## SB03UD

### Solution of discrete-time Lyapunov equations and condition and error bounds estimation

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

Purpose

```  To solve the real discrete-time Lyapunov matrix equation

op(A)'*X*op(A) - X = scale*C,

estimate the conditioning, and compute an error bound on the
solution X, where op(A) = A or A' (A**T), the matrix A is N-by-N,
the right hand side C and the solution X are N-by-N symmetric
matrices (C = C', X = X'), and scale is an output scale factor,
set less than or equal to 1 to avoid overflow in X.

```
Specification
```      SUBROUTINE SB03UD( JOB, FACT, TRANA, UPLO, LYAPUN, N, SCALE, A,
\$                   LDA, T, LDT, U, LDU, C, LDC, X, LDX, SEPD,
\$                   RCOND, FERR, WR, WI, IWORK, DWORK, LDWORK,
\$                   INFO )
C     .. Scalar Arguments ..
CHARACTER          FACT, JOB, LYAPUN, TRANA, UPLO
INTEGER            INFO, LDA, LDC, LDT, LDU, LDWORK, LDX, N
DOUBLE PRECISION   FERR, RCOND, SCALE, SEPD
C     .. Array Arguments ..
INTEGER            IWORK( * )
DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), DWORK( * ),
\$                   T( LDT, * ), U( LDU, * ), WI( * ), WR( * ),
\$                   X( LDX, * )

