MB04WU - SLICOT Library Routine Documentation

## MB04WU

### Generating an orthogonal basis spanning an isotropic subspace (unblocked version)

[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 ) ]

Specification

          SUBROUTINE MB04WU( 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.
              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).

Error Indicator

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

References

      [1] Bunse-Gerstner, A.
          Matrix factorizations for symplectic QR-like methods.
          Linear Algebra Appl., 83, pp. 49-77, 1986.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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