### Benchmark examples of (generalized) discrete-time Lyapunov equations

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

Purpose

```  To generate benchmark examples of (generalized) discrete-time
Lyapunov equations

T           T
A  X  A  -  E  X E  =  Y .

In some examples, the right hand side has the form

T
Y  =  - B  B

and the solution can be represented as a product of Cholesky
factors

T
X  =  U  U .

E, A, Y, X, and U are real N-by-N matrices, and B is M-by-N. Note
that E can be the identity matrix. For some examples, B, X, or U
are not provided.

This routine is an implementation of the benchmark library
DTLEX (Version 1.0) described in [1].

```
Specification
```      SUBROUTINE BB04AD(DEF, NR, DPAR, IPAR, VEC, N, M, E, LDE, A, LDA,
1                  Y, LDY, B, LDB, X, LDX, U, LDU, NOTE, DWORK,
2                  LDWORK, INFO)
C     .. Scalar Arguments ..
CHARACTER         DEF
CHARACTER*70      NOTE
INTEGER           INFO, LDA, LDB, LDE, LDU, LDWORK, LDX, LDY, M, N
C     .. Array Arguments ..
LOGICAL           VEC(8)
INTEGER           IPAR(*), NR(*)
DOUBLE PRECISION  A(LDA,*), B(LDB,*), DPAR(*), DWORK(LDWORK),
1                  E(LDE,*), U(LDU,*), X(LDX,*), Y(LDY,*)

```
Arguments

Mode Parameters

```  DEF     CHARACTER*1
Specifies the kind of values used as parameters when
generating parameter-dependent and scalable examples
(i.e., examples with NR(1) = 2, 3, or 4):
DEF = 'D' or 'd': Default values are used.
DEF = 'N' or 'n': Values set in DPAR and IPAR are used.
This parameter is not referenced if NR(1) = 1.
Note that the scaling parameter of examples with
NR(1) = 3 or 4 is considered as a regular parameter in
this context.

```
Input/Output Parameters
```  NR      (input) INTEGER array, dimension 2
Specifies the index of the desired example according
to [1].
NR(1) defines the group:
1 : parameter-free problems of fixed size
2 : parameter-dependent problems of fixed size
3 : parameter-free problems of scalable size
4 : parameter-dependent problems of scalable size
NR(2) defines the number of the benchmark example
within a certain group according to [1].

DPAR    (input/output) DOUBLE PRECISION array, dimension 2
On entry, if DEF = 'N' or 'n' and the desired example
depends on real parameters, then the array DPAR must
contain the values for these parameters.
For an explanation of the parameters see [1].
For Example 4.1, DPAR(1) and DPAR(2) define 'r' and 's',
respectively.
For Example 4.2, DPAR(1) and DPAR(2) define 'lambda' and
's', respectively.
For Examples 4.3 and 4.4, DPAR(1) defines the parameter
't'.
On exit, if DEF = 'D' or 'd' and the desired example
depends on real parameters, then the array DPAR is
overwritten by the default values given in [1].

IPAR    (input/output) INTEGER array of DIMENSION at least 1
On entry, if DEF = 'N' or 'n' and the desired example
depends on integer parameters, then the array IPAR must
contain the values for these parameters.
For an explanation of the parameters see [1].
For Examples 4.1, 4.2, and 4.3, IPAR(1) defines 'n'.
For Example 4.4, IPAR(1) defines 'q'.
On exit, if DEF = 'D' or 'd' and the desired example
depends on integer parameters, then the array IPAR is
overwritten by the default values given in [1].

VEC     (output) LOGICAL array, dimension 8
Flag vector which displays the availability of the output
data:
VEC(1) and VEC(2) refer to N and M, respectively, and are
always .TRUE.
VEC(3) is .TRUE. iff E is NOT the identity matrix.
VEC(4) and VEC(5) refer to A and Y, respectively, and are
always .TRUE.
VEC(6) is .TRUE. iff B is provided.
VEC(7) is .TRUE. iff the solution matrix X is provided.
VEC(8) is .TRUE. iff the Cholesky factor U is provided.

N       (output) INTEGER
The actual state dimension, i.e., the order of the
matrices E and A.

M       (output) INTEGER
The number of rows in the matrix B. If B is not provided
for the desired example, M = 0 is returned.

E       (output) DOUBLE PRECISION array, dimension (LDE,N)
The leading N-by-N part of this array contains the
matrix E.
NOTE that this array is overwritten (by the identity
matrix), if VEC(3) = .FALSE.

LDE     INTEGER
The leading dimension of array E.  LDE >= N.

A       (output) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array contains the
matrix A.

LDA     INTEGER
The leading dimension of array A.  LDA >= N.

Y       (output) DOUBLE PRECISION array, dimension (LDY,N)
The leading N-by-N part of this array contains the
matrix Y.

LDY     INTEGER
The leading dimension of array Y.  LDY >= N.

B       (output) DOUBLE PRECISION array, dimension (LDB,N)
The leading M-by-N part of this array contains the
matrix B.

LDB     INTEGER
The leading dimension of array B.  LDB >= M.

X       (output) DOUBLE PRECISION array, dimension (LDX,N)
The leading N-by-N part of this array contains the
matrix X.

LDX     INTEGER
The leading dimension of array X.  LDX >= N.

U       (output) DOUBLE PRECISION array, dimension (LDU,N)
The leading N-by-N part of this array contains the
matrix U.

LDU     INTEGER
The leading dimension of array U.  LDU >= N.

NOTE    (output) CHARACTER*70
String containing short information about the chosen
example.

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (LDWORK)

LDWORK  INTEGER
The length of the array DWORK.
For Examples 4.1 and 4.2., LDWORK >= 2*IPAR(1) is
required.
For the other examples, no workspace is needed, i.e.,
LDWORK >= 1.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value; in particular, INFO = -3 or -4 indicates
that at least one of the parameters in DPAR or
IPAR, respectively, has an illegal value.

