Es=Q'*E*Z. + + + + + Formas de Schur Ordenadas: + + + [As,Es,Z,dim] = schur(A,E,'c') + retorna a forma real generalizada de Schur do feixe + s*E-A. Ainda, as primeiras dim colunas de + Z geram uma base para o autoespaço direito + associado aos autovalores com partes reais negativas + (autoespaço de "tempo contínuo" generalizado). + + + [As,Es,Z,dim] = schur(A,E,'d') + + retorna a forma real generalizada de Schur do feixe + s*E-A. Ainda, as dim primeiras colunas de + Z formam uma base para o autoespaço direito + associado aos autovalores de magnitude menor que 1 (autoespaço + de "tempo discreto" generalizado). + + + [As,Es,Z,dim] = + schur(A,E,extern2) + + + retorna a forma real generalizada de Schur do feixe + s*E-A. Ainda, as dim primeiras colunas de + Z formam uma base para o autoespaço direito + associado aos autovalores do feixe que são selecionados de + acordo com a regra que é dada pela "external" + extern2. (veja "external" para detalhes). + Esta external pode ser descrita por uma função do Scilab ou + por um "procedure" de C ou FORTRAN. + + + + Função do Scilab + + + Se extern2 é descrita por uma + função do Scilab, deve ter a seqüência de chamamento: + s=extern2(Alpha,Beta), onde + Alpha e Beta + definem um autovalor generalizado e s + um booleano. + + + + + Um "Procedure" C ou FORTRAN + + + Se a "external" extern2 é + descrita por um "procedure" C ou FORTRAN, deve ter a + seqüência de chamamento: + + + int extern2(double *AlphaR, double + *AlphaI, double *Beta) + + + + se A e E são + matrizes de reais e + + + int extern2(double *AlphaR, double + *AlphaI, double *BetaR, double *BetaI) + + + + se A ou E é + matriz de complexos. Alpha, e + Beta definem o autovalor + generalizado. Um valor verdadeiro ou diferente de zero + siginfica autovalor generalizado selecionado. + + + + + + + + + + + + + Referências + As computações da forma de Schur matricial são baseadas nas rotinas + de Lapack DGEES e ZGEES. + + As computações da forma de Schur de feixes são baseadas nas rotinas + de Lapack DGGES e ZGGES. + + + + Exemplos + + + + Ver Também + + + spec + + + bdiag + + + ricc + + + pbig + + + psmall + + + + diff --git a/modules/linear_algebra/help/pt_BR/eigen/spec.xml b/modules/linear_algebra/help/pt_BR/eigen/spec.xml new file mode 100755 index 000000000..00be379ad --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/eigen/spec.xml @@ -0,0 +1,277 @@ + + + + + spec + autovalores de matrizes e feixes + + + Seqüência de Chamamento + evals=spec(A) + [R,diagevals]=spec(A) + + evals=spec(A,B) + [alpha,beta]=spec(A,B) + [alpha,beta,Z]=spec(A,B) + [alpha,beta,Q,Z]=spec(A,B) + + + + Parâmetros + + + A + + matriz quadrada de reais ou complexos + + + + B + + matriz quadrada de reais ou complexos com as mesmas dimensões + que A + + + + + evals + + vetor de reais ou complexos, os autovalores + + + + diagevals + + matriz diagonal de reais ou complexos (autovalores ao longo da + diagonal) + + + + + alpha + + vetor de reais ou complexos, al./be fornece os + autovalores + + + + + beta + + vetor de reais, al./be fornece os autovalores + + + + R + + matriz quadrada de reais ou complexos invertível, autovetores + direitos da matriz + + + + + Q + + matriz quadrada de reais ou complexos invertível, autovetores + esquerdos do feixe + + + + + Z + + matriz quadrada de reais ou complexos invertível, autovetores + direitos do feixe + + + + + + + Descrição + + + evals=spec(A) + + + retorna no vetor evals os + autovalores. + + + + + [R,diagevals] =spec(A) + + + retorna na matriz diagonal evals os + autovalores e em R os autovetores + direitos. + + + + + evals=spec(A,B) + + retorna o espectro do feixe de matrizes A - s B, i.e. as + raízes da matriz de polinômios s B - A. + + + + + [alpha,beta] = spec(A,B) + + + retorna o espectro do feixe de matrizes A- s + B + + ,i.e. as raízes da matriz de polinômios A - s + B + + .Auto valores generalizados alpha e beta são tais que a + matriz A - alpha./beta B é uma matriz singular. + Os autovalores são dados por al./be e se + beta(i) = 0 o i-ésimo autovalor está no infinito. + (Para B = eye(A), alpha./beta é + spec(A)). É usualmente representado pelo par + (alpha,beta), já que há uma interpretação razoável para beta=0, e + até mesmo para os dois sendo zero. + + + + + [alpha,beta,R] = spec(A,B) + + + retorna, ainda, a matriz R de autovetores + direitos generalizados do feixe. + + + + + [al,be,Q,Z] = spec(A,B) + + + rretorna ainda a matriz Q e + Z de autovetores esquerdos e direitos + generalizados do feixe. + + + + + + + Referências + As computações de autovalores de matrizes são baseadas nas rotinas + Lapack + + + + DGEEV e ZGEEV quando as matrizes não são simétricas, + + + DSYEV e ZHEEV quando as matrizes são simétricas. + + + Uma matriz de complexos simétrica tem termos fora da diagonal + conjugados e termos diagonais reais. + + As computações de autovalores de feixes são baseadas nas rotinas + Lapack DGGEV e ZGGEV. + + + + Matrizes de reais e de complexos + Deve-se notar que o tipo das variáveis de saída, tais como evals ou + R por exemplo, não é necessariamente o mesmo das que das matrizes de + entrada A e B. No parágrafo seguinte, análisamos o tipo das variáveis de + saída no caso onde nos casos onde se computa os autovalores e autovetores + de uma única matriz A. + + + + Matriz A de reais + + + Simétrica + Os autovetores e autovalores são reais. + + + Não simétrica + Os autovetores e autovalores são complexos. + + + + + Matriz A de complexos + + + Simétrica + Os autovalores são reais, mas os autovetores são + complexos. + + + + Não simétrica + Os autovetores e autovalores são complexos. + + + + + + + Exemplos + + + + Ver Também + + + poly + + + det + + + schur + + + bdiag + + + colcomp + + + + diff --git a/modules/linear_algebra/help/pt_BR/eigen/sva.xml b/modules/linear_algebra/help/pt_BR/eigen/sva.xml new file mode 100755 index 000000000..0a88f269d --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/eigen/sva.xml @@ -0,0 +1,83 @@ + + + + + sva + aproximação em valores singulares + + + Seqüência de Chamamento + [U,s,V]=sva(A,k) + [U,s,V]=sva(A,tol) + + + + Parâmetros + + + A + + matriz de reais ou complexos + + + + k + + inteiro + + + + tol + + número real não-negativo + + + + + + Descrição + Aproximação em valores singulares. + + [U,S,V]=sva(A,k) com k inteiro + >=1, retorna U,S e V tais que + B=U*S*V' é a melhor aproximação L2 de + A com + posto(B)=k. + + + [U,S,V]=sva(A,tol) com tol + real retorna U,S e V tais que + B=U*S*V' e a norma-L2 de A-B é, no + máximo, tol. + + + + Exemplos + + + + Ver Também + + + svd + + + + diff --git a/modules/linear_algebra/help/pt_BR/eigen/svd.xml b/modules/linear_algebra/help/pt_BR/eigen/svd.xml new file mode 100755 index 000000000..71998511e --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/eigen/svd.xml @@ -0,0 +1,132 @@ + + + + + svd + decomposição em valores singulares + + + Seqüência de Chamamento + s=svd(X) + [U,S,V]=svd(X) + [U,S,V]=svd(X,0) (obsolete) + [U,S,V]=svd(X,"e") + [U,S,V,rk]=svd(X [,tol]) + + + + Parâmetros + + + X + + matriz de reais ou complexos + + + + s + + vetor de reais (valores singulares) + + + + S + + matriz diagonal de reais (valores singulares) + + + + U,V + + matrizes quadradas ortogonais ou unitárias (vetores + singulares) + + + + + tol + + número real + + + + + + Descrição + + [U,S,V] = svd(X) produz uma matriz diagonal + S , com dimensão igual a de X e com + elementos da diagonal não-negativos em ordem decrescente, e matrizes + unitárias U e V tais que X + = U*S*V' + + . + + + [U,S,V] = svd(X,0) produz a decomposição com + "economia de tamanho". Se X é m-por-n com m > n, + então apenas as primeiras n colunas de U são computadas + e S é n-por-n. + + + s= svd(X) por si mesmo retorna um vetor + s contendo os valores singulares. + + + [U,S,V,rk]=svd(X,tol) fornece também + rk, o posto numérico de X i.e. i.e. + o número de valores singulares maiores que tol. + + + O valor default de tol é o mesmo que em + rank. + + + + Exemplos + + + + Ver Também + + + rank + + + qr + + + colcomp + + + rowcomp + + + sva + + + spec + + + + + Função Usada + Decomposições svd são baseadas nas rotinas Lapack DGESVD para + matrizes de reais e ZGESVD no caso de matrizes de complexos. + + + diff --git a/modules/linear_algebra/help/pt_BR/factorization/CHAPTER b/modules/linear_algebra/help/pt_BR/factorization/CHAPTER new file mode 