Purpose
To solve for X the discrete-time Sylvester equation
X + AXB = C,
where A, B, C and X are general N-by-N, M-by-M, N-by-M and
N-by-M matrices respectively. A Hessenberg-Schur method, which
reduces A to upper Hessenberg form, H = U'AU, and B' to real
Schur form, S = Z'B'Z (with U, Z orthogonal matrices), is used.
Specification
SUBROUTINE SB04QD( N, M, A, LDA, B, LDB, C, LDC, Z, LDZ, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDWORK, LDZ, M, N
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), Z(LDZ,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrix A. N >= 0.
M (input) INTEGER
The order of the matrix B. M >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the coefficient matrix A of the equation.
On exit, the leading N-by-N upper Hessenberg part of this
array contains the matrix H, and the remainder of the
leading N-by-N part, together with the elements 2,3,...,N
of array DWORK, contain the orthogonal transformation
matrix U (stored in factored form).
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading M-by-M part of this array must
contain the coefficient matrix B of the equation.
On exit, the leading M-by-M part of this array contains
the quasi-triangular Schur factor S of the matrix B'.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,M).
C (input/output) DOUBLE PRECISION array, dimension (LDC,M)
On entry, the leading N-by-M part of this array must
contain the coefficient matrix C of the equation.
On exit, the leading N-by-M part of this array contains
the solution matrix X of the problem.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,N).
Z (output) DOUBLE PRECISION array, dimension (LDZ,M)
The leading M-by-M part of this array contains the
orthogonal matrix Z used to transform B' to real upper
Schur form.
LDZ INTEGER
The leading dimension of array Z. LDZ >= MAX(1,M).
Workspace
IWORK INTEGER array, dimension (4*N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, and DWORK(2), DWORK(3),..., DWORK(N) contain
the scalar factors of the elementary reflectors used to
reduce A to upper Hessenberg form, as returned by LAPACK
Library routine DGEHRD.
LDWORK INTEGER
The length of the array DWORK.
LDWORK = MAX(1, 2*N*N + 9*N, 5*M, N + M).
For optimum performance LDWORK should be larger.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, 1 <= i <= M, the QR algorithm failed to
compute all the eigenvalues of B (see LAPACK Library
routine DGEES);
> M: if a singular matrix was encountered whilst solving
for the (INFO-M)-th column of matrix X.
Method
The matrix A is transformed to upper Hessenberg form H = U'AU by
the orthogonal transformation matrix U; matrix B' is transformed
to real upper Schur form S = Z'B'Z using the orthogonal
transformation matrix Z. The matrix C is also multiplied by the
transformations, F = U'CZ, and the solution matrix Y of the
transformed system
Y + HYS' = F
is computed by back substitution. Finally, the matrix Y is then
multiplied by the orthogonal transformation matrices, X = UYZ', in
order to obtain the solution matrix X to the original problem.
References
[1] Golub, G.H., Nash, S. and Van Loan, C.F.
A Hessenberg-Schur method for the problem AX + XB = C.
IEEE Trans. Auto. Contr., AC-24, pp. 909-913, 1979.
[2] Sima, V.
Algorithms for Linear-quadratic Optimization.
Marcel Dekker, Inc., New York, 1996.
Numerical Aspects
3 3 2 2 The algorithm requires about (5/3) N + 10 M + 5 N M + 2.5 M N operations and is backward stable.Further Comments
NoneExample
Program Text
* SB04QD EXAMPLE PROGRAM TEXT
* Copyright (c) 2002-2010 NICONET e.V.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX
PARAMETER ( NMAX = 20, MMAX = 20 )
INTEGER LDA, LDB, LDC, LDZ
PARAMETER ( LDA = NMAX, LDB = MMAX, LDC = NMAX,
$ LDZ = MMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = 4*NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 1, 2*NMAX*NMAX+9*NMAX, 5*MMAX,
$ NMAX+MMAX ) )
* .. Local Scalars ..
INTEGER I, INFO, J, M, N
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,MMAX),
$ DWORK(LDWORK), Z(LDZ,MMAX)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL SB04QD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,M )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,M ), I = 1,N )
* Find the solution matrix X.
CALL SB04QD( N, M, A, LDA, B, LDB, C, LDC, Z, LDZ, IWORK,
$ DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,M )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, M
WRITE ( NOUT, FMT = 99996 ) ( Z(I,J), J = 1,M )
40 CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB04QD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB04QD = ',I2)
99997 FORMAT (' The solution matrix X is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The orthogonal matrix Z is ')
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' M is out of range.',/' M = ',I5)
END
Program Data
SB04QD EXAMPLE PROGRAM DATA 3 3 1.0 2.0 3.0 6.0 7.0 8.0 9.0 2.0 3.0 7.0 2.0 3.0 2.0 1.0 2.0 3.0 4.0 1.0 271.0 135.0 147.0 923.0 494.0 482.0 578.0 383.0 287.0Program Results
SB04QD EXAMPLE PROGRAM RESULTS The solution matrix X is 2.0000 3.0000 6.0000 4.0000 7.0000 1.0000 5.0000 3.0000 2.0000 The orthogonal matrix Z is 0.8337 0.5204 -0.1845 0.3881 -0.7900 -0.4746 0.3928 -0.3241 0.8606
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