## SB04ND

### Solution of continuous-time Sylvester equations with one matrix in real Schur form and the other matrix in Hessenberg form (Hessenberg-Schur method)

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

```  To solve for X the continuous-time Sylvester equation

AX + XB = C,

with at least one of the matrices A or B in Schur form and the
other in Hessenberg or Schur form (both either upper or lower);
A, B, C and X are N-by-N, M-by-M, N-by-M, and N-by-M matrices,
respectively.

```
Specification
```      SUBROUTINE SB04ND( ABSCHU, ULA, ULB, N, M, A, LDA, B, LDB, C,
\$                   LDC, TOL, IWORK, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER         ABSCHU, ULA, ULB
INTEGER           INFO, LDA, LDB, LDC, LDWORK, M, N
DOUBLE PRECISION  TOL
C     .. Array Arguments ..
INTEGER           IWORK(*)
DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*)

```
Arguments

Mode Parameters

```  ABSCHU  CHARACTER*1
Indicates whether A and/or B is/are in Schur or
Hessenberg form as follows:
= 'A':  A is in Schur form, B is in Hessenberg form;
= 'B':  B is in Schur form, A is in Hessenberg form;
= 'S':  Both A and B are in Schur form.

ULA     CHARACTER*1
Indicates whether A is in upper or lower Schur form or
upper or lower Hessenberg form as follows:
= 'U':  A is in upper Hessenberg form if ABSCHU = 'B' and
upper Schur form otherwise;
= 'L':  A is in lower Hessenberg form if ABSCHU = 'B' and
lower Schur form otherwise.

ULB     CHARACTER*1
Indicates whether B is in upper or lower Schur form or
upper or lower Hessenberg form as follows:
= 'U':  B is in upper Hessenberg form if ABSCHU = 'A' and
upper Schur form otherwise;
= 'L':  B is in lower Hessenberg form if ABSCHU = 'A' and
lower Schur form otherwise.

```
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) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
coefficient matrix A of the equation.

LDA     INTEGER
The leading dimension of array A.  LDA >= MAX(1,N).

B       (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading M-by-M part of this array must contain the
coefficient matrix B of the equation.

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, if INFO = 0, 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).

```
Tolerances
```  TOL     DOUBLE PRECISION
The tolerance to be used to test for near singularity in
the Sylvester equation. If the user sets TOL > 0, then the
given value of TOL is used as a lower bound for the
reciprocal condition number; a matrix whose estimated
condition number is less than 1/TOL is considered to be
nonsingular. If the user sets TOL <= 0, then a default
tolerance, defined by TOLDEF = EPS, is used instead, where
EPS is the machine precision (see LAPACK Library routine
DLAMCH).
This parameter is not referenced if ABSCHU = 'S',
ULA = 'U', and ULB = 'U'.

```
Workspace
```  IWORK   INTEGER array, dimension (2*MAX(M,N))
This parameter is not referenced if ABSCHU = 'S',
ULA = 'U', and ULB = 'U'.

DWORK   DOUBLE PRECISION array, dimension (LDWORK)
This parameter is not referenced if ABSCHU = 'S',
ULA = 'U', and ULB = 'U'.

LDWORK  INTEGER
The length of the array DWORK.
LDWORK = 0, if ABSCHU = 'S', ULA = 'U', and ULB = 'U';
LDWORK = 2*MAX(M,N)*(4 + 2*MAX(M,N)), otherwise.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value;
= 1:  if a (numerically) singular matrix T was encountered
during the computation of the solution matrix X.
That is, the estimated reciprocal condition number
of T is less than or equal to TOL.

```
Method
```  Matrices A and B are assumed to be in (upper or lower) Hessenberg
or Schur form (with at least one of them in Schur form). The
solution matrix X is then computed by rows or columns via the back
substitution scheme proposed by Golub, Nash and Van Loan (see
[1]), which involves the solution of triangular systems of
equations that are constructed recursively and which may be nearly
singular if A and -B have close eigenvalues. If near singularity
is detected, then the routine returns with the Error Indicator
(INFO) set to 1.

```
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.

```
Numerical Aspects
```                                         2         2
The algorithm requires approximately 5M N + 0.5MN  operations in
2         2
the worst case and 2.5M N + 0.5MN  operations in the best case
(where M is the order of the matrix in Hessenberg form and N is
the order of the matrix in Schur form) and is mixed stable (see
[1]).

```
```  None
```
Example

Program Text

```*     SB04ND EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          NMAX, MMAX
PARAMETER        ( NMAX = 20, MMAX = 20 )
INTEGER          LDA, LDB, LDC
PARAMETER        ( LDA = NMAX, LDB = MMAX, LDC = NMAX )
INTEGER          LDWORK
PARAMETER        ( LDWORK = 2*( MAX( NMAX,MMAX ) )*
\$                        ( 4+2*( MAX( NMAX,MMAX ) ) ) )
INTEGER          LIWORK
PARAMETER        ( LIWORK = 2*MAX( NMAX,MMAX ) )
*     .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER          I, INFO, J, M, N
CHARACTER*1      ABSCHU, ULA, ULB
*     .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,MMAX),
\$                 DWORK(LDWORK)
INTEGER          IWORK(LIWORK)
*     .. External Subroutines ..
EXTERNAL         SB04ND
*     .. 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, TOL, ULA, ULB, ABSCHU
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) 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 = 99994 ) 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 SB04ND( ABSCHU, ULA, ULB, N, M, A, LDA, B, LDB, C,
\$                   LDC, TOL, 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
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB04ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB04ND = ',I2)
99997 FORMAT (' The solution matrix X is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' N is out of range.',/' N = ',I5)
99994 FORMAT (/' M is out of range.',/' M = ',I5)
END
```
Program Data
``` SB04ND EXAMPLE PROGRAM DATA
5     3     0.0     U     U     B
17.0  24.0   1.0   8.0  15.0
23.0   5.0   7.0  14.0  16.0
0.0   6.0  13.0  20.0  22.0
0.0   0.0  19.0  21.0   3.0
0.0   0.0   0.0   2.0   9.0
8.0   1.0   6.0
0.0   5.0   7.0
0.0   9.0   2.0
62.0 -12.0  26.0
59.0 -10.0  31.0
70.0  -6.0   9.0
35.0  31.0  -7.0
36.0 -15.0   7.0
```
Program Results
``` SB04ND EXAMPLE PROGRAM RESULTS

The solution matrix X is
0.0000   0.0000   1.0000
1.0000   0.0000   0.0000
0.0000   1.0000   0.0000
1.0000   1.0000  -1.0000
2.0000  -2.0000   1.0000
```