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diff --git a/src/fortran/lapack/dtgsen.f b/src/fortran/lapack/dtgsen.f new file mode 100644 index 0000000..917a7b0 --- /dev/null +++ b/src/fortran/lapack/dtgsen.f @@ -0,0 +1,723 @@ + SUBROUTINE DTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, + $ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL, + $ PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO ) +* +* -- LAPACK routine (version 3.1) -- +* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. +* November 2006 +* +* Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. +* +* .. Scalar Arguments .. + LOGICAL WANTQ, WANTZ + INTEGER IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK, + $ M, N + DOUBLE PRECISION PL, PR +* .. +* .. Array Arguments .. + LOGICAL SELECT( * ) + INTEGER IWORK( * ) + DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), + $ B( LDB, * ), BETA( * ), DIF( * ), Q( LDQ, * ), + $ WORK( * ), Z( LDZ, * ) +* .. +* +* Purpose +* ======= +* +* DTGSEN reorders the generalized real Schur decomposition of a real +* matrix pair (A, B) (in terms of an orthonormal equivalence trans- +* formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues +* appears in the leading diagonal blocks of the upper quasi-triangular +* matrix A and the upper triangular B. The leading columns of Q and +* Z form orthonormal bases of the corresponding left and right eigen- +* spaces (deflating subspaces). (A, B) must be in generalized real +* Schur canonical form (as returned by DGGES), i.e. A is block upper +* triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper +* triangular. +* +* DTGSEN also computes the generalized eigenvalues +* +* w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j) +* +* of the reordered matrix pair (A, B). +* +* Optionally, DTGSEN computes the estimates of reciprocal condition +* numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), +* (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) +* between the matrix pairs (A11, B11) and (A22,B22) that correspond to +* the selected cluster and the eigenvalues outside the cluster, resp., +* and norms of "projections" onto left and right eigenspaces w.r.t. +* the selected cluster in the (1,1)-block. +* +* Arguments +* ========= +* +* IJOB (input) INTEGER +* Specifies whether condition numbers are required for the +* cluster of eigenvalues (PL and PR) or the deflating subspaces +* (Difu and Difl): +* =0: Only reorder w.r.t. SELECT. No extras. +* =1: Reciprocal of norms of "projections" onto left and right +* eigenspaces w.r.t. the selected cluster (PL and PR). +* =2: Upper bounds on Difu and Difl. F-norm-based estimate +* (DIF(1:2)). +* =3: Estimate of Difu and Difl. 1-norm-based estimate +* (DIF(1:2)). +* About 5 times as expensive as IJOB = 2. +* =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic +* version to get it all. +* =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above) +* +* WANTQ (input) LOGICAL +* .TRUE. : update the left transformation matrix Q; +* .FALSE.: do not update Q. +* +* WANTZ (input) LOGICAL +* .TRUE. : update the right transformation matrix Z; +* .FALSE.: do not update Z. +* +* SELECT (input) LOGICAL array, dimension (N) +* SELECT specifies the eigenvalues in the selected cluster. +* To select a real eigenvalue w(j), SELECT(j) must be set to +* .TRUE.. To select a complex conjugate pair of eigenvalues +* w(j) and w(j+1), corresponding to a 2-by-2 diagonal block, +* either SELECT(j) or SELECT(j+1) or both must be set to +* .TRUE.; a complex conjugate pair of eigenvalues must be +* either both included in the cluster or both excluded. +* +* N (input) INTEGER +* The order of the matrices A and B. N >= 0. +* +* A (input/output) DOUBLE PRECISION array, dimension(LDA,N) +* On entry, the upper quasi-triangular matrix A, with (A, B) in +* generalized real Schur canonical form. +* On exit, A is overwritten by the reordered matrix A. +* +* LDA (input) INTEGER +* The leading dimension of the array A. LDA >= max(1,N). +* +* B (input/output) DOUBLE PRECISION array, dimension(LDB,N) +* On entry, the upper triangular matrix B, with (A, B) in +* generalized real Schur canonical form. +* On exit, B is overwritten by the reordered matrix B. +* +* LDB (input) INTEGER +* The leading dimension of the array B. LDB >= max(1,N). +* +* ALPHAR (output) DOUBLE PRECISION array, dimension (N) +* ALPHAI (output) DOUBLE PRECISION array, dimension (N) +* BETA (output) DOUBLE PRECISION array, dimension (N) +* On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will +* be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i +* and BETA(j),j=1,...,N are the diagonals of the complex Schur +* form (S,T) that would result if the 2-by-2 diagonal blocks of +* the real generalized Schur form of (A,B) were further reduced +* to triangular form using complex unitary transformations. +* If ALPHAI(j) is zero, then the j-th eigenvalue is real; if +* positive, then the j-th and (j+1)-st eigenvalues are a +* complex conjugate pair, with ALPHAI(j+1) negative. +* +* Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N) +* On entry, if WANTQ = .TRUE., Q is an N-by-N matrix. +* On exit, Q has been postmultiplied by the left orthogonal +* transformation matrix which reorder (A, B); The leading M +* columns of Q form orthonormal bases for the specified pair of +* left eigenspaces (deflating subspaces). +* If WANTQ = .FALSE., Q is not referenced. +* +* LDQ (input) INTEGER +* The leading dimension of the array Q. LDQ >= 1; +* and if WANTQ = .TRUE., LDQ >= N. +* +* Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N) +* On entry, if WANTZ = .TRUE., Z is an N-by-N matrix. +* On exit, Z has been postmultiplied by the left orthogonal +* transformation matrix which reorder (A, B); The leading M +* columns of Z form orthonormal bases for the specified pair of +* left eigenspaces (deflating subspaces). +* If WANTZ = .FALSE., Z is not referenced. +* +* LDZ (input) INTEGER +* The leading dimension of the array Z. LDZ >= 1; +* If WANTZ = .TRUE., LDZ >= N. +* +* M (output) INTEGER +* The dimension of the specified pair of left and right eigen- +* spaces (deflating subspaces). 0 <= M <= N. +* +* PL (output) DOUBLE PRECISION +* PR (output) DOUBLE PRECISION +* If IJOB = 1, 4 or 5, PL, PR are lower bounds on the +* reciprocal of the norm of "projections" onto left and right +* eigenspaces with respect to the selected cluster. +* 0 < PL, PR <= 1. +* If M = 0 or M = N, PL = PR = 1. +* If IJOB = 0, 2 or 3, PL and PR are not referenced. +* +* DIF (output) DOUBLE PRECISION array, dimension (2). +* If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl. +* If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on +* Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based +* estimates of Difu and Difl. +* If M = 0 or N, DIF(1:2) = F-norm([A, B]). +* If IJOB = 0 or 1, DIF is not referenced. +* +* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) +* IF IJOB = 0, WORK is not referenced. Otherwise, +* on exit, if INFO = 0, WORK(1) returns the optimal LWORK. +* +* LWORK (input) INTEGER +* The dimension of the array WORK. LWORK >= 4*N+16. +* If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)). +* If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)). +* +* If LWORK = -1, then a