## MB04WD

### Generating an orthogonal basis spanning an isotropic subspace

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

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.

Specification

          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(*)

Arguments

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'.

Input/Output Parameters

      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).

Workspace

      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.

Error Indicator

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

References

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

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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