MB04QB - SLICOT Library Routine Documentation

## MB04QB

### Applying a product of symplectic reflectors and Givens rotations to two general real matrices

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

Purpose

      To overwrite general real m-by-n matrices C and D, or their
      transposes, with

                [ op(C) ]
          Q  *  [       ]   if TRANQ = 'N', or
                [ op(D) ]

           T    [ op(C) ]
          Q  *  [       ]   if TRANQ = 'T',
                [ op(D) ]

      where Q is defined as the 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) ).

      Blocked version.

Specification

          SUBROUTINE MB04QB( TRANC, TRAND, TRANQ, STOREV, STOREW, M, N, K,
         $                   V, LDV, W, LDW, C, LDC, D, LDD, CS, TAU, DWORK,
         $                   LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         STOREV, STOREW, TRANC, TRAND, TRANQ
          INTEGER           INFO, K, LDC, LDD, LDV, LDW, LDWORK, M, N
    C     .. Array Arguments ..
          DOUBLE PRECISION  C(LDC,*), CS(*), D(LDD,*), DWORK(*), TAU(*),
         $                  V(LDV,*), W(LDW,*)

Arguments

Mode Parameters

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

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

      TRANQ   CHARACTER*1
              = 'N':  apply Q;
              = 'T':  apply Q'.

      STOREV  CHARACTER*1
              Specifies how the vectors which define the concatenated
              Householder reflectors contained in V are stored:
              = 'C':  columnwise;
              = 'R':  rowwise.

      STOREW  CHARACTER*1
              Specifies how the vectors which define the concatenated
              Householder reflectors contained in W are stored:
              = 'C':  columnwise;
              = 'R':  rowwise.

Input/Output Parameters

      M       (input) INTEGER
              The number of rows of the matrices op(C) and op(D).
              M >= 0.

      N       (input) INTEGER
              The number of columns of the matrices op(C) and op(D).
              N >= 0.

      K       (input) INTEGER
              The number of elementary reflectors whose product defines
              the matrix Q.  M >= K >= 0.

      V       (input) DOUBLE PRECISION array, dimension
                      (LDV,K) if STOREV = 'C',
                      (LDV,M) if STOREV = 'R'
              On entry with STOREV = 'C', the leading M-by-K part of
              this array must contain in its columns the vectors which
              define the elementary reflectors F(i).
              On entry with STOREV = 'R', the leading K-by-M part of
              this array must contain in its rows the vectors which
              define the elementary reflectors F(i).

      LDV     INTEGER
              The leading dimension of the array V.
              LDV >= MAX(1,M),  if STOREV = 'C';
              LDV >= MAX(1,K),  if STOREV = 'R'.

      W       (input) DOUBLE PRECISION array, dimension
                      (LDW,K) if STOREW = 'C',
                      (LDW,M) if STOREW = 'R'
              On entry with STOREW = 'C', the leading M-by-K part of
              this array must contain in its columns the vectors which
              define the elementary reflectors H(i).
              On entry with STOREW = 'R', the leading K-by-M part of
              this array must contain in its rows the vectors which
              define the elementary reflectors H(i).

      LDW     INTEGER
              The leading dimension of the array W.
              LDW >= MAX(1,M),  if STOREW = 'C';
              LDW >= MAX(1,K),  if STOREW = 'R'.

      C       (input/output) DOUBLE PRECISION array, dimension
                      (LDC,N) if TRANC = 'N',
                      (LDC,M) if TRANC = 'T' or TRANC = 'C'
              On entry with TRANC = 'N', the leading M-by-N part of
              this array must contain the matrix C.
              On entry with TRANC = 'C' or TRANC = 'T', the leading
              N-by-M part of this array must contain the transpose of
              the matrix C.
              On exit with TRANC = 'N', the leading M-by-N part of
              this array contains the updated matrix C.
              On exit with TRANC = 'C' or TRANC = 'T', the leading
              N-by-M part of this array contains the transpose of the
              updated matrix C.

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

      D       (input/output) DOUBLE PRECISION array, dimension
                      (LDD,N) if TRAND = 'N',
                      (LDD,M) if TRAND = 'T' or TRAND = 'C'
              On entry with TRAND = 'N', the leading M-by-N part of
              this array must contain the matrix D.
              On entry with TRAND = 'C' or TRAND = 'T', the leading
              N-by-M part of this array must contain the transpose of
              the matrix D.
              On exit with TRAND = 'N', the leading M-by-N part of
              this array contains the updated matrix D.
              On exit with TRAND = 'C' or TRAND = 'T', the leading
              N-by-M part of this array contains the transpose of the
              updated matrix D.

      LDD     INTEGER
              The leading dimension of the array D.
              LDD >= MAX(1,M),  if TRAND = 'N';
              LDD >= MAX(1,N),  if TRAND = 'T' or TRAND = '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 = -20,  DWORK(1)  returns the minimum
              value of LDWORK.

      LDWORK  INTEGER
              The length of the array DWORK.  LDWORK >= MAX(1,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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