workspace query is assumed; the routine +* only calculates the optimal size of the WORK array, returns +* this value as the first entry of the WORK array, and no error +* message related to LWORK is issued by XERBLA. +* +* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) +* IF IJOB = 0, IWORK is not referenced. Otherwise, +* on exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. +* +* LIWORK (input) INTEGER +* The dimension of the array IWORK. LIWORK >= 1. +* If IJOB = 1, 2 or 4, LIWORK >= N+6. +* If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6). +* +* If LIWORK = -1, then a workspace query is assumed; the +* routine only calculates the optimal size of the IWORK array, +* returns this value as the first entry of the IWORK array, and +* no error message related to LIWORK is issued by XERBLA. +* +* INFO (output) INTEGER +* =0: Successful exit. +* <0: If INFO = -i, the i-th argument had an illegal value. +* =1: Reordering of (A, B) failed because the transformed +* matrix pair (A, B) would be too far from generalized +* Schur form; the problem is very ill-conditioned. +* (A, B) may have been partially reordered. +* If requested, 0 is returned in DIF(*), PL and PR. +* +* Further Details +* =============== +* +* DTGSEN first collects the selected eigenvalues by computing +* orthogonal U and W that move them to the top left corner of (A, B). +* In other words, the selected eigenvalues are the eigenvalues of +* (A11, B11) in: +* +* U'*(A, B)*W = (A11 A12) (B11 B12) n1 +* ( 0 A22),( 0 B22) n2 +* n1 n2 n1 n2 +* +* where N = n1+n2 and U' means the transpose of U. The first n1 columns +* of U and W span the specified pair of left and right eigenspaces +* (deflating subspaces) of (A, B). +* +* If (A, B) has been obtained from the generalized real Schur +* decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the +* reordered generalized real Schur form of (C, D) is given by +* +* (C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)', +* +* and the first n1 columns of Q*U and Z*W span the corresponding +* deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). +* +* Note that if the selected eigenvalue is sufficiently ill-conditioned, +* then its value may differ significantly from its value before +* reordering. +* +* The reciprocal condition numbers of the left and right eigenspaces +* spanned by the first n1 columns of U and W (or Q*U and Z*W) may +* be returned in DIF(1:2), corresponding to Difu and Difl, resp. +* +* The Difu and Difl are defined as: +* +* Difu[(A11, B11), (A22, B22)] = sigma-min( Zu ) +* and +* Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)], +* +* where sigma-min(Zu) is the smallest singular value of the +* (2*n1*n2)-by-(2*n1*n2) matrix +* +* Zu = [ kron(In2, A11) -kron(A22', In1) ] +* [ kron(In2, B11) -kron(B22', In1) ]. +* +* Here, Inx is the identity matrix of size nx and A22' is the +* transpose of A22. kron(X, Y) is the Kronecker product between +* the matrices X and Y. +* +* When DIF(2) is small, small changes in (A, B) can cause large changes +* in the deflating subspace. An approximate (asymptotic) bound on the +* maximum angular error in the computed deflating subspaces is +* +* EPS * norm((A, B)) / DIF(2), +* +* where EPS is the machine precision. +* +* The reciprocal norm of the projectors on the left and right +* eigenspaces associated with (A11, B11) may be returned in PL and PR. +* They are computed as follows. First we compute L and R so that +* P*(A, B)*Q is block diagonal, where +* +* P = ( I -L ) n1 Q = ( I R ) n1 +* ( 0 I ) n2 and ( 0 I ) n2 +* n1 n2 n1 n2 +* +* and (L, R) is the solution to the generalized Sylvester equation +* +* A11*R - L*A22 = -A12 +* B11*R - L*B22 = -B12 +* +* Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2). +* An approximate (asymptotic) bound on the average absolute error of +* the selected eigenvalues is +* +* EPS * norm((A, B)) / PL. +* +* There are also global error bounds which valid for perturbations up +* to a certain restriction: A lower bound (x) on the smallest +* F-norm(E,F) for which an eigenvalue of (A11, B11) may move and +* coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), +* (i.e. (A + E, B + F), is +* +* x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)). +* +* An approximate bound on x can be computed from DIF(1:2), PL and PR. +* +* If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed +* (L', R') and unperturbed (L, R) left and right deflating subspaces +* associated with the selected cluster in the (1,1)-blocks can be +* bounded as +* +* max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2)) +* max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2)) +* +* See LAPACK User's Guide section 4.11 or the following references +* for more information. +* +* Note that if the default method for computing the Frobenius-norm- +* based estimate DIF is not wanted (see DLATDF), then the parameter +* IDIFJB (see below) should be changed from 3 to 4 (routine DLATDF +* (IJOB = 2 will be used)). See DTGSYL for more details. +* +* Based on contributions by +* Bo Kagstrom and Peter Poromaa, Department of Computing Science, +* Umea University, S-901 87 Umea, Sweden. +* +* References +* ========== +* +* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the +* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in +* M.S. Moonen et al (eds), Linear Algebra for Large Scale and +* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. +* +* [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified +* Eigenvalues of a Regular Matrix Pair (A, B) and Condition +* Estimation: Theory, Algorithms and Software, +* Report UMINF - 94.04, Department of Computing Science, Umea +* University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working +* Note 87. To appear in Numerical Algorithms, 1996. +* +* [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software +* for Solving the Generalized Sylvester Equation and Estimating the +* Separation between Regular Matrix Pairs, Report UMINF - 93.23, +* Department of Computing Science, Umea University, S-901 87 Umea, +* Sweden, December 1993, Revised April 1994, Also as LAPACK Working +* Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, +* 1996. +* +* ===================================================================== +* +* .. Parameters .. + INTEGER IDIFJB + PARAMETER ( IDIFJB = 3 ) + DOUBLE PRECISION ZERO, ONE + PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) +* .. +* .. Local Scalars .. + LOGICAL LQUERY, PAIR, SWAP, WANTD, WANTD1, WANTD2, + $ WANTP + INTEGER I, IERR, IJB, K, KASE, KK, KS, LIWMIN, LWMIN, + $ MN2, N1, N2 + DOUBLE PRECISION DSCALE, DSUM, EPS, RDSCAL, SMLNUM +* .. +* .. Local Arrays .. + INTEGER ISAVE( 3 ) +* .. +* .. External Subroutines .. + EXTERNAL DLACN2, DLACPY, DLAG2, DLASSQ, DTGEXC, DTGSYL, + $ XERBLA +* .. +* .. External Functions .. + DOUBLE PRECISION DLAMCH + EXTERNAL DLAMCH +* .. +* .. Intrinsic Functions .. + INTRINSIC MAX, SIGN, SQRT +* .. +* .. Executable Statements .. +* +* Decode and test the input parameters +* + INFO = 0 + LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) +* + IF( IJOB.LT.0 .OR. IJOB.GT.5 ) THEN + INFO = -1 + ELSE IF( N.LT.0 ) THEN + INFO = -5 + ELSE IF( LDA.LT.MAX( 1, N ) ) THEN + INFO = -7 + ELSE IF( LDB.LT.MAX( 1, N ) ) THEN + INFO = -9 + ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN + INFO = -14 + ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN + INFO = -16 + END IF +* + IF( INFO.NE.0 ) THEN + CALL XERBLA( 'DTGSEN', -INFO ) + RETURN + END IF +* +* Get machine constants +* + EPS = DLAMCH( 'P' ) + SMLNUM = DLAMCH( 'S' ) / EPS + IERR = 0 +* + WANTP = IJOB.EQ.1 .OR. IJOB.GE.4 + WANTD1 = IJOB.EQ.2 .OR. IJOB.EQ.4 + WANTD2 = IJOB.EQ.3 .OR. IJOB.EQ.5 + WANTD = WANTD1 .OR. WANTD2 +* +* Set M to the dimension of the specified pair of deflating +* subspaces. +* + M = 0 + PAIR = .FALSE. + DO 10 K = 1, N + IF( PAIR ) THEN + PAIR = .FALSE. + ELSE + IF( K.LT.N ) THEN + IF( A( K+1, K ).EQ.ZERO ) THEN + IF( SELECT( K ) ) + $ M = M + 1 + ELSE + PAIR = .TRUE. + IF( SELECT( K ) .OR. SELECT( K+1 ) ) + $ M = M + 2 + END IF + ELSE + IF( SELECT( N ) ) + $ M = M + 1 + END IF + END IF + 10 CONTINUE +* + IF( IJOB.EQ.1 .OR. IJOB.EQ.2 .OR. IJOB.EQ.4 ) THEN + LWMIN = MAX( 1, 4*N+16, 2*M*( N-M ) ) + LIWMIN = MAX( 1, N+6 ) + ELSE IF( IJOB.EQ.3 .OR. IJOB.EQ.5 ) THEN + LWMIN = MAX( 1, 4*N+16, 4*M*( N-M ) ) + LIWMIN = MAX( 1, 2*M*( N-M ), N+6 ) + ELSE + LWMIN = MAX( 1, 4*N+16 ) + LIWMIN = 1 + END IF +* + WORK( 1 ) = LWMIN + IWORK( 1 ) = LIWMIN +* + IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN + INFO = -22 + ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN + INFO = -24 + END IF +* + IF( INFO.NE.0 ) THEN + CALL XERBLA( 'DTGSEN', -INFO ) + RETURN + ELSE IF( LQUERY ) THEN + RETURN + END IF +* +* Quick return if possible. +* + IF( M.EQ.N .OR. M.EQ.0 ) THEN + IF( WANTP ) THEN + PL = ONE + PR = ONE + END IF + IF( WANTD ) THEN + DSCALE = ZERO + DSUM = ONE + DO 20 I = 1, N + CALL DLASSQ( N, A( 1, I ), 1, DSCALE, DSUM ) + CALL DLASSQ( N, B( 1, I ), 1, DSCALE, DSUM ) + 20 CONTINUE + DIF( 1 ) = DSCALE*SQRT( DSUM ) + DIF( 2 ) = DIF( 1 ) + END IF + GO TO 60 + END IF +* +* Collect the selected blocks at the top-left corner of (A, B). +* + KS = 0 + PAIR = .FALSE. + DO 30 K = 1, N + IF( PAIR ) THEN + PAIR = .FALSE. + ELSE +* + SWAP = SELECT( K ) + IF( K.LT.N ) THEN + IF( A( K+1, K ).NE.ZERO ) THEN + PAIR = .TRUE. + SWAP = SWAP .OR. SELECT( K+1 ) + END IF + END IF +* + IF( SWAP ) THEN + KS = KS + 1 +* +* Swap the K-th block to position KS. +* Perform the reordering of diagonal blocks in (A, B) +* by orthogonal transformation matrices and update +* Q and Z accordingly (if requested): +* + KK = K + IF( K.NE.KS ) + $ CALL DTGEXC( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, + $ Z, LDZ, KK, KS, WORK, LWORK, IERR ) +* + IF( IERR.GT.0 ) THEN +* +* Swap is rejected: exit. +* + INFO = 1 + IF( WANTP ) THEN + PL = ZERO + PR = ZERO + END IF + IF( WANTD ) THEN + DIF( 1 ) = ZERO + DIF( 2 ) = ZERO + END IF + GO TO 60 + END IF +* + IF( PAIR ) + $ KS = KS + 1 + END IF + END IF + 30 CONTINUE + IF( WANTP ) THEN +* +* Solve generalized Sylvester equation for R and L +* and compute PL and PR. +* + N1 = M + N2 = N - M + I = N1 + 1 + IJB = 0 + CALL DLACPY( 'Full', N1, N2, A( 1, I ), LDA, WORK, N1 ) + CALL DLACPY( 'Full', N1, N2, B( 1, I ), LDB, WORK( N1*N2+1 ), + $ N1 ) + CALL DTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA, WORK, + $ N1, B, LDB, B( I, I ), LDB, WORK( N1*N2+1 ), N1, + $ DSCALE, DIF( 1 ), WORK( N1*N2*2+1 ), + $ LWORK-2*N1*N2, IWORK, IERR ) +* +* Estimate the reciprocal of norms of "projections" onto left +* and right eigenspaces. +* + RDSCAL = ZERO + DSUM = ONE + CALL DLASSQ( N1*N2, WORK, 1, RDSCAL, DSUM ) + PL = RDSCAL*SQRT( DSUM ) + IF( PL.EQ.ZERO ) THEN + PL = ONE + ELSE + PL = DSCALE / ( SQRT( DSCALE*DSCALE / PL+PL )*SQRT( PL ) ) + END IF + RDSCAL = ZERO + DSUM = ONE + CALL DLASSQ( N1*N2, WORK( N1*N2+1 ), 1, RDSCAL, DSUM ) + PR = RDSCAL*SQRT( DSUM ) + IF( PR.EQ.ZERO ) THEN + PR = ONE + ELSE + PR = DSCALE / ( SQRT( DSCALE*DSCALE / PR+PR )*SQRT( PR ) ) + END IF + END IF +* + IF( WANTD ) THEN +* +* Compute estimates of Difu and Difl. +* + IF( WANTD1 ) THEN + N1 = M + N2 = N - M + I = N1 + 1 + IJB = IDIFJB +* +* Frobenius norm-based Difu-estimate. +* + CALL DTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA, WORK, + $ N1, B, LDB, B( I, I ), LDB, WORK( N1*N2+1 ), + $ N1, DSCALE, DIF( 1 ), WORK( 2*N1*N2+1 ), + $ LWORK-2*N1*N2, IWORK, IERR ) +* +* Frobenius norm-based Difl-estimate. +* + CALL DTGSYL( 'N', IJB, N2, N1, A( I, I ), LDA, A, LDA, WORK, + $ N2, B( I, I ), LDB, B, LDB, WORK( N1*N2+1 ), + $ N2, DSCALE, DIF( 2 ), WORK( 2*N1*N2+1 ), + $ LWORK-2*N1*N2, IWORK, IERR ) + ELSE +* +* +* Compute 1-norm-based estimates of Difu and Difl using +* reversed communication with DLACN2. In each step a +* generalized Sylvester equation or a transposed variant +* is solved. +* + KASE = 0 + N1 = M + N2 = N - M + I = N1 + 1 + IJB = 0 + MN2 = 2*N1*N2 +* +* 1-norm-based estimate of Difu. +* + 40 CONTINUE + CALL DLACN2( MN2, WORK( MN2+1 ), WORK, IWORK, DIF( 1 ), + $ KASE, ISAVE ) + IF( KASE.NE.0 ) THEN + IF( KASE.EQ.1 ) THEN +* +* Solve generalized Sylvester equation. +* + CALL DTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA, + $ WORK, N1, B, LDB, B( I, I ), LDB, + $ WORK( N1*N2+1 ), N1, DSCALE, DIF( 1 ), + $ WORK( 2*N1*N2+1 ), LWORK-2*N1*N2, IWORK, + $ IERR ) + ELSE +* +* Solve the transposed variant. +* + CALL DTGSYL( 'T', IJB, N1, N2, A, LDA, A( I, I ), LDA, + $ WORK, N1, B, LDB, B( I, I ), LDB, + $ WORK( N1*N2+1 ), N1, DSCALE, DIF( 1 ), + $ WORK( 2*N1*N2+1 ), LWORK-2*N1*N2, IWORK, + $ IERR ) + END IF + GO TO 40 + END IF + DIF( 1 ) = DSCALE / DIF( 1 ) +* +* 1-norm-based estimate of Difl. +* + 50 CONTINUE + CALL DLACN2( MN2, WORK( MN2+1 ), WORK, IWORK, DIF( 2 ), + $ KASE, ISAVE ) + IF( KASE.NE.0 ) THEN + IF( KASE.EQ.1 ) THEN +* +* Solve generalized Sylvester equation. +* + CALL DTGSYL( 'N', IJB, N2, N1, A( I, I ), LDA, A, LDA, + $ WORK, N2, B( I, I ), LDB, B, LDB, + $ WORK( N1*N2+1 ), N2, DSCALE, DIF( 2 ), + $ WORK( 2*N1*N2+1 ), LWORK-2*N1*N2, IWORK, + $ IERR ) + ELSE +* +* Solve the transposed variant. +* + CALL DTGSYL( 'T', IJB, N2, N1, A( I, I ), LDA, A, LDA, + $ WORK, N2, B( I, I ), LDB, B, LDB, + $ WORK( N1*N2+1 ), N2, DSCALE, DIF( 2 ), + $ WORK( 2*N1*N2+1 ), LWORK-2*N1*N2, IWORK, + $ IERR ) + END IF + GO TO 50 + END IF + DIF( 2 ) = DSCALE / DIF( 2 ) +* + END IF + END IF +* + 60 CONTINUE +* +* Compute generalized eigenvalues of reordered pair (A, B) and +* normalize the generalized Schur form. +* + PAIR = .FALSE. + DO 80 K = 1, N + IF( PAIR ) THEN + PAIR = .FALSE. + ELSE +* + IF( K.LT.N ) THEN + IF( A( K+1, K ).NE.ZERO ) THEN + PAIR = .TRUE. + END IF + END IF +* + IF( PAIR ) THEN +* +* Compute the eigenvalue(s) at position K. +* + WORK( 1 ) = A( K, K ) + WORK( 2 ) = A( K+1, K ) + WORK( 3 ) = A( K, K+1 ) + WORK( 4 ) = A( K+1, K+1 ) + WORK( 5 ) = B( K, K ) + WORK( 6 ) = B( K+1, K ) + WORK( 7 ) = B( K, K+1 ) + WORK( 8 ) = B( K+1, K+1 ) + CALL DLAG2( WORK, 2, WORK( 5 ), 2, SMLNUM*EPS, BETA( K ), + $ BETA( K+1 ), ALPHAR( K ), ALPHAR( K+1 ), + $ ALPHAI( K ) ) + ALPHAI( K+1 ) = -ALPHAI( K ) +* + ELSE +* + IF( SIGN( ONE, B( K, K ) ).LT.ZERO ) THEN +* +* If B(K,K) is negative, make it positive +* + DO 70 I = 1, N + A( K, I ) = -A( K, I ) + B( K, I ) = -B( K, I ) + Q( I, K ) = -Q( I, K ) + 70 CONTINUE + END IF +* + ALPHAR( K ) = A( K, K ) + ALPHAI( K ) = ZERO + BETA( K ) = B( K, K ) +* + END IF + END IF + 80 CONTINUE +* + WORK( 1 ) = LWMIN + IWORK( 1 ) = LIWMIN +* + RETURN +* +* End of DTGSEN +* + END |