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c Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
c Copyright (C) 1989-2008 - INRIA - Serge STEER
c
c This file must be used under the terms of the CeCILL.
c This source file is licensed as described in the file COPYING, which
c you should have received as part of this distribution. The terms
c are also available at
c http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt
subroutine recbez(A,n,B,m,best,ipb,w,err)
c!but
c calcule la factorisation de bezout de deux polynomes A,B c'est a dire
c les polynomes x,y,u, v et pgcd de degre minimal et tels que :
c [x u]
c [A B] * [ ] = [0 Pgcd]
c [y v]
c!liste d'appel
c subroutine recbez(A,n,B,m,best,ipb,w,err)
c double precision A(n+1),B(m+1),w(*),best(*),err
c
c A : vector of A polynomial coefficients, in increasing degrees order
c n : degree of A polynomial
c B : vector of B polynomial coefficients, in increasing degrees order
c m : degree of B polynomial
c
c best : Array where the result is stored
c (2*(n+m)+min(n,m)+3) [gcd u,v,x,y]
c gcd=best(1:ipb(2)-1),
c u=best(ipb(2):ipb(3)-1) v=best(ipb(3):ipb(4)-1)
c x=best(ipb(4):ipb(5)-1) y=best(ipb(5):ipb(6)-1)
c ipb : integer array that gives the way gcd u,v,x,y are stored in best
c w: working array w(7*n0+3*n0*n0) ou n0=max(n,m)+1
c err : estimated equation error
c! method
c If A and B are polynomials of degree m and n : A in R[x] , B in R[x]
c n m
c The application: (P,Q) in R[x] x R[x] --> P*A+Q*B in R[x]
c m-l n-l n+m-l
c
c is linear. It can be represented by a linear application from
c R^(n+m-2*l+2) to R^(n+m-l+1) X--> X*S. Let call S the matrix S
c of this linear application. If l=1 this is the Sylvester matrix.
c example for A of degree 5 (n=5) and B of degree 4 (m=4) and l=2.
c [A_0 A_1 A_2 A_3 A_4 A_5 0 0]
c [ 0 A_0 A_1 A_2 A_3 A_4 A_5 0]
c [ 0 0 A_0 A_1 A_2 A_3 A_4 A_5]
c S = [B_0 B_1 B_2 B_3 B_4 0 0 0]
c [ 0 B_0 B_1 B_2 B_3 B_4 0 0]
c [ 0 0 B_0 B_1 B_2 B_3 B_4 0]
c [ 0 0 0 B_0 B_1 B_2 B_3 B_4]
c
c X = [P_0 P_1 P_2 Q_0 Q_1 Q_2 Q_3]
c The greater l for which S becomes not full rank is the GCD degree.
c The kernel of S then gives the x and y value
c The GCD is the element of the image of S that has the smaller
c degree.
c The determination of the S rank is a difficult problem. To avoid
c the use of a tolerance The algorithm try all l from min(m,n)-1 to
c 1. and keeps the solution that minimise a criterion based on
c forward equation errors ||A*x+B*y|| and ||A*u+B*V-GCD|| and
c backward equation errors ||y*CGD+A|| and ||x*GCD-B||
c Finally instead of computing QR decomposition of each S matrices a
c recursive method is used that updates previous QR decomposition
c using givens rotations. This algorithm also requires to store only
c a matrix which has max(m,n)+1 columns.
c
c! called routines
c dest dcopy bezstp dlamch
double precision A(n+1),B(m+1),w(*),best(*),err
double precision dlamch
integer ipb(6)
c
integer nn,mm
c
err=dlamch('o')
nn = n
mm = m
c
c compute the true degree for A and B taking care of 0 higer degree
c
nn=n+1
01 nn=nn-1
if(nn.lt.0) goto 30
if(A(nn+1).eq.0.0d+0) goto 01
c
mm=m+1
02 mm=mm-1
if(mm.lt.0) goto 30
if(B(mm+1).eq.0.0d+0) goto 02
c
n0=max(nn,mm)+1
n02=n0*(n0+1)
c leading dimension of the S matrix
ldS=n0+1
c leading dimension of the Q matrix of the QR factorization
ldQ=n0+1
c allocate working arrays
c start index of S array in w
iS=1
c start index of Q array in w
iQ=iS+n02
c start index of working array for bezstp
iw=iQ+2*n02
ifree=iw+7*n0
c initialize S (n0+1) x n0 matrix array. Take care that it is a
c compact form of the S matrix described in the method part.