100755 index 000000000..e6daeb8eb --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/factorization/CHAPTER @@ -0,0 +1,2 @@ +title = Factorization + diff --git a/modules/linear_algebra/help/pt_BR/factorization/givens.xml b/modules/linear_algebra/help/pt_BR/factorization/givens.xml new file mode 100755 index 000000000..66b9e5a73 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/factorization/givens.xml @@ -0,0 +1,91 @@ + + + + + givens + transformação de Givens + + + Seqüência de Chamamento + U=givens(xy) + U=givens(x,y) + [U,c]=givens(xy) + [U,c]=givens(x,y) + + + + Parâmetros + + + x,y + + dois números reais ou complexos + + + + xy + + vetor coluna de reais ou complexos de tamanho 2 + + + + U + + matriz 2x2 unitária + + + + c + + vetor coluna de reais ou complexos de tamanho 2 + + + + + + Descrição + + U= givens(x, y) ou U = + givens(xy) + + com xy = [x;y] retorna uma matriz + unitária 2x2 U + tal que: + + + U*xy=[r;0]=c. + + + + Note que givens(x,y) e + givens([x;y]) são equivalentes. + + + + + Exemplos + + + + Ver Também + + + qr + + + + diff --git a/modules/linear_algebra/help/pt_BR/factorization/householder.xml b/modules/linear_algebra/help/pt_BR/factorization/householder.xml new file mode 100755 index 000000000..8faaa7029 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/factorization/householder.xml @@ -0,0 +1,79 @@ + + + + + householder + matriz de reflexão ortogonal de Householder + + + Seqüência de Chamamento + u=householder(v [,w]) + + + Parâmetros + + + v + + vetor coluna de reais ou complexos + + + + w + + vetor coluna de reais ou complexos com o mesmo tamanho que + v. Valor padrão é + eye(v) + + + + + u + + vetor coluna de reais ou complexos + + + + + + Descrição + + Dados dois vetores coluna v, + w + + de mesmo tamanho, householder(v,w) retorna + um vetor coluna unitário u, tal que + (eye()-2*u*u')*v + + éproporcional a w. + (eye()-2*u*u') é a matriz de reflexão ortogonal de + Householder . + + + O valor padrão de w é eye(v). + Neste caso, o vetor (eye()-2*u*u')*v é o + vetor eye(v)*norm(v). + + + + Ver Também + + + qr + + + givens + + + + diff --git a/modules/linear_algebra/help/pt_BR/factorization/sqroot.xml b/modules/linear_algebra/help/pt_BR/factorization/sqroot.xml new file mode 100755 index 000000000..4c5eab66e --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/factorization/sqroot.xml @@ -0,0 +1,64 @@ + + + + + sqroot + fatoração hermitiana W*W' + + + Seqüência de Chamamento + sqroot(X) + + + Parâmetros + + + X + + matriz simétrica, não-negativa definida de reais ou + complexos + + + + + + + Descrição + + Retorna W tal que X=W*W' (usa SVD). + + + + Exemplos + + + + Ver Também + + + chol + + + svd + + + + diff --git a/modules/linear_algebra/help/pt_BR/kernel/CHAPTER b/modules/linear_algebra/help/pt_BR/kernel/CHAPTER new file mode 100755 index 000000000..be67920e1 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/kernel/CHAPTER @@ -0,0 +1,2 @@ +title = Kernel + diff --git a/modules/linear_algebra/help/pt_BR/kernel/colcomp.xml b/modules/linear_algebra/help/pt_BR/kernel/colcomp.xml new file mode 100755 index 000000000..5a281554c --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/kernel/colcomp.xml @@ -0,0 +1,112 @@ + + + + + colcomp + compressão de colunas, núcleo + + + Seqüência de Chamamento + [W,rk]=colcomp(A [,flag] [,tol]) + + + Parâmetros + + + A + + matriz de reais ou complexos + + + + flag + + string + + + + tol + + número real + + + + W + + matriz quadrada não-singular (mudança de base) + + + + rk + + + inteiro (posto de A) + + + + + + + Descrição + + Compressão de colunas de A: Ac = + A*W + + éde colunas comprimidas ,i.e., + + + Ac=[0,Af] com o posto-coluna de + Af cheio, posto(Af) = + posto(A) = rk. + + + flag e tol são parâmetros + opcionais: flag = 'qr' ou 'svd' (o + padrão é 'svd'). + + + tol = parâmetro de tolerância (de ordem + %eps como valor padrão). + + + As ma-rk primeiras colunas de + W geram o núcleo de A quando + size(A)=(na,ma) + + + + Exemplos + + + + Ver Também + + + rowcomp + + + fullrf + + + fullrfk + + + kernel + + + + diff --git a/modules/linear_algebra/help/pt_BR/kernel/fullrf.xml b/modules/linear_algebra/help/pt_BR/kernel/fullrf.xml new file mode 100755 index 000000000..5bf4e8220 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/kernel/fullrf.xml @@ -0,0 +1,106 @@ + + + + + fullrf + fatoração de posto completo (ou cheio) + + + Seqüência de Chamamento + [Q,M,rk]=fullrf(A,[tol]) + + + Parâmetros + + + A + + matriz de reais ou complexos + + + + tol + + número real (limiar para determinação do posto) + + + + Q,M + + matriz de reais ou complexos + + + + rk + + + inteiro(posto de A) + + + + + + + Descrição + + Fatoração de posto cheio : fullrf retorna + Q e M tais que A = + Q*M + + com Im(Q)=Im(A) e + Nuc(M)=Nuc(A), Q + de posto-coluna cheio, M de posto-linha cheio e + rk = rank(A) = #columns(Q) = #rows(M). + + + tol é um parâmetro real opcional (valor real + padrão é sqrt(%eps)). O posto rk de + A é definido como o número de valores singulares + maiores que norm(A)*tol. + + + Se A é simétrica, fullrf retorna + M=Q'. + + + + Exemplos + + + + Ver Também + + + svd + + + qr + + + fullrfk + + + rowcomp + + + colcomp + + + + diff --git a/modules/linear_algebra/help/pt_BR/kernel/fullrfk.xml b/modules/linear_algebra/help/pt_BR/kernel/fullrfk.xml new file mode 100755 index 000000000..9f7ba85f3 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/kernel/fullrfk.xml @@ -0,0 +1,77 @@ + + + + + fullrfk + fatoração de posto completo de A^k + + + Seqüência de Chamamento + [Bk,Ck]=fullrfk(A,k) + + + Parâmetros + + + A + + matriz de reais ou de complexos + + + + k + + inteiro + + + + Bk,Ck + + matrizes de reais ou de complexos + + + + + + Descrição + Esta função calcula a fatoração de posto completo (ou cheio) de + A^k i.e. Bk*Ck=A^k onde + Bk é de posto-coluna cheio e Ck de + posto-linha cheio. Tem-se + Im(Bk)=Im(A^k) e + Nuc(Ck)=Nuc(A^k). + + + Para k=1, fullrfk é + equivalente a fullrf. + + + + Exemplos + + + + Ver Também + + + fullrf + + + range + + + + diff --git a/modules/linear_algebra/help/pt_BR/kernel/im_inv.xml b/modules/linear_algebra/help/pt_BR/kernel/im_inv.xml new file mode 100755 index 000000000..07f69c4b8 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/kernel/im_inv.xml @@ -0,0 +1,110 @@ + + + + + im_inv + imagem inversa + + + Seqüência de Chamamento + [X,dim]=im_inv(A,B [,tol]) + [X,dim,Y]=im_inv(A,B, [,tol]) + + + + Parâmetros + + + A,B + + duas matirzes de reais ou complexos com igual número de + colunas + + + + + X + + matriz quadrada ortogonal ou unitária de ordem igual ao número + de colunas de A + + + + + dim + + inteiro (dimensão do subespaço) + + + + Y + + matriz ortogonal de ordem igual ao número de linhas de + A e B. + + + + + + + Descrição + + [X,dim]=im_inv(A,B) + computa(A^-1)(B) ,i.e, vetores cujas imagens através de + A estão em Im(B) + + + As dim primeiras colunas de X + geram (A^-1)(B) + + + tol é um limiar usado para testar a inclusão de + subespaço ; o valor padrão é tol = 100*%eps. Se + Y é retornado, então [Y*A*X,Y*B] é + particionado como segue: + [A11,A12;0,A22],[B1;0] + + + onde B1 tem posto-linha cheio (igual a + posto(B)) e A22 tem posto-coluna + cheio e tem dim colunas. + + + + Exemplos + + + + Ver Também + + + rowcomp + + + spaninter + + + spanplus + + + linsolve + + + + diff --git a/modules/linear_algebra/help/pt_BR/kernel/kernel.xml b/modules/linear_algebra/help/pt_BR/kernel/kernel.xml new file mode 100755 index 000000000..93f8e850e --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/kernel/kernel.xml @@ -0,0 +1,99 @@ + + + + + kernel + núcleo de uma matriz + + + Seqüência de Chamamento + W=kernel(A [,tol,[,flag]) + + + Parâmetros + + + A + + matriz de reais ou complexos completa ou matriz de reais + esparsa + + + + + flag + + + string 'svd' (padrão) ou + 'qr' + + + + + tol + + número real + + + + W + + matriz de posto-coluna completo + + + + + + Descrição + + W=kernel(A) retorna o núcleo (espaço nulo) de + A. Se A tem posto-coluna completo, então uma matriz + vazia [] é retornada. + + + flag e tol são parâmetros + opcionais: flag = 'qr' ou'svd' (o + padrão é 'svd'). + + + tol = parâmetro de tolerância (de ordem + %eps como valor padrão). + + + + Exemplos + + + + Ver Também + + + colcomp + + + fullrf + + + fullrfk + + + linsolve + + + + diff --git a/modules/linear_algebra/help/pt_BR/kernel/range.xml