## SB02MT

### Conversion of optimal problems with coupling weighting terms to standard problems

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

Purpose

```  To compute the following matrices

-1
G = B*R  *B',

-          -1
A = A - B*R  *L',

-          -1
Q = Q - L*R  *L',

where A, B, Q, R, L, and G are N-by-N, N-by-M, N-by-N, M-by-M,
N-by-M, and N-by-N matrices, respectively, with Q, R and G
symmetric matrices.

When R is well-conditioned with respect to inversion, standard
algorithms for solving linear-quadratic optimization problems will
then also solve optimization problems with coupling weighting
matrix L. Moreover, a gain in efficiency is possible using matrix
G in the deflating subspace algorithms (see SLICOT Library routine
SB02OD) or in the Newton's algorithms (see SLICOT Library routine
SG02CD).

```
Specification
```      SUBROUTINE SB02MT( JOBG, JOBL, FACT, UPLO, N, M, A, LDA, B, LDB,
\$                   Q, LDQ, R, LDR, L, LDL, IPIV, OUFACT, G, LDG,
\$                   IWORK, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER         FACT, JOBG, JOBL, UPLO
INTEGER           INFO, LDA, LDB, LDG, LDL, LDQ, LDR, LDWORK, M,
\$                  N, OUFACT
C     .. Array Arguments ..
INTEGER           IPIV(*), IWORK(*)
DOUBLE PRECISION  A(LDA,*), B(LDB,*), DWORK(*), G(LDG,*),
\$                  L(LDL,*), Q(LDQ,*), R(LDR,*)

```
Arguments

Mode Parameters

```  JOBG    CHARACTER*1
Specifies whether or not the matrix G is to be computed,
as follows:
= 'G':  Compute G;
= 'N':  Do not compute G.

JOBL    CHARACTER*1
Specifies whether or not the matrix L is zero, as follows:
= 'Z':  L is zero;
= 'N':  L is nonzero.

FACT    CHARACTER*1
Specifies how the matrix R is given (factored or not), as
follows:
= 'N':  Array R contains the matrix R;
= 'C':  Array R contains the Cholesky factor of R;
= 'U':  Array R contains the factors of the symmetric
indefinite UdU' or LdL' factorization of R.

UPLO    CHARACTER*1
Specifies which triangle of the matrices R, Q (if
JOBL = 'N'), and G (if JOBG = 'G') is stored, as follows:
= 'U':  Upper triangle is stored;
= 'L':  Lower triangle is stored.

