## MB04RV

### Solution of a generalized complex Sylvester equation with matrix pairs in generalized complex Schur form

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

Purpose

      To solve the generalized complex Sylvester equation

               A * R - L * B = scale * C,                            (1)
               D * R - L * E = scale * F,

      using Level 1 and 2 BLAS, where R and L are unknown M-by-N
      matrices, and (A, D), (B, E) and (C, F) are given matrix pairs of
      size M-by-M, N-by-N and M-by-N, respectively. A, B, D and E are
      complex upper triangular (i.e., (A,D) and (B,E) are in generalized
      Schur form).

      The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an
      output scaling factor chosen to avoid overflow.

      This routine is intended to be called only by SLICOT Library
      routine MB04RW. For efficiency purposes, the computations are
      aborted when the absolute value of an element of R or L is greater
      than a given value PMAX.

Specification

          SUBROUTINE MB04RV( M, N, PMAX, A, LDA, B, LDB, C, LDC, D, LDD, E,
         $                   LDE, F, LDF, SCALE, INFO )
    C     .. Scalar Arguments ..
          INTEGER            INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N
          DOUBLE PRECISION   PMAX, SCALE
    C     .. Array Arguments ..
          COMPLEX*16         A( LDA, * ), B( LDB, * ), C( LDC, * ),
         $                   D( LDD, * ), E( LDE, * ), F( LDF, * )

Arguments

Input/Output Parameters

      M       (input) INTEGER
              The order of the matrices A and D, and the row dimension
              of C, F, R and L.  M >= 0.

      N       (input) INTEGER
              The order of the matrices B and E, and the column
              dimension of C, F, R and L.  N >= 0.

      PMAX    (input) DOUBLE PRECISION
              An upper bound for the "absolute value" of the elements of
              the solution (R, L). (See FURTHER COMMENTS.)
              PMAX >= 1.0D0.

      A       (input) COMPLEX*16 array, dimension (LDA, M)
              On entry, the leading M-by-M upper triangular part of this
              array must contain the matrix A in the generalized complex
              Schur form, as returned by LAPACK routine ZGGES.

      LDA     INTEGER
              The leading dimension of the array A.  LDA >= max(1, M).

      B       (input) COMPLEX*16 array, dimension (LDB, N)
              On entry, the leading N-by-N upper triangular part of this
              array must contain the matrix B in the generalized complex
              Schur form.

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

      C       (input/output) COMPLEX*16 array, dimension (LDC, N)
              On entry, the leading M-by-N part of this array must
              contain the right-hand-side of the first matrix equation
              in (1).
              On exit, if INFO = 0, the leading M-by-N part of this
              array contains the solution R.

      LDC     INTEGER
              The leading dimension of the array C.  LDC >= max(1, M).

      D       (input) COMPLEX*16 array, dimension (LDD, M)
              On entry, the leading M-by-M upper triangular part of this
              array must contain the matrix D in the generalized complex
              Schur form. The diagonal elements are non-negative real.

      LDD     INTEGER
              The leading dimension of the array D.  LDD >= max(1, M).

      E       (input) COMPLEX*16 array, dimension (LDE, N)
              On entry, the leading N-by-N upper triangular part of this
              array must contain the matrix E in the generalized complex
              Schur form. The diagonal elements are non-negative real.

      LDE     INTEGER
              The leading dimension of the array E.  LDE >= max(1, N).

      F       (input/output) COMPLEX*16 array, dimension (LDF, N)
              On entry, the leading M-by-N part of this array must
              contain the right-hand-side of the second matrix equation
              in (1).
              On exit, if INFO = 0, the leading M-by-N part of this
              array contains the solution L.

      LDF     INTEGER
              The leading dimension of the array F.  LDF >= max(1, M).

      SCALE   (output) DOUBLE PRECISION
              On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
              R and L (C and F on entry) will hold the solutions to a
              slightly perturbed system but the input matrices A, B, D
              and E have not been changed. If SCALE = 0, R and L will
              hold the solutions to the homogeneous system with C = 0
              and F = 0. Normally, SCALE = 1.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              = 1:  an element of R or L had the absolute value greater
                    than the given value PMAX.
              = 2:  the matrix pairs (A, D) and (B, E) have common or
                    very close eigenvalues. The matrix Z in section
                    METHOD is (almost) singular.

Method

      The routine uses an adaptation of the method for solving
      generalized Sylvester equations [1], which controls the magnitude
      of the individual elements of the computed solution [2].

      In matrix notation, solving equation (1) corresponds to solve
      Zx = scale * b, where Z is defined as

             Z = [ kron(In, A)  -kron(B', Im) ]                      (2)
                 [ kron(In, D)  -kron(E', Im) ],

      Ik is the identity matrix of size k and X' is the transpose of X.
      kron(X, Y) is the Kronecker product between the matrices X and Y.

References

      [1] Kagstrom, B. and Westin, L.
          Generalized Schur Methods with Condition Estimators for
          Solving the Generalized Sylvester Equation.
          IEEE Trans. Auto. Contr., 34, pp. 745-751, 1989.
      [2] Kagstrom, B. and Westin, L.
          GSYLV - Fortran Routines for the Generalized Schur Method with
          Dif Estimators for Solving the Generalized Sylvester Equation.
          Report UMINF-132.86, Institute of Information Processing,
          Univ. of Umea, Sweden, July 1987.

Further Comments

      For efficiency reasons, the "absolute value" of a complex number x
      is computed as |real(x)| + |imag(x)|.

Example

Program Text

      None

Program Data

      None

Program Results

      None

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