```
References
```  [1]  D. Kressner, V. Mehrmann, and T. Penzl.
DTLEX - a Collection of Benchmark Examples for Discrete-
Time Lyapunov Equations.
SLICOT Working Note 1999-7, 1999.

```
Numerical Aspects
```  None

```
```  None
```
Example

Program Text

```C     BB04AD EXAMPLE PROGRAM TEXT
C     Copyright (c) 2002-2017 NICONET e.V.
C
C     .. Parameters ..
INTEGER          NIN, NOUT
PARAMETER        (NIN = 5, NOUT = 6)
INTEGER          NMAX, MMAX
PARAMETER        (NMAX = 100, MMAX = 100)
INTEGER          LDE, LDA, LDY, LDB, LDX, LDU, LDWORK
PARAMETER        (LDE = NMAX, LDA = NMAX, LDY = NMAX, LDB = MMAX,
1                  LDX = NMAX, LDU = NMAX, LDWORK = 2*NMAX)
C     .. Local Scalars ..
CHARACTER        DEF
INTEGER          INFO, N, M, I, J, LDPAR, LIPAR
CHARACTER*70     NOTE
C     .. Local Arrays ..
DOUBLE PRECISION E(LDE,NMAX), A(LDA, NMAX), Y(LDY, NMAX),
1                 B(LDB,NMAX), X(LDX, NMAX), U(LDU, NMAX),
2                 DPAR(2), DWORK(LDWORK)
INTEGER          NR(2), IPAR(1)
LOGICAL          VEC(8)
C     .. External Functions ..
LOGICAL          LSAME
EXTERNAL         LSAME
C     .. External Subroutines ..
C     .. Executable Statements ..
WRITE (NOUT, FMT = 99999)
C     Skip the heading in the data file and read the data.
READ (NIN, FMT = *) DEF
READ (NIN, FMT = *) (NR(I), I = 1, 2)
IF (LSAME(DEF,'N')) THEN
READ (NIN, FMT = *) LDPAR
IF (LDPAR .GT. 0)  READ (NIN, FMT = *) (DPAR(I), I = 1, LDPAR)
READ (NIN, FMT = *) LIPAR
IF (LIPAR .GT. 0)  READ (NIN, FMT = *) (IPAR(I), I = 1, LIPAR)
END IF
C     Generate benchmark example
CALL BB04AD(DEF, NR, DPAR, IPAR, VEC, N, M, E, LDE, A, LDA, Y,
1            LDY, B, LDB, X, LDX, U, LDU, NOTE, DWORK, LDWORK,
2            INFO)
C
IF (INFO .NE. 0) THEN
WRITE (NOUT, FMT = 99998) INFO
ELSE
WRITE (NOUT, FMT = *) NOTE
WRITE (NOUT, FMT = 99997) N
WRITE (NOUT, FMT = 99996) M
IF (VEC(3)) THEN
WRITE (NOUT, FMT = 99995)
DO 10  I = 1, N
WRITE (NOUT, FMT = 99985) (E(I,J), J = 1, N)
10        CONTINUE
ELSE
WRITE (NOUT, FMT = 99994)
END IF
WRITE (NOUT,FMT = 99993)
DO 20  I = 1, N
WRITE (NOUT, FMT = 99985) (A(I,J), J = 1, N)
20      CONTINUE
IF (VEC(6)) THEN
