MB04WD

Generating an orthogonal basis spanning an isotropic subspace

Purpose

  To generate a matrix Q with orthogonal columns (spanning an
  isotropic subspace), which is defined as the first n columns
  of a product of symplectic reflectors and Givens rotations,

      Q = diag( H(1),H(1) ) G(1) diag( F(1),F(1) )
          diag( H(2),H(2) ) G(2) diag( F(2),F(2) )
                            ....
          diag( H(k),H(k) ) G(k) diag( F(k),F(k) ).

  The matrix Q is returned in terms of its first 2*M rows

                   [  op( Q1 )   op( Q2 ) ]
               Q = [                      ].
                   [ -op( Q2 )   op( Q1 ) ]

  Blocked version of the SLICOT Library routine MB04WU.

      SUBROUTINE MB04WD( TRANQ1, TRANQ2, M, N, K, Q1, LDQ1, Q2, LDQ2,
     $                   CS, TAU, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER         TRANQ1, TRANQ2
      INTEGER           INFO, K, LDQ1, LDQ2, LDWORK, M, N
C     .. Array Arguments ..
      DOUBLE PRECISION  CS(*), DWORK(*), Q1(LDQ1,*), Q2(LDQ2,*), TAU(*)

Mode Parameters

  TRANQ1  CHARACTER*1
          Specifies the form of op( Q1 ) as follows:
          = 'N':  op( Q1 ) = Q1;
          = 'T':  op( Q1 ) = Q1';
          = 'C':  op( Q1 ) = Q1'.

  TRANQ2  CHARACTER*1
          Specifies the form of op( Q2 ) as follows:
          = 'N':  op( Q2 ) = Q2;
          = 'T':  op( Q2 ) = Q2';
          = 'C':  op( Q2 ) = Q2'.

  M       (input) INTEGER
          The number of rows of the matrices Q1 and Q2. M >= 0.

  N       (input) INTEGER
          The number of columns of the matrices Q1 and Q2.
          M >= N >= 0.

  K       (input) INTEGER
          The number of symplectic Givens rotations whose product
          partly defines the matrix Q. N >= K >= 0.

  Q1      (input/output) DOUBLE PRECISION array, dimension
                  (LDQ1,N) if TRANQ1 = 'N',
                  (LDQ1,M) if TRANQ1 = 'T' or TRANQ1 = 'C'
          On entry with TRANQ1 = 'N', the leading M-by-K part of
          this array must contain in its i-th column the vector
          which defines the elementary reflector F(i).
          On entry with TRANQ1 = 'T' or TRANQ1 = 'C', the leading
          K-by-M part of this array must contain in its i-th row
          the vector which defines the elementary reflector F(i).
          On exit with TRANQ1 = 'N', the leading M-by-N part of this
          array contains the matrix Q1.
          On exit with TRANQ1 = 'T' or TRANQ1 = 'C', the leading
          N-by-M part of this array contains the matrix Q1'.

  LDQ1    INTEGER
          The leading dimension of the array Q1.
          LDQ1 >= MAX(1,M),  if TRANQ1 = 'N';
          LDQ1 >= MAX(1,N),  if TRANQ1 = 'T' or TRANQ1 = 'C'.

  Q2      (input/output) DOUBLE PRECISION array, dimension
                  (LDQ2,N) if TRANQ2 = 'N',
                  (LDQ2,M) if TRANQ2 = 'T' or TRANQ2 = 'C'
          On entry with TRANQ2 = 'N', the leading M-by-K part of
          this array must contain in its i-th column the vector
          which defines the elementary reflector H(i) and, on the
          diagonal, the scalar factor of H(i).
          On entry with TRANQ2 = 'T' or TRANQ2 = 'C', the leading
          K-by-M part of this array must contain in its i-th row the
          vector which defines the elementary reflector H(i) and, on
          the diagonal, the scalar factor of H(i).
          On exit with TRANQ2 = 'N', the leading M-by-N part of this
          array contains the matrix Q2.
          On exit with TRANQ2 = 'T' or TRANQ2 = 'C', the leading
          N-by-M part of this array contains the matrix Q2'.

  LDQ2    INTEGER
          The leading dimension of the array Q2.
          LDQ2 >= MAX(1,M),  if TRANQ2 = 'N';
          LDQ2 >= MAX(1,N),  if TRANQ2 = 'T' or TRANQ2 = 'C'.

  CS      (input) DOUBLE PRECISION array, dimension (2*K)
          On entry, the first 2*K elements of this array must
          contain the cosines and sines of the symplectic Givens
          rotations G(i).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          On entry, the first K elements of this array must
          contain the scalar factors of the elementary reflectors
          F(i).

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0,  DWORK(1)  returns the optimal
          value of LDWORK, MAX(M+N,8*N*NB + 15*NB*NB), where NB is
          the optimal block size determined by the function UE01MD.
          On exit, if  INFO = -13,  DWORK(1)  returns the minimum
          value of LDWORK.

  LDWORK  INTEGER
          The length of the array DWORK.  LDWORK >= MAX(1,M+N).

          If LDWORK = -1, then a 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 related to LDWORK
          is issued by XERBLA.

  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value.

  [1] Kressner, D.
      Block algorithms for orthogonal symplectic factorizations.
      BIT, 43 (4), pp. 775-790, 2003.

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Program Text

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