TG01WD

Reduction of the descriptor dynamics matrix pair to generalized real Schur form

Purpose

  To reduce the pair (A,E) to a real generalized Schur form
  by using an orthogonal equivalence transformation
  (A,E) <-- (Q'*A*Z,Q'*E*Z) and to apply the transformation
  to the matrices B and C: B <-- Q'*B and C <-- C*Z.

      SUBROUTINE TG01WD( N, M, P, A, LDA, E, LDE, B, LDB, C, LDC,
     $                   Q, LDQ, Z, LDZ, ALPHAR, ALPHAI, BETA, DWORK,
     $                   LDWORK, INFO )
C     .. Scalar Arguments ..
      INTEGER           INFO, LDA, LDB, LDC, LDE, LDQ, LDWORK, LDZ,
     $                  M, N, P
C     .. Array Arguments ..
      DOUBLE PRECISION  A(LDA,*), ALPHAI(*), ALPHAR(*), B(LDB,*),
     $                  BETA(*),  C(LDC,*),  DWORK(*),  E(LDE,*),
     $                  Q(LDQ,*), Z(LDZ,*)

Input/Output Parameters

  N       (input) INTEGER
          The order of the original state-space representation,
          i.e., the order of the matrices A and E.  N >= 0.

  M       (input) INTEGER
          The number of system inputs, or of columns of B.  M >= 0.

  P       (input) INTEGER
          The number of system outputs, or of rows of C.  P >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading N-by-N part of this array must
          contain the original state dynamics matrix A.
          On exit, the leading N-by-N part of this array contains
          the matrix Q' * A * Z in an upper quasi-triangular form.
          The elements below the first subdiagonal are set to zero.

  LDA     INTEGER
          The leading dimension of array A.  LDA >= MAX(1,N).

  E       (input/output) DOUBLE PRECISION array, dimension (LDE,N)
          On entry, the leading N-by-N part of this array must
          contain the original descriptor matrix E.
          On exit, the leading N-by-N part of this array contains
          the matrix Q' * E * Z in an upper triangular form.
          The elements below the diagonal are set to zero.

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

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
          On entry, the leading N-by-M part of this array must
          contain the input matrix B.
          On exit, the leading N-by-M part of this array contains
          the transformed input matrix Q' * B.

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

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the leading P-by-N part of this array must
          contain the output matrix C.
          On exit, the leading P-by-N part of this array contains
          the transformed output matrix C * Z.

  LDC     INTEGER
          The leading dimension of array C.  LDC >= MAX(1,P).

  Q       (output) DOUBLE PRECISION array, dimension (LDQ,N)
          The leading N-by-N part of this array contains the left
          orthogonal transformation matrix used to reduce (A,E) to
          the real generalized Schur form.
          The columns of Q are the left generalized Schur vectors
          of the pair (A,E).

  LDQ     INTEGER
          The leading dimension of array Q.  LDQ >= max(1,N).

  Z       (output) DOUBLE PRECISION array, dimension (LDZ,N)
          The leading N-by-N part of this array contains the right
          orthogonal transformation matrix used to reduce (A,E) to
          the real generalized Schur form.
          The columns of Z are the right generalized Schur vectors
          of the pair (A,E).

  LDZ     INTEGER
          The leading dimension of array Z.  LDZ >= max(1,N).

  ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
  ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
  BETA    (output) DOUBLE PRECISION array, dimension (N)
          On exit, if INFO = 0, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j),
          j=1,...,N, will be the generalized eigenvalues.
          ALPHAR(j) + ALPHAI(j)*i, and BETA(j), j=1,...,N, are the
          diagonals of the complex Schur form that would result if
          the 2-by-2 diagonal blocks of the real Schur form of
          (A,E) were further reduced to triangular form using
          2-by-2 complex unitary transformations.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real;
          if positive, then the j-th and (j+1)-st eigenvalues are a
          complex conjugate pair, with ALPHAI(j+1) negative.

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

  LDWORK  INTEGER
          The dimension of working array DWORK.  LDWORK >= 8*N+16.
          For optimum performance LDWORK should be larger.

  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          > 0:  if INFO = i, the QZ algorithm failed to compute
                the generalized real Schur form; elements i+1:N of
                ALPHAR, ALPHAI, and BETA should be correct.

  The pair (A,E) is reduced to a real generalized Schur form using
  an orthogonal equivalence transformation (A,E) <-- (Q'*A*Z,Q'*E*Z)
  and the transformation is applied to the matrices B and C:
  B <-- Q'*B and C <-- C*Z.

                                  3
  The algorithm requires about 25N  floating point operations.

  None

Program Text

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