b/modules/linear_algebra/help/pt_BR/kernel/range.xml new file mode 100755 index 000000000..a82b20aa8 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/kernel/range.xml @@ -0,0 +1,95 @@ + + + + + range + Imagem (gerado) de A^k + + + Seqüência de Chamamento + [X,dim]=range(A,k) + + + Parâmetros + + + A + + matriz de reais ou complexos que se assume quadrada se k>1 + + + + + k + + inteiro + + + + X + + matriz ortonormal + + + + dim + + inteiro (dimensão de subespaço) + + + + + + Descrição + + Computação da imagem de A^k ; as primeiras dim + colunas de X geram a imagem de A^k. + As últimas linhas de X geram o complemento ortogonal da + imagem. X*X' é a matriz identidade. + + + + Exemplos + + + + Ver Também + + + fullrfk + + + rowcomp + + + + + Função Usada + + A função range é baseada na função rowcomp que usa decomposição svd (decomposição em valores singulares). + + + diff --git a/modules/linear_algebra/help/pt_BR/kernel/rowcomp.xml b/modules/linear_algebra/help/pt_BR/kernel/rowcomp.xml new file mode 100755 index 000000000..a01b2c735 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/kernel/rowcomp.xml @@ -0,0 +1,132 @@ + + + + + rowcomp + compressão de linhas, imagem + + + Seqüência de Chamamento + [W,rk]=rowcomp(A [,flag [,tol]]) + + + Parâmetros + + + A + + matriz de reais ou de complexos + + + + flag + + string opcional, com valores possíveis + 'svd' ou 'qr'. O valor padrão + é 'svd'. + + + + + tol + + número real não-negativo opcional. O valor padrão é + sqrt(%eps)*norm(A,1). + + + + + W + + matriz quadrada não-singular (matriz mudança de base) + + + + rk + + + inteiro (posto de A) + + + + + + + Descrição + + Compressão de linhas de A. Ac = + W*A + + éuma matriz de linhas comprimidas, i.e. + Ac=[Af;0] com Af de posto-linha + cheio. + + + flag e tol são parâmetros + opcionais: flag='qr' ou 'svd' (o + padrão é 'svd'). + + + tol é um parâmetro de tolerância. + + + As rk primeiras colunas de W' + geram a imagem de A. + + + As rk primeiras linhas (do topo) de + W geram a imagem de linha de + A. + + + Um vetor não nulo x pertence à + Im(A) se,e só se, W*x é de linhas + comprimidas de acordo com Ac i.e, a norma de seus + últimos componentes é pequena com relação a dos seus primeiros + componentes. + + + + Exemplos + + + + Ver Também + + + colcomp + + + fullrf + + + fullrfk + + + + + Função Usada + + A função rowcomp é baseada nas decomposições + svd (decomposição em valores singulares) ou + qr . + + + diff --git a/modules/linear_algebra/help/pt_BR/linear/CHAPTER b/modules/linear_algebra/help/pt_BR/linear/CHAPTER new file mode 100755 index 000000000..7d9d9cf49 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/linear/CHAPTER @@ -0,0 +1,2 @@ +title = Linear Equations + diff --git a/modules/linear_algebra/help/pt_BR/linear/aff2ab.xml b/modules/linear_algebra/help/pt_BR/linear/aff2ab.xml new file mode 100755 index 000000000..8aa01b88e --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/linear/aff2ab.xml @@ -0,0 +1,162 @@ + + + + + aff2ab + Conversão de uma função linear (afim) para forma + A,b + + + + Seqüência de Chamamento + [A,b]=aff2ab(afunction,dimX,D [,flag]) + + + Parâmetros + + + afunction + + + uma função do Scilab Y =fct(X,D) onde + X, D, Y são lists de + matrizes + + + + + dimX + + + uma matriz de inteiros p x 2 (p é o número + de matrizes em X) + + + + + D + + + uma list de matrizes de reais (ou qualquer + outro objeto Scilab válido). + + + + + flag + + + parâmetro opcional (flag='f' ou + flag='sp') + + + + + A + + uma matriz de reais + + + + b + + um vetor de reais tendo a mesma dimensão de linha que + A + + + + + + + Descrição + + aff2ab retorna a representação matricial de uma + função afim (na base canônica). + + + afunction é uma função com sintaxe imposta: + Y=afunction(X,D) onde X=list(X1,X2,...,Xp) + + é uma lista de p matrizes de reais, e + Y=list(Y1,...,Yq) + + éuma lista de q matrizes reais que dependem + linearmente das Xi's. A entrada (opcional) + D + + contém parâmetros necessários para computar Y como uma função + de X (geralmente é uma lista de matrizes). + + + dimX é uma matriz p x 2: + dimX(i)=[nri,nci] é o número real de linhas e colunas + da matriz Xi. Estas dimensões determinam + na, a dimensão de coluna da matriz resultante + A: na=nr1*nc1 +...+ nrp*ncp. + + + Se o parâmetro opcional flag='sp' a matriz + resultante A é retornada como uma esparsa. + + Esta função é útil para resolver um sistema de equações lineares + onde as incógnitas são matrizes. + + + + Exemplos + + + + Ver Também + + + linsolve + + + + diff --git a/modules/linear_algebra/help/pt_BR/linear/chol.xml b/modules/linear_algebra/help/pt_BR/linear/chol.xml new file mode 100755 index 000000000..33b11e42f --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/linear/chol.xml @@ -0,0 +1,85 @@ + + + + + chol + Cholesky factorization + + + Seqüência de Chamamento + [R]=chol(X) + + + Parâmetros + + + X + + uma matriz simétrica e positiva definida de reais ou + complexos. + + + + + + + Descrição + + Se X é positiva definida, então R = + chol(X) + + produz uma matriz triangular superior + R tal que R'*R = X. + + + chol(X) usa apenas a diagonal e o triângulo + superior de X. O triângulo inferior é assumido como + sendo a transposta (ou complexo conjugado) da superior. + + + + Referências + A decomposição de Cholesky é baseada nas rotinas de Lapack DPOTRF + para matrizes de reais e ZPOTRF no caso de matrizes de complexos. + + + + Exemplos + + + + Ver Também + + + spchol + + + qr + + + svd + + + bdiag + + + fullrf + + + + diff --git a/modules/linear_algebra/help/pt_BR/linear/inv.xml b/modules/linear_algebra/help/pt_BR/linear/inv.xml new file mode 100755 index 000000000..6eaa5f99d --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/linear/inv.xml @@ -0,0 +1,109 @@ + + + + + inv + inversa de uma matriz + + + Seqüência de Chamamento + inv(X) + + + Parâmetros + + + X + + matriz quadrada de reais ou complexos, matriz de polinômios, + matriz de razões de polinômios em representação de transferência ou + espaço de estados + + + + + + + Descrição + + inv(X) é a inversa da matriz quadrada + X. Uma aviso é impresso na tela se X + possui má escala ou é quase singular. + + Para matrizes de polinômios ou matrizes razões de polinômios em + representação de transferência, inv(X) é equivalente a + invr(X). + + Para sistemas lineares na representação de espaço de estados (lista + syslin), invr(X) é equivalente a + invsyslin(X). + + + + Referências + A função inv para matrizes de números é baseada nas rotinas de + Lapack DGETRF, DGETRI para matrizes de reais e ZGETRF, ZGETRI para o caso + de matrizes de complexos. Para matrizes de polinômios e matrizes de + funções racionais, inv é baseado na função + invr do Scilab. + + + + Exemplos + + + + Ver Também + + + slash + + + backslash + + + pinv + + + qr + + + lufact + + + lusolve + + + invr + + + coff + + + coffg + + + + diff --git a/modules/linear_algebra/help/pt_BR/linear/linsolve.xml b/modules/linear_algebra/help/pt_BR/linear/linsolve.xml new file mode 100755 index 000000000..dffda99b7 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/linear/linsolve.xml @@ -0,0 +1,129 @@ + + + + + linsolve + solucionador de equações lineares + + + Seqüência de Chamamento + [x0,kerA]=linsolve(A,b [,x0]) + + + Parâmetros + + + A + + + uma matriz na x ma de reais (possivelmente + esparsa) + + + + + b + + + um vetor na x 1 (mesma dimensão de linha de + A) + + + + + x0 + + um vetor de reais + + + + kerA + + + uma matriz ma x k de reais + + + + + + + Descrição + + linsolve computa todas as soluções para + A*x+b=0 + + . + + + x0 é uma solução particular (se houver) e + kerA= núcleo de A. Qualquer + x=x0+kerA*w com w arbitrário + satisfaz A*x+b=0. + + + Se um compatible x0 compatível é dado na entrada, + x0 é retornado. Senão, um x0, + compatível é retornado, se houver. + + + + Exemplos + + + + Ver Também + + + inv + + + pinv + + + colcomp + + + im_inv + + + backslash + + + umfpack + + + + diff --git a/modules/linear_algebra/help/pt_BR/linear/lsq.xml b/modules/linear_algebra/help/pt_BR/linear/lsq.xml new file mode 100755 index 000000000..b3f9fc62f --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/linear/lsq.xml @@ -0,0 +1,116 @@ + + + + + lsq + problemas de mínimos quadrados lineares + + + Seqüência de Chamamento + X=lsq(A,B [,tol]) + + + Parâmetros + + + A + + matriz de reais ou de complexos (m x n) + + + + B + + matriz de reais ou de complexos (m x p) + + + + tol + + escalar positivo, usado para determinar o posto efetivo de A + (definido como sendo a ordem da maior submatriz triangular R11 + regente na fatoração QR com pivoteamento de A, cujo número de + condicionamento estimado <= 1/tol. O valor padrão de tol é + sqrt(%eps). + + + + + X + + matriz de reais ou complexos (n x p) + + + + + + Descrição + + X=lsq(A,B) computa a solução de mínimo quadrado + de menor norma da equação A*X=B, enquanto X=A + \ B + + computa uma solução de mínimo quadrado com no máximo + posto(A) componentes não-nulos por coluna. + + + + Referências + + lsq é baseado nas funções Lapack DGELSY para + matrizes de reais e ZGELSY para matrizes de complexos. + + + + Exemplos + + + + Ver Também + + + backslash + + + inv + + + pinv + + + rank + + + + diff --git a/modules/linear_algebra/help/pt_BR/linear/lu.xml b/modules/linear_algebra/help/pt_BR/linear/lu.xml new file mode 100755 index 000000000..d2418e3f7 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/linear/lu.xml @@ -0,0 +1,124 @@ + + + + + lu + fatores LU de eliminação Gaussiana + + + Seqüência de Chamamento + [L,U]= lu(A) + [L,U,E]= lu(A) + + + + Parâmetros + + + A + + matriz de reais ou complexos (m x n) + + + + L + + matriz de reais ou complexos (m x min(m,n)) + + + + U + + matriz de reais ou complexos (min(m,n) x n ) + + + + E + + uma matriz de permutação (n x n) + + + + + + Descrição + + [L,U]= lu(A) produz duas matrizes + L e U tais que A = + L*U + + com U triangular superior e + E*L triangular infeiror para uma matriz de permutação + E. + + + Se A tem posto k, as linhas de + k+1 a n de U são + zeros. + + + [L,U,E]= lu(A) produz três matrizes + L, U e E tais que + E*A = L*U com U triangular superior + e E*L l triangular inferior para uma matriz de + permutação E. + + + Se A é uma matriz de reais, usando as funções + lufact e luget é possível obter as + matrizes de permutação e, também, quando A não é de + posto cheio, a compressão de colunas da matriz + L. + + + + Exemplos + + + + Ver Também + + + lufact + + + luget + + + lusolve + + + qr + + + svd + + + + + Função Usada + As decomposições de lu são baseadas nas rotinas de Lapack DGETRF + para matrizes reais e ZGETRF para o caso de matrizes complexas. + + + diff --git a/modules/linear_algebra/help/pt_BR/linear/pinv.xml b/modules/linear_algebra/help/pt_BR/linear/pinv.xml new file mode 100755 index 000000000..9d1a5f795 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/linear/pinv.xml @@ -0,0 +1,83 @@ + + + + + pinv + pseudo-inversa + + + Seqüência de Chamamento + pinv(A,[tol]) + + + Parâmetros + + + A + + matriz de reais ou complexos + + + + tol + + número real + + + + + + Descrição + + X= pinv(A) produz uma matriz + X de mesma dimensão que A' tal + que: + + + A*X*A = A, X*A*X = X e ambas + A*X e X*A são Hermitianas. + + A computação é baseada em SVD e qualquer valor singular abaixo da + tolerância é tratado como zero: esta tolerância é acessada por + X=pinv(A,tol). + + + + Exemplos + + + + Ver Também + + + rank + + + svd + + + qr + + + + + Função Usada + + pinv é baseada na decomposição em valores + singulares (função do Scilab svd). + + + diff --git a/modules/linear_algebra/help/pt_BR/linear/qr.xml b/modules/linear_algebra/help/pt_BR/linear/qr.xml new file mode 100755 index 000000000..bf20c82ce --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/linear/qr.xml @@ -0,0 +1,200 @@ + + + + + qr + QR decomposição + + + Seqüência de Chamamento + [Q,R]=qr(X [,"e"]) + [Q,R,E]=qr(X [,"e"]) + [Q,R,rk,E]=qr(X [,tol]) + + + + Parâmetros + + + X + + matriz de reais ou complexos + + + + tol + + número real não-negativo + + + + Q + + matriz quadrada ortogonal ou unitária + + + + R + + + matriz com as mesmas dimensões de X + + + + + E + + matriz de permutação + + + + rk + + + inteiro (posto QR de X) + + + + + + + Descrição + + + [Q,R] = qr(X) + + + pproduz uma matriz triangular superior R de + mesma dimensão que X e uma matriz ortogonal + (unitária no caso de matriz de complexos) Q tais + que X = Q*R. [Q,R] = qr(X,"e") + produz um "economia de tamanho": Se X é m-por-n + com m > n, então, apenas as primeiras n colunas de + Q são computadas assim como as primeiras n linhas + de R. + + + De Q*R = X , segue que a k-ésima coluna da + matriz X, é expressa como combinação linear das k + primeiras colunas de Q (com coeficientes + R(1,k), ..., R(k,k) ). As k primeiras colunas de + Q formam uma base ortogonal para o subespaço + gerado pelas k priemiras colunas de X. Se a + coluna k de X (i.e. + X(:,k) ) é uma combinação linear das + p primeiras colunas de X, + então, as entradas de R(p+1,k), ..., R(k,k) são + zeros. Neste caso, R é trapezoidal superior. Se + X tem posto rk, as linhas + R(rk+1,:), R(rk+2,:), ... são zeros. + + + + + [Q,R,E] = qr(X) + + produz uma matriz de permutação (de colunas) + E, uma matriz triangular superior + R com elementos na diagonal decrescentes e uma + matriz ortogonal (ou unitaria) Q tais que + X*E = Q*R. Se rk é o posto de + X, as rk primeiras entradas ao + longo da diagonal de R, i.e. R(1,1), + R(2,2), ..., R(rk,rk) + + são todas diferentes de zero. + [Q,R,E] = qr(X,"e") produz uma "economia de + tamanho": Se X ié m-por-n com m > n, então, + apenas as n primeiras colunas de Q são computadas + tanto quanto as n priemiras linhas de R. + + + + + [Q,R,rk,E] = qr(X ,tol) + + + retorna rk = estimativa do posto de + X i.e. rk é o número elementos + da diagonal de R que são maiores que um dado + limiar tol. + + + + + [Q,R,rk,E] = qr(X) + + + retorna rk = estimativa do posto de + X i.e. rk é o número de + elementos da diagonal de R que são maiores que + tol=R(1,1)*%eps*max(size(R)). Veja + rankqr para uma fatoração QR que revela o posto + usando o número de condicionamento de R. + + + + + + + Exemplos + + + + Ver Também + + + rankqr + + + rank + + + svd + + + rowcomp + + + colcomp + + + + + Funções Usadas + A decomposição qr é baseada nas rotinas de Lapack DGEQRF, DGEQPF, + DORGQR para as matrizes de reais ZGEQRF, ZGEQPF, ZORGQR para as matrizes + de complexos. + + + diff --git a/modules/linear_algebra/help/pt_BR/linear/rankqr.xml b/modules/linear_algebra/help/pt_BR/linear/rankqr.xml new file mode 100755 index 000000000..a09aa75b1 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/linear/rankqr.xml @@ -0,0 +1,147 @@ + + + + + rankqr + fatoração QR com revelação do posto + + + Seqüência de Chamamento + [Q,R,JPVT,RANK,SVAL]=rankqr(A, [RCOND,JPVT]) + + + Parâmetros + + + A + + matriz de reais ou complexos + + + + RCOND + + número real usado para determinar o posto efetivo de + A, que é definido como sendo a ordem da maior + submatriz regente triangular R11 na fatoração QR + com pivoteamento de A, cujo número de + condicionamento estimado é < 1/RCOND. + + + + + JPVT + + + vetor de inteiros nas entradas, se JPVT(i) + não é 0, a i-ésimo coluna de + A + + épermtutada para a frente de AP, + senão, a coluna i é uma coluna livre. Na saída, + se JPVT(i) = k, então a + i-ésima coluna de A*P era a + k-ésima coluna de A. + + + + + RANK + + + posto efetivo de A, i.e., a ordem da + submatriz R11. É o mesmo que a ordem da submatriz + T1 na fatoração ortogonal completa de + A. + + + + + SVAL + + vetor de reais com 3 componentes; as estimativas de alguns dos + valores singulares do fator triangular R. + + + SVAL(1) é o maior valor singular de + R(1:RANK,1:RANK); + + + SVAL(2) é o menor valor singular de + R(1:RANK,1:RANK); + + + SVAL(3) é o menor valor singular de + R(1:RANK+1,1:RANK+1), se RANK + < MIN(M,N), ou de + R(1:RANK,1:RANK), caso contrário. + + + + + + + Descrição + Computa (opcionalmente) uma fatoração QR com revelação do posto de + uma matriz de reais geral M-por-N, ou de complexos A, + que pode ser deficiente de posto, e estima seu posto efetivo usando + estimativa de condição incremental. + + A rotina usa uma fatoração QR com pivoteamento de colunas: + + + com R11 definida como a maior submatriz regente + cujo número de condição estimado é menor que 1/RCOND. A + ordem de R11, RANK, é o posto + efetivo deA. + + Se a fatoração triangular revela o posto (que será o caso se as + colunas regentes forem bem condicionadas), então + SVAL(1) também será uma estimativa para o maior valor + singular de A, e SVAL(2) e + SVAL(3) serão estimativas para o + RANK-ésimo e (RANK+1)-ésimo valores + singulares de A, respectivamente. + + Examinando-se estes valores, pode-se confirmar que o posto é bem + definido a respeito do valor escolhido de RCOND. A + razão SVAL(1)/SVAL(2) é uma estimativa do número de + condicionamento de R(1:RANK,1:RANK). + + + + Exemplos + + + + Ver Também + + + qr + + + rank + + + + + Funções Usadas + Rotinas da biblioteca Slicot MB03OD, ZB03OD. + + diff --git a/modules/linear_algebra/help/pt_BR/markov/CHAPTER b/modules/linear_algebra/help/pt_BR/markov/CHAPTER new file mode 100755 index 000000000..c29eb913c --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/markov/CHAPTER @@ -0,0 +1,2 @@ +title = Markov Matrices + diff --git a/modules/linear_algebra/help/pt_BR/markov/classmarkov.xml b/modules/linear_algebra/help/pt_BR/markov/classmarkov.xml new file mode 100755 index 000000000..affa1f588 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/markov/classmarkov.xml @@ -0,0 +1,102 @@ + + + + + classmarkov + classes transientes e recorrentes da matriz de + Markov + + + + Seqüência de Chamamento + [perm,rec,tr,indsRec,indsT]=classmarkov(M) + + + Parâmetros + + + M + + matriz de Markov N x N de reais. A soma das entradas em cada + linha deve ser acrescida em uma unidade + + + + + perm + + vetor de permutação de inteiros + + + + rec, tr + + vetor de inteiros, número (número de estados em cada classe + recorrente, número de estados transientes) + + + + + indsRec,indsT + + vetor de inteiros (índices dos estados recorrentes e + transientes) + + + + + + + Descrição + + Retorna um vetor de permutação perm tal + que + + + + Cada Mii é uma matriz de Markov de dimensão + rec(i) i=1,..,r. Q é uma submatriz + de Markov de dimensão tr. Estados de 1 a sum(rec) são + recorrentes e estados de r+1 a n são transientes. Tem-se + perm=[indsRec,indsT] onde indsRec é um vetor de tamanho + sum(rec) e indsT é um vetor de tamanho tr. + + + + Exemplos + + + + Ver Também + + + genmarkov + + + eigenmarkov + + + + diff --git a/modules/linear_algebra/help/pt_BR/markov/eigenmarkov.xml b/modules/linear_algebra/help/pt_BR/markov/eigenmarkov.xml new file mode 100755 index 000000000..8926b8738 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/markov/eigenmarkov.xml @@ -0,0 +1,83 @@ + + + + + eigenmarkov + Autovetores esquerdo e direito normalizados de + Markov + + + + Seqüência de Chamamento + [M,Q]=eigenmarkov(P) + + + Parâmetros + + + P + + matriz de Markov N x N de reais. A soma das entradas de cada + linha deve ser acrescida de uma unidade + + + + + M + + matriz de reais de N colunas + + + + Q + + matriz de reais de N linhas + + + + + + Descrição + Retorna os autovetores esquerdo e direito normalizados associados ao + autovalor 1 da matriz P de transição de Markov. Se a multiplicidade deste + autovalor é m e P é N x N, M é uma matriz m x N e Q é uma matriz N x m. + M(k,:) é o vetor de distribuição de probabilidade associado ao k-ésimo + conjunto ergódico (classe recorrente). M(k,x) é zero se x não está na + k-ésima classe recorrente. Q(x,k) é a probabilidade de se terminar na + k-ésima classe recorrente começando de x. Se P^k + converge para k (sem autovalores no círculo unitário, + exceto 1), então o limite é Q*M (auto-projeção). + + + + Exemplos + + + + Ver Também + + + genmarkov + + + classmarkov + + + + diff --git a/modules/linear_algebra/help/pt_BR/markov/genmarkov.xml b/modules/linear_algebra/help/pt_BR/markov/genmarkov.xml new file mode 100755 index 000000000..a8da3164d --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/markov/genmarkov.xml @@ -0,0 +1,89 @@ + + + + + genmarkov + gera uma matriz de Markov aleatória com classes recorrentes e + transientes + + + + Seqüência de Chamamento + M=genmarkov(rec,tr) + M=genmarkov(rec,tr,flag) + + + + Parâmetros + + + rec + + vetor linha de inteiros (sua dimensão é o número de classes + recorrentes) + + + + + tr + + inteiro (número de estados transientes) + + + + M + + matriz de Markov de reais. A soma das entradas de cada linha + deve ser acrecsida de uma unidade + + + + + flag + + + string 'perm'. Se fornecido, uma permutação + dos estados é feita. + + + + + + + Descrição + Retorna em M uma matriz de probabilidade de transição de Markov + aleatória com rec(1),...rec($) entradas respectivamente + e tr estados transientes. + + + + Exemplos + + + + Ver Também + + + classmarkov + + + eigenmarkov + + + + diff --git a/modules/linear_algebra/help/pt_BR/matrix/CHAPTER b/modules/linear_algebra/help/pt_BR/matrix/CHAPTER new file mode 100755 index 000000000..bb89125cd --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/matrix/CHAPTER @@ -0,0 +1,2 @@ +title = Matrix Analysis + diff --git a/modules/linear_algebra/help/pt_BR/matrix/cond.xml b/modules/linear_algebra/help/pt_BR/matrix/cond.xml new file mode 100755 index 000000000..78d153da3 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/matrix/cond.xml @@ -0,0 +1,59 @@ + + + + + cond + número de condicionamento de uma matriz + + + Seqüência de Chamamento + cond(X) + + + Parâmetros + + + X + + rmatriz quadrada de reais ou complexos + + + + + + Descrição + + Número de condicionamento em norma-2. cond(X) é a + razão entre o maior e o menor valor singular de + X. + + + + Exemplos + + + + Ver Também + + + rcond + + + svd + + + + diff --git a/modules/linear_algebra/help/pt_BR/matrix/det.xml b/modules/linear_algebra/help/pt_BR/matrix/det.xml new file mode 100755 index 000000000..d294693e7 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/matrix/det.xml @@ -0,0 +1,94 @@ + + + + + det + determinante + + + Seqüência de Chamamento + det(X) + [e,m]=det(X) + + + + Parâmetros + + + X + + matriz quadrada de reais ou complexos, matriz de polinômios ou + de razões de polinômios + + + + + m + + número real ou complexo, a mantissa de base 10 do + determinante + + + + + e + + inteiro, o expoente de base 10 do determinante + + + + + + Descrição + + det(X) ( m*10^e é o + determinante da matriz quadrada X). + + + Para uma matriz de polinômios, det(X) é + equivalente a determ(X). + + + Para matrizes de razões de polinômios det(X) é + equivalente a detr(X). + + + + Referências + As computações da função det são baseadas nas rotinas do LAPACK + DGETRF para matrizes de reais e ZGETRF para o caso de matrizes de + complexos. + + + + Exemplos + + + + Ver Também + + + detr + + + determ + + + + diff --git a/modules/linear_algebra/help/pt_BR/matrix/orth.xml b/modules/linear_algebra/help/pt_BR/matrix/orth.xml new file mode 100755 index 000000000..d352ff212 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/matrix/orth.xml @@ -0,0 +1,76 @@ + + + + + orth + base ortogonal + + + Seqüência de Chamamento + Q=orth(A) + + + Parâmetros + + + A + + matriz de reais ou complexos + + + + Q + + matriz de reais ou complexos + + + + + + Descrição + + Q=orth(A) retorna Q, uma base + ortogonal para o gerado de A. Im(Q) + = Im(A) e Q'*Q=eye. + + + O número de colunas de Q é o posto de + A como determinado pelo algoritmo QR. + + + + Exemplos + + + + Ver Também + + + qr + + + rowcomp + + + colcomp + + + range + + + + diff --git a/modules/linear_algebra/help/pt_BR/matrix/rank.xml b/modules/linear_algebra/help/pt_BR/matrix/rank.xml new file mode 100755 index 000000000..3cff123f5 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/matrix/rank.xml @@ -0,0 +1,88 @@ + + + + + rank + posto + + + Seqüência de Chamamento + [i]=rank(X) + [i]=rank(X,tol) + + + + Parâmetros + + + X + + matriz de reais ou complexos + + + + tol + + número real não-negativo + + + + + + Descrição + + rank(X) é o posto numérico de + X i.e. o número de valores singulares de X que são + maiores que norm(size(X),'inf') * norm(X) * + %eps + + . + + + rank(X,tol) é o número de valores singulares de + X que são maiores que tol. + + + Note que o valor padrão de tol é proporcional a + norm(X). Como conseqüência, + rank([1.d-80,0;0,1.d-80]) é 2 !. + + + + Exemplos + + + + Ver Também + + + svd + + + qr + + + rowcomp + + + colcomp + + + lu + + + + diff --git a/modules/linear_algebra/help/pt_BR/matrix/rcond.xml b/modules/linear_algebra/help/pt_BR/matrix/rcond.xml new file mode 100755 index 000000000..0fe708628 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/matrix/rcond.xml @@ -0,0 +1,77 @@ + + + + + rcond + número de condicionamento inverso + + + Seqüência de Chamamento + rcond(X) + + + Parâmetros + + + X + + matriz quadrada de reais ou complexos + + + + + + Descrição + + rcond(X) é uma estimativa para a recíproca da + condição de X na norma-1. + + + Se X é bem condicionada, + rcond(X) é próximo a 1. Senão, + rcond(X) é próximo a 0. + + + [r,z]=rcond(X) ajusta r a + rcond(X) e retorna z tal que + norm(X*z,1) = r*norm(X,1)*norm(z,1) + + + Portanto, se rcond é pequeno, + z é um vetor do núcleo. + + + + Exemplos + + + + Ver Também + + + svd + + + cond + + + inv + + + + diff --git a/modules/linear_algebra/help/pt_BR/matrix/rref.xml b/modules/linear_algebra/help/pt_BR/matrix/rref.xml new file mode 100755 index 000000000..c785ce98e --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/matrix/rref.xml @@ -0,0 +1,73 @@ + + + + + rref + computa a matriz-linha reduzida a forma escada por + transformações de LU + + + + Seqüência de Chamamento + R=rref(A) + + + Parâmetros + + + A + + matriz m x n com entradas escalares + + + + R + + matriz m x n, forma escada de A + + + + + + Descrição + + rref computa a forma escada de linhas reduzidas + da matriz dada pela decomposição esquerda LU. Se for necessária a + transformação usada, basta chamar X=rref([A,eye(m,m)]) + A forma escada de linhas reduzidas R é + X(:,1:n) e a transformação esquerda + L ié dada por X(:,n+1:n+m) tal como + L*A=R + + + + Exemplos + + + + Ver Também + + + lu + + + qr + + + + diff --git a/modules/linear_algebra/help/pt_BR/matrix/trace.xml b/modules/linear_algebra/help/pt_BR/matrix/trace.xml new file mode 100755 index 000000000..4e168bcc2 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/matrix/trace.xml @@ -0,0 +1,60 @@ + + + + + trace + traço de uma matriz + + + Seqüência de Chamamento + trace(X) + + + Parâmetros + + + X + + matriz de reais ou complexos, matriz de polinômios ou de + razões de polinômios. + + + + + + + Descrição + + trace(X) é o traço da matriz + X. + + + É o mesmo que sum(diag(X)). + + + + Exemplos + + + + Ver Também + + + det + + + + diff --git a/modules/linear_algebra/help/pt_BR/pencil/CHAPTER b/modules/linear_algebra/help/pt_BR/pencil/CHAPTER new file mode 100755 index 000000000..86d1da116 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/pencil/CHAPTER @@ -0,0 +1,2 @@ +title = Matrix Pencil + diff --git a/modules/linear_algebra/help/pt_BR/pencil/companion.xml b/modules/linear_algebra/help/pt_BR/pencil/companion.xml new file mode 100755 index 000000000..1778f00ce --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/pencil/companion.xml @@ -0,0 +1,79 @@ + + + + + companion + matriz companheira + + + Seqüência de Chamamento + A=companion(p) + + + Parâmetros + + + p + + polinômio ou vetor de polinômios + + + + A + + matriz quadrada + + + + + + Descrição + + Retorna uma matriz quadrada A com o polinômio + característico igual a p se p é + mônico. Se p não é mônico, o polinômio característico + de A é igual a p/c onde + c é o coeficiente do termo de maior grau em + p. + + + Se p é um vetor de polinômios mônicos, + A é diagonal em blocos, e o polinômio característico do + i-ésimo bloco é p(i). + + + + Exemplos + + + + Ver Também + + + spec + + + poly + + + randpencil + + + + diff --git a/modules/linear_algebra/help/pt_BR/pencil/ereduc.xml b/modules/linear_algebra/help/pt_BR/pencil/ereduc.xml new file mode 100755 index 000000000..92a68a905 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/pencil/ereduc.xml @@ -0,0 +1,123 @@ + + + + + ereduc + computa de forma escada de colunas da matriz por + transformações qz + + + + Seqüência de Chamamento + [E,Q,Z [,stair [,rk]]]=ereduc(X,tol) + + + Parâmetros + + + X + + matriz m x n de entradas reais + + + + tol + + escalar real positivo + + + + E + + matriz em forma escada de colunas + + + + Q + + matriz unitária m x m + + + + Z + + matriz unitária n x n + + + + stair + + vetor de índices, + + + * + + + ISTAIR(i) = + j se o elemento da + borda E(i,j) é uma quina. + + + + + * + + + ISTAIR(i) = - j se o elemento da + borda E(i,j) não é uma quina. + + + + + + (i=1,...,M) + + + + + rk + + inteiro, posto estimado da matriz + + + + + + Descrição + + Dada uma matriz X mx n (não necessariamente + regular), a função ereduc calcula a matriz unitária transformada + E=Q*X*Z que está na forma escada de colunas (forma + trapezoidal). Ainda, o posto da matriz X é + determinado. + + + + Exemplos + + + + Ver Também + + + fstair + + + + + Autores + Th.G.J. Beelen (Philips Glass Eindhoven). SLICOT + + diff --git a/modules/linear_algebra/help/pt_BR/pencil/fstair.xml b/modules/linear_algebra/help/pt_BR/pencil/fstair.xml new file mode 100755 index 000000000..e78b6eae9 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/pencil/fstair.xml @@ -0,0 +1,175 @@ + + + + + fstair + computa a forma escada de feixe de colunas por transformações + qz + + + + Seqüência de Chamamento + [AE,EE,QE,ZE,blcks,muk,nuk,muk0,nuk0,mnei]=fstair(A,E,Q,Z,stair,rk,tol) + + + Parâmetros + + + A + + matriz m x n com entradas reais + + + + tol + + escalar real positivo + + + + E + + matriz de forma escada de colunas + + + + Q + + matriz unitária m x m + + + + Z + + matriz unitária n x n + + + + stair + + vetor de índices (ver ereduc) + + + + rk + + inteiro, posto estimado da matriz + + + + AE + + matriz m x n com entradas reais + + + + EE + + matriz de forma escada de colunas + + + + QE + + matriz unitária m x m + + + + ZE + + matriz unitária n x n + + + + nblcks + + é o número de submatrizes com posto linha completo >= 0 + detectado na matriz A + + + + + muk: + + array (vetor ou matriz) de inteiros de dimensão (n). Contém + as dimensões de coluna mu(k) (k=1,...,nblcks) das submatrizes com + posto coluna cheio no feixe sE(eps)-A(eps) + + + + + nuk: + + array de inteiros de dimensão (m+1). Contém as dimensões de + linha nu(k) (k=1,...,nblcks) das submatrizes com posto linha cheio + no feixe sE(eps)-A(eps) + + + + + muk0: + + array de inteiros de dimensão (n). Contém as dimensões de + coluna mu(k) (k=1,...,nblcks) das submatrizes com o posto-coluna + cheio no feixe sE(eps,inf)-A(eps,inf) + + + + + nuk: + + array de inteiros de dimensão (m+1). Contém as dimensões de + linha nu(k) (k=1,...,nblcks) das submatrizes com posto-linha cheio + no feixe sE(eps,inf)-A(eps,inf) + + + + + mnei: + + array de inteiros dimensão (4). mnei(1) = dimensão de linha + de sE(eps)-A(eps) + + + + + + + Descrição + + Dado o feixe sE-A onde a matriz + E está na forma escada de colunas, a função + fstair computa, de acordo com as necessidades do + usuário, um feixe unitário transformado QE(sEE-AE)ZE + que é mais ou menos similar à forma generalizada de Schur do feixe + sE-A. A função também produz parte da estrutura de + Kronecker para um dado feixe. + + + Q,Z são as matrizes unitárias usadas para + computar o feixe onde E está na forma escada de colunas (ver + ereduc) + + + + Ver Também + + + quaskro + + + ereduc + + + + diff --git a/modules/linear_algebra/help/pt_BR/pencil/glever.xml b/modules/linear_algebra/help/pt_BR/pencil/glever.xml new file mode 100755 index 000000000..1a4b7f839 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/pencil/glever.xml @@ -0,0 +1,119 @@ + + + + + glever + inverso do feixe de matrizes + + + Seqüência de Chamamento + [Bfs,Bis,chis]=glever(E,A [,s]) + + + Parâmetros + + + E, A + + duas matrizes de reais quadradas de igual dimensão + + + + s + + + string (o valor padrão é 's') + + + + + Bfs,Bis + + duas matrizes de polinômios + + + + chis + + polinômio + + + + + + Descrição + Computação de + + (s*E-A)^-1 + + pelo algoritmo generalizado de Leverrier para um feixe de + matrizes. + + + + chis = polinômio característico (até uma + constante multiplicativa). + + + Bfs = nmatriz de polinômios como + numerador. + + + Bis = matriz de polinômios ( - expansão de + (s*E-A)^-1 ao infinito). + + + Note o sinal - antes de Bis. + + + + Cuidado + + Esta função usa cleanp para simplificar + Bfs,Bis e chis. + + + + Exemplos + + + + Ver Também + + + rowshuff + + + det + + + invr + + + coffg + + + pencan + + + penlaur + + + + diff --git a/modules/linear_algebra/help/pt_BR/pencil/kroneck.xml b/modules/linear_algebra/help/pt_BR/pencil/kroneck.xml new file mode 100755 index 000000000..2ca6a403d --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/pencil/kroneck.xml @@ -0,0 +1,161 @@ + + + + + kroneck + forma de Kronecker de feixe de matrizes + + + Seqüência de Chamamento + [Q,Z,Qd,Zd,numbeps,numbeta]=kroneck(F) + [Q,Z,Qd,Zd,numbeps,numbeta]=kroneck(E,A) + + + + Parâmetros + + + F + + + feixe