## SB03MY

### Solving a continuous-time Lyapunov equation with matrix A quasi-triangular

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

Purpose

```  To solve the real Lyapunov matrix equation

op(A)'*X + X*op(A) = scale*C

where op(A) = A or A' (A**T), A is upper quasi-triangular and C is
symmetric (C = C'). (A' denotes the transpose of the matrix A.)
A is N-by-N, the right hand side C and the solution X are N-by-N,
and scale is an output scale factor, set less than or equal to 1
to avoid overflow in X. The solution matrix X is overwritten
onto C.

A must be in Schur canonical form (as returned by LAPACK routines
DGEES or DHSEQR), that is, block upper triangular with 1-by-1 and
2-by-2 diagonal blocks; each 2-by-2 diagonal block has its
diagonal elements equal and its off-diagonal elements of opposite
sign.

```
Specification
```      SUBROUTINE SB03MY( TRANA, N, A, LDA, C, LDC, SCALE, INFO )
C     .. Scalar Arguments ..
CHARACTER          TRANA
INTEGER            INFO, LDA, LDC, N
DOUBLE PRECISION   SCALE
C     .. Array Arguments ..
DOUBLE PRECISION   A( LDA, * ), C( LDC, * )

```
Arguments

Mode Parameters

```  TRANA   CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N':  op(A) = A    (No transpose);
= 'T':  op(A) = A**T (Transpose);
= 'C':  op(A) = A**T (Conjugate transpose = Transpose).

```
Input/Output Parameters
```  N       (input) INTEGER
The order of the matrices A, X, and C.  N >= 0.

A       (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
upper quasi-triangular matrix A, in Schur canonical form.
The part of A below the first sub-diagonal is not
referenced.

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

C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading N-by-N part of this array must
contain the symmetric matrix C.
On exit, if INFO >= 0, the leading N-by-N part of this
array contains the symmetric solution matrix X.

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

SCALE   (output) DOUBLE PRECISION
The scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value;
= 1:  if A and -A have common or very close eigenvalues;
perturbed values were used to solve the equation
(but the matrix A is unchanged).

```
Method
```  Bartels-Stewart algorithm is used. A set of equivalent linear
algebraic systems of equations of order at most four are formed
and solved using Gaussian elimination with complete pivoting.

```
References
```  [1] Bartels, R.H. and Stewart, G.W.  T
Solution of the matrix equation A X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.

```
Numerical Aspects
```                            3
The algorithm requires 0(N ) operations.

```
Further Comments
```  None
```
Example

Program Text

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

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