```
Arguments

Mode Parameters

```  JOB     CHARACTER*1
Specifies the computation to be performed, as follows:
= 'X':  Compute the solution only;
= 'S':  Compute the separation only;
= 'C':  Compute the reciprocal condition number only;
= 'E':  Compute the error bound only;
= 'A':  Compute all: the solution, separation, reciprocal
condition number, and the error bound.

FACT    CHARACTER*1
Specifies whether or not the real Schur factorization
of the matrix A is supplied on entry, as follows:
= 'F':  On entry, T and U (if LYAPUN = 'O') contain the
factors from the real Schur factorization of the
matrix A;
= 'N':  The Schur factorization of A will be computed
and the factors will be stored in T and U (if
LYAPUN = 'O').

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).

UPLO    CHARACTER*1
Specifies which part of the symmetric matrix C is to be
used, as follows:
= 'U':  Upper triangular part;
= 'L':  Lower triangular part.

LYAPUN  CHARACTER*1
Specifies whether or not the original or "reduced"
Lyapunov equations should be solved, as follows:
= 'O':  Solve the original Lyapunov equations, updating
the right-hand sides and solutions with the
matrix U, e.g., X <-- U'*X*U;
= 'R':  Solve reduced Lyapunov equations only, without
updating the right-hand sides and solutions.
This means that a real Schur form T of A appears
in the equation, instead of A.

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

SCALE   (input or output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'E', SCALE is an input argument:
the scale factor, set by a Lyapunov solver.
0 <= SCALE <= 1.
If JOB = 'X' or JOB = 'A', SCALE is an output argument:
the scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
If JOB = 'S', this argument is not used.

A       (input) DOUBLE PRECISION array, dimension (LDA,N)
If FACT = 'N' or (LYAPUN = 'O' and JOB <> 'X'), the
leading N-by-N part of this array must contain the
original matrix A.
If FACT = 'F' and (LYAPUN = 'R' or JOB = 'X'), A is
not referenced.

LDA     INTEGER
The leading dimension of the array A.
LDA >= MAX(1,N), if FACT = 'N' or LYAPUN = 'O' and
JOB <> 'X';
LDA >= 1,        otherwise.

T       (input/output) DOUBLE PRECISION array, dimension
(LDT,N)
If FACT = 'F', then on entry the leading N-by-N upper
Hessenberg part of this array must contain the upper
quasi-triangular matrix T in Schur canonical form from a
Schur factorization of A.
If FACT = 'N', then this array need not be set on input.
On exit, (if INFO = 0 or INFO = N+1, for FACT = 'N') the
leading N-by-N upper Hessenberg part of this array
contains the upper quasi-triangular matrix T in Schur
canonical form from a Schur factorization of A.
The contents of array T is not modified if FACT = 'F'.

LDT     INTEGER
The leading dimension of the array T.  LDT >= MAX(1,N).

U       (input or output) DOUBLE PRECISION array, dimension
(LDU,N)
If LYAPUN = 'O' and FACT = 'F', then U is an input
argument and on entry, the leading N-by-N part of this
array must contain the orthogonal matrix U from a real
Schur factorization of A.
If LYAPUN = 'O' and FACT = 'N', then U is an output
argument and on exit, if INFO = 0 or INFO = N+1, it
contains the orthogonal N-by-N matrix from a real Schur
factorization of A.
If LYAPUN = 'R', the array U is not referenced.

LDU     INTEGER
The leading dimension of the array U.
LDU >= 1,        if LYAPUN = 'R';
LDU >= MAX(1,N), if LYAPUN = 'O'.

C       (input) DOUBLE PRECISION array, dimension (LDC,N)
If JOB <> 'S' and UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the matrix C of the original Lyapunov
equation (with matrix A), if LYAPUN = 'O', or of the
reduced Lyapunov equation (with matrix T), if
LYAPUN = 'R'.
If JOB <> 'S' and UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the matrix C of the original Lyapunov
equation (with matrix A), if LYAPUN = 'O', or of the
reduced Lyapunov equation (with matrix T), if
LYAPUN = 'R'.
The remaining strictly triangular part of this array is
used as workspace.
If JOB = 'X', then this array may be identified with X
in the call of this routine.
If JOB = 'S', the array C is not referenced.

LDC     INTEGER
The leading dimension of the array C.
LDC >= 1,        if JOB = 'S';
LDC >= MAX(1,N), otherwise.

X       (input or output) DOUBLE PRECISION array, dimension
(LDX,N)
If JOB = 'C' or 'E', then X is an input argument and on
entry, the leading N-by-N part of this array must contain
the symmetric solution matrix X of the original Lyapunov
equation (with matrix A), if LYAPUN = 'O', or of the
reduced Lyapunov equation (with matrix T), if
LYAPUN = 'R'.
If JOB = 'X' or 'A', then X is an output argument and on
exit, if INFO = 0 or INFO = N+1, the leading N-by-N part
of this array contains the symmetric solution matrix X of
of the original Lyapunov equation (with matrix A), if
LYAPUN = 'O', or of the reduced Lyapunov equation (with
matrix T), if LYAPUN = 'R'.
If JOB = 'S', the array X is not referenced.

LDX     INTEGER
The leading dimension of the array X.
LDX >= 1,        if JOB = 'S';
LDX >= MAX(1,N), otherwise.

SEPD    (output) DOUBLE PRECISION
If JOB = 'S' or JOB = 'C' or JOB = 'A', and INFO = 0 or
INFO = N+1, SEPD contains the estimated separation of the
matrices op(A) and op(A)', sepd(op(A),op(A)').
If N = 0, or X = 0, or JOB = 'X' or JOB = 'E', SEPD is not
referenced.

RCOND   (output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'A', an estimate of the reciprocal
condition number of the continuous-time Lyapunov equation.
If N = 0 or X = 0, RCOND is set to 1 or 0, respectively.
If JOB = 'X' or JOB = 'S' or JOB = 'E', RCOND is not
referenced.

FERR    (output) DOUBLE PRECISION
If JOB = 'E' or JOB = 'A', and INFO = 0 or INFO = N+1,
FERR contains an estimated forward error bound for the
solution X. If XTRUE is the true solution, FERR bounds the
relative error in the computed solution, measured in the
Frobenius norm:  norm(X - XTRUE)/norm(XTRUE).
If N = 0 or X = 0, FERR is set to 0.
If JOB = 'X' or JOB = 'S' or JOB = 'C', FERR is not
referenced.

WR      (output) DOUBLE PRECISION array, dimension (N)
WI      (output) DOUBLE PRECISION array, dimension (N)
If FACT = 'N', and INFO = 0 or INFO = N+1, WR and WI
contain the real and imaginary parts, respectively, of the
eigenvalues of A.
If FACT = 'F', WR and WI are not referenced.

