SB04QY

Constructing and solving a linear algebraic system whose coefficient matrix (stored compactly) has zeros below the first subdiagonal

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

Purpose

```  To construct and solve a linear algebraic system of order M whose
coefficient matrix is in upper Hessenberg form. Such systems
appear when solving discrete-time Sylvester equations using the
Hessenberg-Schur method.

```
Specification
```      SUBROUTINE SB04QY( N, M, IND, A, LDA, B, LDB, C, LDC, D, IPR,
\$                   INFO )
C     .. Scalar Arguments ..
INTEGER           INFO, IND, LDA, LDB, LDC, M, N
C     .. Array Arguments ..
INTEGER           IPR(*)
DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), D(*)

```
Arguments

Input/Output Parameters

```  N       (input) INTEGER
The order of the matrix B.  N >= 0.

M       (input) INTEGER
The order of the matrix A.  M >= 0.

IND     (input) INTEGER
The index of the column in C to be computed.  IND >= 1.

A       (input) DOUBLE PRECISION array, dimension (LDA,M)
The leading M-by-M part of this array must contain an
upper Hessenberg matrix.

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

B       (input) DOUBLE PRECISION array, dimension (LDB,N)
The leading N-by-N part of this array must contain a
matrix in real Schur form.

LDB     INTEGER
The leading dimension of array B.  LDB >= MAX(1,N).

C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading M-by-N part of this array must
contain the coefficient matrix C of the equation.
On exit, the leading M-by-N part of this array contains
the matrix C with column IND updated.

LDC     INTEGER
The leading dimension of array C.  LDC >= MAX(1,M).

```
Workspace
```  D       DOUBLE PRECISION array, dimension (M*(M+1)/2+2*M)

IPR     INTEGER array, dimension (2*M)

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
> 0:  if INFO = IND, a singular matrix was encountered.

```
Method
```  A special linear algebraic system of order M, with coefficient
matrix in upper Hessenberg form is constructed and solved. The
coefficient matrix is stored compactly, row-wise.

```
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.
Marcel Dekker, Inc., New York, 1996.

```
Numerical Aspects
```  None.

```
```  None
```
Example

Program Text

```  None
```
Program Data
```  None
```
Program Results
```  None
```