call dset(n02,0.0d+0,w(iS),1)
c the last 2 rows form the current S matrix
c lS points to the beginning of last row
c and lS-1 to the beginning of the previous one
lS=iS-1+ldS
c initialize with x and y of degree 0
c [0 ]
c S= [coeff(A)]
c [coeff(B)]
c no
call dcopy(nn+1,A,1,w(lS-1),ldS)
call dcopy(mm+1,B,1,w(lS),ldS)
c initialize U (n0+1) x (2*n0) array
c [0...0|0...0 0 0]
c U= [0...0|0...0 1 0]
c [0...0|0...0 0 1]
C n0 n0
call dset(2*n02,0.0d+0,w(iQ),1)
lQ=iQ-1+ldQ+2*n0*ldQ
c the last 2 rows form the current Q matrix
call dset(2,1.0d+0,w(lQ-1-2*ldQ),ldQ+1)
c loop on GCD degree degree(gcd)=n0+1-l
do 20 l=1,n0
lS=lS-1
lQ=lQ-1-2*ldQ
lx=lS-1+ldS
lv=lQ-1-2*ldQ
c . lS : points on the upper left corner of the current A matrix,
c . lQ : points on the upper left corner of the current Q matrix
call bezstp(A,nn,B,mm,w(lS),ldS,w(lQ),ldQ,l,w(lx),
$ w(lv),w(iw),best,ipb,err)
20 continue
return
30 continue
c Handle special cases at least one polynomial is a scalar
err=0.0d+0
ipb(1)=1
if(min(nn,mm).eq.0) goto 70
CRES=nn-mm
if (CRES .lt. 0) then
goto 40
elseif (CRES .eq. 0) then
goto 50
else
goto 60
endif
40 continue
c A=0
np=mm
call dcopy(mm+1,B,1,best(1),1)
ipb(2)=ipb(1)+mm+1
best(ipb(2))=0.0d+0
ipb(3)=ipb(2)+1
best(ipb(3))=1.0d+0
ipb(4)=ipb(3)+1
best(ipb(4))=1.0d+0
ipb(5)=ipb(4)+1
best(ipb(5))=0.0d+0
ipb(6)=ipb(5)+1
return
50 continue
c A=0,B=0 la decomposition n'est pas definie
np=0
best(1)=0.0d+0
ipb(2)=ipb(1)+1
best(ipb(2))=1.0d+0
ipb(3)=ipb(2)+1
best(ipb(3))=0.0d+0
ipb(4)=ipb(3)+1
best(ipb(4))=0.0d+0
ipb(5)=ipb(4)+1
best(ipb(5))=1.0d+0
ipb(6)=ipb(5)+1
return
60 continue
c B=0
np=nn
call dcopy(nn+1,A,1,best(1),1)
ipb(2)=ipb(1)+nn+1
best(ipb(2))=1.0d+0
ipb(3)=ipb(2)+1
best(ipb(3))=0.0d+0
ipb(4)=ipb(3)+1
best(ipb(4))=0.0d+0
ipb(5)=ipb(4)+1
best(ipb(5))=1.0d+0
ipb(6)=ipb(5)+1
return
c
70 best(1)=1.0d+0
ipb(2)=2
CRES=nn-mm
if (CRES .lt. 0) then
goto 90
elseif (CRES .eq. 0) then
goto 95
else
goto 100
endif
90 continue
c A=cte
best(ipb(2))=1.0d+0/A(1)
ipb(3)=ipb(2)+1
best(ipb(3))=0.0d+0
ipb(4)=ipb(3)+1
call dcopy(mm+1,B,1,best(ipb(4)),1)
ipb(5)=ipb(4)+mm+1
best(ipb(5))=-A(1)
ipb(6)=ipb(5)+1
return
95 continue
c A=cte,B=cte
if(abs(A(1)).gt.abs(B(1))) goto 90
c
100 continue
c B=cte
best(ipb(2))=0.0d+0
ipb(3)=ipb(2)+1
best(ipb(3))=1.0d+0/B(1)
ipb(4)=ipb(3)+1
best(ipb(4))=-B(1)
ipb(5)=ipb(4)+1
call dcopy(nn+1,A,1,best(ipb(5)),1)
ipb(6)=ipb(5)+nn+1
return
c
end
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