```
Workspace
```  IWORK   INTEGER array, dimension (N*N)
This array is not referenced if JOB = 'X'.

DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = N+1, DWORK(1) returns the
optimal value of LDWORK.

LDWORK  INTEGER
The length of the array DWORK.
If JOB = 'X', then
LDWORK >= MAX(1,N*N,2*N),       if FACT = 'F';
LDWORK >= MAX(1,N*N,3*N),       if FACT = 'N'.
If JOB = 'S', then
LDWORK >= MAX(3,2*N*N).
If JOB = 'C', then
LDWORK >= MAX(3,2*N*N) + N*N.
If JOB = 'E', or JOB = 'A', then
LDWORK >= MAX(3,2*N*N) + N*N + 2*N.
For optimum performance LDWORK should sometimes 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, i <= N, the QR algorithm failed to
complete the reduction to Schur canonical form (see
LAPACK Library routine DGEES); on exit, the matrix
T(i+1:N,i+1:N) contains the partially converged
Schur form, and the elements i+1:n of WR and WI
contain the real and imaginary parts, respectively,
of the converged eigenvalues; this error is unlikely
to appear;
= N+1:  if the matrix T has almost reciprocal eigenvalues;
perturbed values were used to solve Lyapunov
equations, but the matrix T, if given (for
FACT = 'F'), is unchanged.

```
Method
```  After reducing matrix A to real Schur canonical form (if needed),
a discrete-time version of the 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.

The condition number of the discrete-time Lyapunov equation is
estimated as

cond = (norm(Theta)*norm(A) + norm(inv(Omega))*norm(C))/norm(X),

where Omega and Theta are linear operators defined by

Omega(W) = op(A)'*W*op(A) - W,
Theta(W) = inv(Omega(op(W)'*X*op(A) + op(A)'*X*op(W))).

The routine estimates the quantities

sepd(op(A),op(A)') = 1 / norm(inv(Omega))

and norm(Theta) using 1-norm condition estimators.

The forward error bound is estimated using a practical error bound
similar to the one proposed in .

```
References
```   Barraud, A.Y.                   T
A numerical algorithm to solve A XA - X = Q.
IEEE Trans. Auto. Contr., AC-22, pp. 883-885, 1977.

 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.

 Higham, N.J.
Perturbation theory and backward error for AX-XB=C.
BIT, vol. 33, pp. 124-136, 1993.

```
Numerical Aspects
```                            3
The algorithm requires 0(N ) operations.
The accuracy of the estimates obtained depends on the solution
accuracy and on the properties of the 1-norm estimator.

```
```  The "separation" sepd of op(A) and op(A)' can also be defined as

sepd( op(A), op(A)' ) = sigma_min( T ),

where sigma_min(T) is the smallest singular value of the
N*N-by-N*N matrix

T = kprod( op(A)', op(A)' ) - I(N**2).

I(N**2) is an N*N-by-N*N identity matrix, and kprod denotes the
Kronecker product. The routine estimates sigma_min(T) by the
reciprocal of an estimate of the 1-norm of inverse(T). The true
reciprocal 1-norm of inverse(T) cannot differ from sigma_min(T) by
more than a factor of N.