de matrizes de reais F=s*E-A + + + + + E,A + + duas matrizes de reais de mesma dimensão + + + + Q,Z + + duas matrizes quadradas ortogonais + + + + Qd,Zd + + dois vetores de inteiros + + + + numbeps,numeta + + dois vetores de inteiros + + + + + + Descrição + + Forma de Kronecker de feixe de matrizes: kroneck + computa duas matrizes ortogonais Q, Z que põem o feixe + F=s*E -A na forma triangular superior: + + + As dimensões dos quatro blocos são dadas por: + + eps=Qd(1) x Zd(1), inf=Qd(2) x + Zd(2) + + ,f = Qd(3) x Zd(3), + eta=Qd(4)xZd(4) + + + O bloco inf contém modos infinitos de + feixes. + + + O bloco f contém modos finitos de feixes. + + A estrutura dos blocos epsilon e eta é dada por + + numbeps(1) = # de blocos eps + de tamanho 0 x 1 + + + numbeps(2) = # de blocos eps + de tamanho 1 x 2 + + + numbeps(3) = # de blocos eps + de tamanho 2 x 3 etc... + + + numbeta(1) = # de blocos eta + de tamanho 1 x 0 + + + numbeta(2) = # de blocos eta + de tamanho 2 x 1 + + + numbeta(3) = # de blocos eta + de tamanho 3 x 2 etc... + + O código foi retirado de T. Beelen (Slicot-WGS group). + + + Exemplos + + + + Ver Também + + + gschur + + + gspec + + + systmat + + + pencan + + + randpencil + + + trzeros + + + + diff --git a/modules/linear_algebra/help/pt_BR/pencil/lyap.xml b/modules/linear_algebra/help/pt_BR/pencil/lyap.xml new file mode 100755 index 000000000..2d51a056c --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/pencil/lyap.xml @@ -0,0 +1,78 @@ + + + + + lyap + equação de Lyapunov + + + Seqüência de Chamamento + [X]=lyap(A,C,'c') + [X]=lyap(A,C,'d') + + + + Parâmetros + + + A, C + + + matrizes quadradas de reais, C deve ser + simétrica + + + + + + + Descrição + + X= lyap(A,C,flag) resolve as equações matriciais + de tempo contínuo ou de tempo discreto de Lyapunov: + + + Perceba que existe uma única solução se e só se um autovalor de + A não é um autovalor de -A + (flag='c') ou 1 sobre um autovalor de + A (flag='d'). + + + + Exemplos + + + + Ver Também + + + sylv + + + ctr_gram + + + obs_gram + + + + diff --git a/modules/linear_algebra/help/pt_BR/pencil/pencan.xml b/modules/linear_algebra/help/pt_BR/pencil/pencan.xml new file mode 100755 index 000000000..fba81a530 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/pencil/pencan.xml @@ -0,0 +1,107 @@ + + + + + pencan + forma canônica de feixe de matrizes + + + Seqüência de Chamamento + [Q,M,i1]=pencan(Fs) + [Q,M,i1]=pencan(E,A) + + + + Parâmetros + + + Fs + + + um feixe regular s*E-A + + + + + E,A + + duas matrizes quadradas de reais + + + + Q,M + + duas matrizes não-singulares de reais + + + + i1 + + inteiro + + + + + + Descrição + + Dado o feixe regular Fs=s*E-A, + pencan retorna as matrizes Q e + M tais que M*(s*E-A)*Q está na forma + "canônica". + + + M*E*Q é uma matriz de blocos + + + + com N nilpotente e i1 = + tamanho da matriz acima I. + + + M*A*Q é uma matriz de blocos: + + + + + Exemplos + + + + Ver Também + + + glever + + + penlaur + + + rowshuff + + + + diff --git a/modules/linear_algebra/help/pt_BR/pencil/penlaur.xml b/modules/linear_algebra/help/pt_BR/pencil/penlaur.xml new file mode 100755 index 000000000..62f4c7757 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/pencil/penlaur.xml @@ -0,0 +1,123 @@ + + + + + penlaur + Laurent coefficients of matrix pencil + + + Seqüência de Chamamento + [Si,Pi,Di,order]=penlaur(Fs) + [Si,Pi,Di,order]=penlaur(E,A) + + + + Parâmetros + + + Fs + + + um feixe regular s*E-A + + + + + E, A + + duas matrizes quadradas de reais + + + + Si,Pi,Di + + três matrizes quadradas de reais + + + + order + + inteiro + + + + + + Descrição + + penlaur computa os primeiros coeficientes de + Laurent de (s*E-A)^-1 no infinito. + + + (s*E-A)^-1 = ... + Si/s - Pi - s*Di + ... em + s = infinito. + + + order = ordem da singularidade + (ordem=índice-1). + + + O feixe de matrizes Fs=s*E-A deve ser + invertível. + + + Para um feixe de índice 0, Pi, Di,... são zero e + Si=inv(E). + + + Para um feixe de índice 1 (order=0),Di =0. + + + Para feixes de índices maiores, os termos -s^2 Di(2), -s^3 + Di(3),... + + são dados por: + + + Di(2)=Di*A*Di, + Di(3)=Di*A*Di*A*Di + + (até Di(order)). + + + + Observação + Versão experimental: há problemas quando se tem mal-condicionamento + deso*E-A + + + + Exemplos + + + + Ver Também + + + glever + + + pencan + + + rowshuff + + + + diff --git a/modules/linear_algebra/help/pt_BR/pencil/quaskro.xml b/modules/linear_algebra/help/pt_BR/pencil/quaskro.xml new file mode 100755 index 000000000..beb65fb07 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/pencil/quaskro.xml @@ -0,0 +1,134 @@ + + + + + quaskro + forma quasi-Kronecker + + + Seqüência de Chamamento + [Q,Z,Qd,Zd,numbeps,numbeta]=quaskro(F) + [Q,Z,Qd,Zd,numbeps,numbeta]=quaskro(E,A) + [Q,Z,Qd,Zd,numbeps,numbeta]=quaskro(F,tol) + [Q,Z,Qd,Zd,numbeps,numbeta]=quaskro(E,A,tol) + + + + Parâmetros + + + F + + + feixe de matrizes de reais F=s*E-A + (s=poly(0,'s')) + + + + + E,A + + duas matrizes reais de iguais dimensões + + + + tol + + número real (tolerância, valor padrão=1.d-10) + + + + Q,Z + + duas matrizes quadradas ortogonais + + + + Qd,Zd + + dois vetores de inteiros + + + + numbeps + + vetor de inteiros + + + + + + Descrição + Forma quasi-Kronecker de um feixe de matrizes: + quaskro computa duas matrizes ortogonais Q, + Z + + que põem o feixe F=s*E -A na forma + triangular superior: + + + As dimensões dos blocos são dadas por: + + eps=Qd(1) x Zd(1), inf=Qd(2) x + Zd(2) + + ,r = Qd(3) x Zd(3) + + + O bloco inf contém os modos infinitos do + feixe. + + + O bloco f contém os modos finitos do feixe + + A estrutura dos blocos epsilon é dada por: + + numbeps(1) = # de blocos eps + de tamanho 0 x 1 + + + numbeps(2) = # de blocos eps + de tamanho 1 x 2 + + + numbeps(3) = # de blocos eps + de tamanho 2 x 3 etc... + + A forma completa (de quatro blocos) de Kronecker é dada pela função + kroneck que chama a função quaskro + sobre o feixe (pertransposto) sE(r)-A(r). + + O código é retirado de T. Beelen. + + + Ver Também + + + kroneck + + + gschur + + + gspec + + + + diff --git a/modules/linear_algebra/help/pt_BR/pencil/randpencil.xml b/modules/linear_algebra/help/pt_BR/pencil/randpencil.xml new file mode 100755 index 000000000..3d0fe44e6 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/pencil/randpencil.xml @@ -0,0 +1,117 @@ + + + + + randpencil + feixe aleatório + + + Seqüência de Chamamento + F=randpencil(eps,infi,fin,eta) + + + Parâmetros + + + eps + + vetor de inteiros + + + + infi + + vetor de inteiros + + + + fin + + vetor de reais, ou polinômio mônico, ou vetor de polinômios + mônicos + + + + + eta + + vetor de inteiros + + + + F + + + feixe de matrizes de reais F=s*E-A + (s=poly(0,'s')) + + + + + + + Descrição + + Função utilitária. F=randpencil(eps,infi,fin,eta) + retorna um feixe aleatório F com dada estrutura de + Kronecker. A estrutura é dada por: eps=[eps1,...,epsk]: + estrutura de blocos epsilon (tamanho eps1x(eps1+1),....) + fin=[l1,...,ln] conjunto de autovalores finitos + (assumidos como reais) (possivelmente []) + infi=[k1,...,kp] tamanho de blocos J no infinito + ki>=1 (infi=[] se não há blocos J). + eta=[eta1,...,etap]: estrutura dos blocos eta (size + eta1+1)xeta1,...) + + + epsi's devem ser >=0, + etai's devem ser >=0, infi's + devem ser >=1. + + + Se fin é um polinômio (mônico), o bloco finito + admite raízes de fin como autovalores. + + + Se fin é um vetor de polinômios, eles são os + divisores elementares finitos de F ,i.e., as raízes de + p(i) são autovalores finitos de + F. + + + + Exemplos + + + + Ver Também + + + kroneck + + + pencan + + + penlaur + + + + diff --git a/modules/linear_algebra/help/pt_BR/pencil/rowshuff.xml b/modules/linear_algebra/help/pt_BR/pencil/rowshuff.xml new file mode 100755 index 000000000..ff28a40ce --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/pencil/rowshuff.xml @@ -0,0 +1,111 @@ + + + + + rowshuff + algoritmo de embaralhamento + + + Seqüência de Chamamento + [Ws,Fs1]=rowshuff(Fs, [alfa]) + + + Parâmetros + + + Fs + + + feixe quadrado de reais Fs = s*E-A + + + + + Ws + + matriz de polinômios + + + + Fs1 + + + feixe quadrado de reais F1s = s*E1 -A1 com + E1 não-singular + + + + + alfa + + + número real (alfa = 0 é o valor + padrão) + + + + + + + Descrição + Algoritmo de embaralhamento: dado o feixe + Fs=s*E-A, retorna Ws=W(s) (matriz quadrada de + polinômios) tal