WRITE (NOUT,FMT = 99992)
DO 30  I = 1, M
WRITE (NOUT, FMT = 99985) (B(I,J), J = 1, N)
30        CONTINUE
ELSE
WRITE (NOUT, FMT = 99991)
END IF
WRITE (NOUT,FMT = 99990)
DO 40  I = 1, N
WRITE (NOUT, FMT = 99985) (Y(I,J), J = 1, N)
40      CONTINUE
IF (VEC(7)) THEN
WRITE (NOUT, FMT = 99989)
DO 50  I = 1, N
WRITE (NOUT, FMT = 99985) (X(I,J), J = 1, N)
50        CONTINUE
ELSE
WRITE (NOUT, FMT = 99988)
END IF
IF (VEC(8)) THEN
WRITE (NOUT, FMT = 99987)
DO 60  I = 1, N
WRITE (NOUT, FMT = 99985) (U(I,J), J = 1, N)
60        CONTINUE
ELSE
WRITE (NOUT, FMT = 99986)
END IF
END IF
C
99999 FORMAT (' BB04AD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT (' INFO on exit from BB04AD = ', I3)
99997 FORMAT (/' Order of matrix A:            N  = ', I3)
99996 FORMAT (' Number of rows in matrix B:   M  = ', I3)
99995 FORMAT (/' E  = ')
99994 FORMAT (/' E is the identity matrix.')
99993 FORMAT (' A  = ')
99992 FORMAT (' B  = ')
99991 FORMAT (' B is not provided.')
99990 FORMAT (' Y  = ')
99989 FORMAT (' X  = ')
99988 FORMAT (' X is not provided.')
99987 FORMAT (' U  = ')
99986 FORMAT (' U is not provided.')
99985 FORMAT (20(1X,F8.4))
C
END
```
Program Data
```BB04AD EXAMPLE PROGRAM DATA
N
4 1
2
.15D1
.15D1
1
5
```
Program Results
``` BB04AD EXAMPLE PROGRAM RESULTS

DTLEX: Example 4.1

Order of matrix A:            N  =   5
Number of rows in matrix B:   M  =   1

E is the identity matrix.
A  =
0.4562   0.0308   0.1990   0.0861   0.0217
0.0637   0.5142  -0.1828   0.0096  -0.1148
0.3139   0.1287   0.3484   0.1653  -0.1975
0.1500   0.0053  -0.1838   0.2501  -0.0687
0.0568  -0.1006  -0.3735  -0.0202   0.2285
B  =
0.3086   0.0247  -0.4691   0.1728  -0.3704
Y  =
-0.0953  -0.0076   0.1448  -0.0533   0.1143
-0.0076  -0.0006   0.0116  -0.0043   0.0091
0.1448   0.0116  -0.2201   0.0811  -0.1738
-0.0533  -0.0043   0.0811  -0.0299   0.0640
0.1143   0.0091  -0.1738   0.0640  -0.1372
X  =
0.0953   0.0076  -0.1448   0.0533  -0.1143
0.0076   0.0006  -0.0116   0.0043  -0.0091
-0.1448  -0.0116   0.2201  -0.0811   0.1738
0.0533   0.0043  -0.0811   0.0299  -0.0640
-0.1143  -0.0091   0.1738  -0.0640   0.1372
U is not provided.
```