## 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 strictly lower triangular part
              (if UPLO = 'U') or strictly 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 strictly lower triangular part (if UPLO
              = 'U') or strictly 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.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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