```
Input/Output Parameters
```  N       (input) INTEGER
The order of the matrices A, Q, and G, and the number of
rows of the matrices B and L.  N >= 0.

M       (input) INTEGER
The order of the matrix R, and the number of columns of
the matrices B and L.  M >= 0.

A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, if JOBL = 'N', the leading N-by-N part of this
array must contain the matrix A.
On exit, if JOBL = 'N', and INFO = 0, the leading N-by-N
-          -1
part of this array contains the matrix A = A - B*R  L'.
If JOBL = 'Z', this array is not referenced.

LDA     INTEGER
The leading dimension of array A.
LDA >= MAX(1,N) if JOBL = 'N';
LDA >= 1        if JOBL = 'Z'.

B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the matrix B.
On exit, if OUFACT = 1, and INFO = 0, the leading N-by-M
-1
part of this array contains the matrix B*chol(R)  .
On exit, B is unchanged if OUFACT <> 1 (hence also when
FACT = 'U').

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

Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if JOBL = 'N', the leading N-by-N upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the upper
triangular part or lower triangular part, respectively, of
the symmetric matrix Q. The stricly lower triangular part
(if UPLO = 'U') or stricly upper triangular part (if
UPLO = 'L') is not referenced.
On exit, if JOBL = 'N' and INFO = 0, the leading N-by-N
upper triangular part (if UPLO = 'U') or lower triangular
part (if UPLO = 'L') of this array contains the upper
triangular part or lower triangular part, respectively, of
-          -1
the symmetric matrix Q = Q - L*R  *L'.
If JOBL = 'Z', this array is not referenced.

LDQ     INTEGER
The leading dimension of array Q.
LDQ >= MAX(1,N) if JOBL = 'N';
LDQ >= 1        if JOBL = 'Z'.

R       (input/output) DOUBLE PRECISION array, dimension (LDR,M)
On entry, if FACT = 'N', the leading M-by-M upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the upper
triangular part or lower triangular part, respectively,
of the symmetric input weighting matrix R.
On entry, if FACT = 'C', the leading M-by-M upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the Cholesky
factor of the positive definite input weighting matrix R
(as produced by LAPACK routine DPOTRF).
On entry, if FACT = 'U', the leading M-by-M upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array must contain the factors of
the UdU' or LdL' factorization, respectively, of the
symmetric indefinite input weighting matrix R (as produced
by LAPACK routine DSYTRF).
If FACT = 'N', the stricly lower triangular part (if UPLO
= 'U') or stricly upper triangular part (if UPLO = 'L') of
this array is used as workspace (filled in by symmetry).
On exit, if OUFACT = 1, and INFO = 0 (or INFO = M+1),
the leading M-by-M upper triangular part (if UPLO = 'U')
or lower triangular part (if UPLO = 'L') of this array
contains the Cholesky factor of the given input weighting
matrix.
On exit, if OUFACT = 2, and INFO = 0 (or INFO = M+1),
the leading M-by-M upper triangular part (if UPLO = 'U')
or lower triangular part (if UPLO = 'L') of this array
contains the factors of the UdU' or LdL' factorization,
respectively, of the given input weighting matrix.
On exit R is unchanged if FACT = 'C' or 'U'.

LDR     INTEGER
The leading dimension of array R.  LDR >= MAX(1,M).

L       (input/output) DOUBLE PRECISION array, dimension (LDL,M)
On entry, if JOBL = 'N', the leading N-by-M part of this
array must contain the matrix L.
On exit, if JOBL = 'N', OUFACT = 1, and INFO = 0, the
leading N-by-M part of this array contains the matrix
-1
L*chol(R)  .
On exit, L is unchanged if OUFACT <> 1 (hence also when
FACT = 'U').
L is not referenced if JOBL = 'Z'.

LDL     INTEGER
The leading dimension of array L.
LDL >= MAX(1,N) if JOBL = 'N';
LDL >= 1        if JOBL = 'Z'.

IPIV    (input/output) INTEGER array, dimension (M)
On entry, if FACT = 'U', this array must contain details
of the interchanges performed and the block structure of
the d factor in the UdU' or LdL' factorization of matrix R
(as produced by LAPACK routine DSYTRF).
On exit, if OUFACT = 2, this array contains details of
the interchanges performed and the block structure of the
d factor in the UdU' or LdL' factorization of matrix R,
as produced by LAPACK routine DSYTRF.
This array is not referenced if FACT = 'C'.

OUFACT  (output) INTEGER
Information about the factorization finally used.
OUFACT = 0:  no factorization of R has been used (M = 0);
OUFACT = 1:  Cholesky factorization of R has been used;
OUFACT = 2:  UdU' (if UPLO = 'U') or LdL' (if UPLO = 'L')
factorization of R has been used.

G       (output) DOUBLE PRECISION array, dimension (LDG,N)
If JOBG = 'G', and INFO = 0, the leading N-by-N upper
triangular part (if UPLO = 'U') or lower triangular part
(if UPLO = 'L') of this array contains the upper
triangular part (if UPLO = 'U') or lower triangular part
-1
(if UPLO = 'L'), respectively, of the matrix G = B*R  B'.
If JOBG = 'N', this array is not referenced.

LDG     INTEGER
The leading dimension of array G.
LDG >= MAX(1,N) if JOBG = 'G';
LDG >= 1        if JOBG = 'N'.

```
Workspace
```  IWORK   INTEGER array, dimension (M)
If FACT = 'C' or FACT = 'U', this array is not referenced.

DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or LDWORK = -1, DWORK(1) returns the
optimal value of LDWORK; if FACT = 'N' and LDWORK is set
as specified below, DWORK(2) contains the reciprocal
condition number of the given matrix R.
On exit, if LDWORK = -2 on input or INFO = -23, then
DWORK(1) returns the minimal value of LDWORK.

LDWORK  INTEGER
The length of the array DWORK.
LDWORK >= 1              if FACT = 'C' or  (FACT = 'U' and
JOBG = 'N' and  JOBL = 'Z');
LDWORK >= MAX(2,3*M)     if FACT = 'N' and  JOBG = 'N' and
JOBL = 'Z';
LDWORK >= MAX(2,3*M,N*M) if FACT = 'N' and (JOBG = 'G' or
JOBL = 'N');
LDWORK >= MAX(1,N*M)     if FACT = 'U' and (JOBG = 'G' or
JOBL = 'N').
For optimum performance LDWORK should be larger than 3*M,
if FACT = 'N'.

If LDWORK = -1, an optimal workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.

If LDWORK = -2, a minimal workspace query is assumed; the
routine only calculates the minimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message is issued by XERBLA.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value;
= i:  if the i-th element (1 <= i <= M) of the d factor is
exactly zero; the UdU' (or LdL') factorization has
been completed, but the block diagonal matrix d is
exactly singular;
= M+1:  if the matrix R is numerically singular.

```
Method
```                         -     -
The matrices G, and/or A and Q are evaluated using the given or
computed symmetric factorization of R.

```
Numerical Aspects
```  The routine should not be used when R is ill-conditioned.

```
```  None
```
Example

Program Text

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