```
Example

Program Text

```*     SB03UD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          NMAX
PARAMETER        ( NMAX = 20 )
INTEGER          LDA, LDC, LDT, LDU, LDX
PARAMETER        ( LDA = NMAX, LDC = NMAX, LDT = NMAX,
\$                   LDU = NMAX, LDX = NMAX )
INTEGER          LIWORK
PARAMETER        ( LIWORK = NMAX*NMAX )
INTEGER          LDWORK
PARAMETER        ( LDWORK = MAX( 3, 2*NMAX*NMAX ) +
\$                                 NMAX*NMAX + 2*NMAX )
*     .. Local Scalars ..
DOUBLE PRECISION FERR, RCOND, SCALE, SEPD
INTEGER          I, INFO, J, N
CHARACTER*1      DICO, FACT, JOB, LYAPUN, TRANA, UPLO
*     .. Local Arrays ..
INTEGER          IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), C(LDC,NMAX), DWORK(LDWORK),
\$                 T(LDT,NMAX), U(LDU,NMAX), X(LDX,NMAX)
*     .. External Functions ..
LOGICAL          LSAME
EXTERNAL         LSAME
*     .. External Subroutines ..
EXTERNAL         SB03UD
*     .. Intrinsic Functions ..
INTRINSIC        MAX
*     .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
DICO = 'D'
*     Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOB, FACT, TRANA, UPLO, LYAPUN
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
IF ( LSAME( JOB, 'C' ) .OR. LSAME( JOB, 'E' ) )
\$                               READ ( NIN, FMT = * ) SCALE
IF ( LSAME( FACT, 'N' ) .OR. ( LSAME( LYAPUN, 'O' ) .AND.
\$                             .NOT.LSAME( JOB, 'X') ) )
\$      READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( FACT, 'F' ) ) THEN
READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( LYAPUN, 'O' ) )
\$         READ ( NIN, FMT = * ) ( ( U(I,J), J = 1,N ), I = 1,N )
END IF
IF ( .NOT.LSAME( JOB, 'S' ) )
\$      READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( JOB, 'C' ) .OR. LSAME( JOB, 'E' ) )
\$      READ ( NIN, FMT = * ) ( ( X(I,J), J = 1,N ), I = 1,N )
*        Solve the discrete-time Lyapunov matrix equation and/or
*        estimate the condition and error bound on the solution.
CALL SB03UD( JOB, FACT, TRANA, UPLO, LYAPUN, N, SCALE, A, LDA,
\$                T, LDT, U, LDU, C, LDC, X, LDX, SEPD, RCOND, FERR,
\$                DWORK(1), DWORK(N+1), IWORK, DWORK(2*N+1),
\$                LDWORK-2*N, INFO )
*
IF ( INFO.EQ.0 ) THEN
IF ( LSAME( JOB, 'X' ) .OR. LSAME( JOB, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( X(I,J), J = 1,N )
10          CONTINUE
WRITE ( NOUT, FMT = 99993 ) SCALE
END IF
IF ( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'C' )
\$                             .OR. LSAME( JOB, 'A' ) )
\$         WRITE ( NOUT, FMT = 99992 ) SEPD
IF ( LSAME( JOB, 'C' ) .OR. LSAME( JOB, 'A' ) )
\$         WRITE ( NOUT, FMT = 99991 ) RCOND
IF ( LSAME( JOB, 'E' ) .OR. LSAME( JOB, 'A' ) )
\$         WRITE ( NOUT, FMT = 99990 ) FERR
ELSE
WRITE ( NOUT, FMT = 99998 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' SB03UD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB03UD =',I2)
99996 FORMAT (' The solution matrix X is')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' Scaling factor = ',F8.4)
99992 FORMAT (/' Estimated separation = ',F8.4)
99991 FORMAT (/' Estimated reciprocal condition number = ',F8.4)
99990 FORMAT (/' Estimated error bound = ',F8.4)
END
```
Program Data
``` SB03UD EXAMPLE PROGRAM DATA
3     A     N     N     U     O
3.0   1.0   1.0
1.0   3.0   0.0
0.0   0.0   3.0
25.0  24.0  15.0
24.0  32.0   8.0
15.0   8.0  40.0
```
Program Results
``` SB03UD EXAMPLE PROGRAM RESULTS

The solution matrix X is
2.0000   1.0000   1.0000
1.0000   3.0000   0.0000
1.0000   0.0000   4.0000

Scaling factor =   1.0000

Estimated separation =   5.2302

Estimated reciprocal condition number =   0.1832

Estimated error bound =   0.0000
```