que: + + + Fs1 = s*E1-A1 = W(s)*(s*E-A) é um feixe com + matriz E1 não-singular. + + + Isto é possível se, e só se, o feixe Fs = s*E-A é + regular (i.e., invertível). O grau de Ws é igual ao + índice do feixe. + + + Os pólos no infinito de Fs asão colocados para + alfa e os zeros de Ws estão em + alfa. + + + Note que (s*E-A)^-1 = (s*E1-A1)^-1 * W(s) = + (W(s)*(s*E-A))^-1 *W(s) + + + + + Exemplos + + + + Ver Também + + + pencan + + + glever + + + penlaur + + + + diff --git a/modules/linear_algebra/help/pt_BR/pencil/sylv.xml b/modules/linear_algebra/help/pt_BR/pencil/sylv.xml new file mode 100755 index 000000000..fcba9fc28 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/pencil/sylv.xml @@ -0,0 +1,77 @@ + + + + + sylv + equação de Sylvester + + + Seqüência de Chamamento + sylv(A,B,C,flag) + + + Parâmetros + + + A,B,C + + três matrizes de reais de dimensões apropriadas + + + + flag + + + string ('c' ou + 'd') + + + + + + + Descrição + + X= sylv(A,B,C,'c') computa X, + solução da equação de "tempo contínuo" de Sylvester. + + + + X=sylv(A,B,C,'d') computa X, + solução da equação de "tempo discreto" de Sylvester. + + + + + Exemplos + + + + Ver Também + + + lyap + + + + diff --git a/modules/linear_algebra/help/pt_BR/proj.xml b/modules/linear_algebra/help/pt_BR/proj.xml new file mode 100755 index 000000000..cef345ed3 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/proj.xml @@ -0,0 +1,73 @@ + + + + + proj + projeção + + + Seqüência de Chamamento + P = proj(X1,X2) + + + Parâmetros + + + X1,X2 + + duas matrizes reais com igual número de colunas + + + + P + + + matriz de projeção de real (P^2=P) + + + + + + + Descrição + + P é a projeção sobre X2 + paralela a X1. + + + + Ver Também + + + + See Also + + + projspec + + + orth + + + fullrf + + + + diff --git a/modules/linear_algebra/help/pt_BR/state_space/CHAPTER b/modules/linear_algebra/help/pt_BR/state_space/CHAPTER new file mode 100755 index 000000000..a0b62cdee --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/state_space/CHAPTER @@ -0,0 +1,2 @@ +title = State-Space Matrices + diff --git a/modules/linear_algebra/help/pt_BR/state_space/coff.xml b/modules/linear_algebra/help/pt_BR/state_space/coff.xml new file mode 100755 index 000000000..f75f38f77 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/state_space/coff.xml @@ -0,0 +1,99 @@ + + + + + coff + resolvente (método do cofator) + + + Seqüência de Chamamento + [N,d]=coff(M [,var]) + + + Parâmetros + + + M + + matriz quadrada de reais + + + + var + + string + + + + N + + matriz de polinômios (com o mesmo tamanho que + M) + + + + + d + + polinômio ( polinômio característico + poly(A,'s')) + + + + + + + Descrição + + coff computa R=(s*eye()-M)^-1 + para M uma matriz de reais. R é dado por + N/d. + + + N = matriz de polinômios como numerador. + + + d = denominador comum. + + + var string ( ('s' se + omitido) + + + + Exemplos + + + + Ver Também + + + coffg + + + ss2tf + + + nlev + + + poly + + + + diff --git a/modules/linear_algebra/help/pt_BR/state_space/nlev.xml b/modules/linear_algebra/help/pt_BR/state_space/nlev.xml new file mode 100755 index 000000000..90ddd6dd2 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/state_space/nlev.xml @@ -0,0 +1,90 @@ + + + + + nlev + algoritmo de Leverrier + + + Seqüência de Chamamento + [num,den]=nlev(A,z [,rmax]) + + + Parâmetros + + + A + + matriz quadrada de reais + + + + z + + string + + + + rmax + + + parâmetro opcional (ver bdiag) + + + + + + + Descrição + + [num,den]=nlev(A,z [,rmax]) computa + (z*eye()-A)^(-1) + + por diagonalização por blocos de A seguido pelo algoritmo de + Leverrier em cada bloco. + + Este algoritmo é melhor que o algoritmo usual de Leverrier, mas + ainda não está perfeito! + + + + Exemplos + + + + Ver Também + + + coff + + + coffg + + + glever + + + ss2tf + + + + + Autores + F. Delebecque., S. Steer INRIA; + + diff --git a/modules/linear_algebra/help/pt_BR/subspaces/CHAPTER b/modules/linear_algebra/help/pt_BR/subspaces/CHAPTER new file mode 100755 index 000000000..d87d9ca5e --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/subspaces/CHAPTER @@ -0,0 +1,3 @@ +title = Subspaces + + diff --git a/modules/linear_algebra/help/pt_BR/subspaces/spaninter.xml b/modules/linear_algebra/help/pt_BR/subspaces/spaninter.xml new file mode 100755 index 000000000..894f73b4b --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/subspaces/spaninter.xml @@ -0,0 +1,98 @@ + + + + + spaninter + interseção de subespaços + + + Seqüência de Chamamento + [X,dim]=spaninter(A,B [,tol]) + + + Parâmetros + + + A, B + + duas matrizes de reais ou de complexos com igual número de + linhas + + + + + X + + matriz quadrada ortogonal ou unitária + + + + dim + + + inteiro, dimensão do subesbaço Im(A) inter + Im(B) + + + + + + + Descrição + + Computa a intereseção de Im(A) e + Im(B). + + + As primeiras dim colunas de X + geram esta interseção i.e. X(:,1:dim) é uma base + ortogonal para + + + Im(A) inter Im(B) + + + Na base X, A e + B são respectivamente representados por: + + + X'*A e X'*B. + + + tol é um limiar (sqrt(%eps) é + o valor padrão). + + + + Exemplos + + + + Ver Também + + + spanplus + + + spantwo + + + + diff --git a/modules/linear_algebra/help/pt_BR/subspaces/spanplus.xml b/modules/linear_algebra/help/pt_BR/subspaces/spanplus.xml new file mode 100755 index 000000000..3bed11264 --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/subspaces/spanplus.xml @@ -0,0 +1,103 @@ + + + + + spanplus + soma de subespaços + + + Seqüência de Chamamento + [X,dim,dima]=spanplus(A,B[,tol]) + + + Parâmetros + + + A, B + + duas matrizes de reais ou complexos com igual número de + linhas + + + + + X + + matriz quadrada ortogonal ou unitária + + + + dim, dima + + inteiros, dimensões de subespaços + + + + tol + + número real não-negativo + + + + + + Descrição + Computa a base X tal que: + + as primeiras dima colunas de X + geram Im(A) e as (dim-dima) colunas + seguintes formam uma base de A+B em relação a + A. + + + As dim primeiras colunas de X + formam uma base para A+B. + + Tem-se a seguinte forma canônica para + [A,B]: + + + + tol é um argumento opcional (ver código da + função). + + + + Exemplos + + + + Ver Também + + + spaninter + + + im_inv + + + spantwo + + + + diff --git a/modules/linear_algebra/help/pt_BR/subspaces/spantwo.xml b/modules/linear_algebra/help/pt_BR/subspaces/spantwo.xml new file mode 100755 index 000000000..96278172b --- /dev/null +++ b/modules/linear_algebra/help/pt_BR/subspaces/spantwo.xml @@ -0,0 +1,119 @@ + + + + + spantwo + soma e interseção de subespaços + + + Seqüência de Chamamento + [Xp,dima,dimb,dim]=spantwo(A,B, [tol]) + + + Parâmetros + + + A, B + + duas matrizes de reais ou complexos com igual número de linhas + + + + + Xp + + matriz quadrada não-singular + + + + dima, dimb, dim + + inteiros, dimensões dos subespaços + + + + tol + + número real não-negativo + + + + + + Descrição + + Dadas duas matrizes A e B com + o mesmo número de linhas, retorna uma matriz quadrada + Xp (não-singular, mas não necessariamente ortogonal) + tal que : + + + + As primeiras dima colunas de + inv(Xp) geram Im(A). + + + As colunas de dim-dimb+1 até + dima de inv(Xp) geram a interseção + de Im(A) e Im(B). + + + As primeiras dim colunas de + inv(Xp) geram + Im(A)+Im(B). + + + As colunas de dim-dimb+1 até + dim de inv(Xp) geram + Im(B). + + + A matrix [A1;A2] tem posto-linha cheio (= + posto(A)), a matrix [B2;B3] tem posto-linha cheio + (=posto(B)), a matriz [A2,B2] tem posto-linha (=posto(A + inter B)) e a matriz [A1,0;A2,B2;0,B3] tem posto-linha + cheio (=posto(A+B)). + + + + Exemplos + + + + Ver Também + + + spanplus + + + spaninter + + + + diff --git a/modules/linear_algebra/help/ru_RU/addchapter.sce b/modules/linear_algebra/help/ru_RU/addchapter.sce new file mode 100755 index 000000000..9c126ad92 --- /dev/null +++ b/modules/linear_algebra/help/ru_RU/addchapter.sce @@ -0,0 +1,11 @@ +// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +// Copyright (C) 2009 - DIGITEO +// +// This file must be used under the terms of the CeCILL. +// This source file is licensed as described in the file COPYING, which +// you should have received as part of this distribution. The terms +// are also available at +// http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt + +add_help_chapter("Linear Algebra",SCI+"/modules/linear_algebra/help/ru_RU",%T); + -- cgit