MB04WR - SLICOT Library Routine Documentation

## MB04WR

### Generating orthogonal symplectic matrices defined as products of symplectic reflectors and Givens rotations

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

Purpose

      To generate orthogonal symplectic matrices U or V, defined as
      products of symplectic reflectors and Givens rotations

      U = diag( HU(1),HU(1) )  GU(1)  diag( FU(1),FU(1) )
          diag( HU(2),HU(2) )  GU(2)  diag( FU(2),FU(2) )
                               ....
          diag( HU(n),HU(n) )  GU(n)  diag( FU(n),FU(n) ),

      V = diag( HV(1),HV(1) )       GV(1)   diag( FV(1),FV(1) )
          diag( HV(2),HV(2) )       GV(2)   diag( FV(2),FV(2) )
                                    ....
          diag( HV(n-1),HV(n-1) )  GV(n-1)  diag( FV(n-1),FV(n-1) ),

      as returned by the SLICOT Library routines MB04TS or MB04TB. The
      matrices U and V are returned in terms of their first N/2 rows:

                  [  U1   U2 ]           [  V1   V2 ]
              U = [          ],      V = [          ].
                  [ -U2   U1 ]           [ -V2   V1 ]

Specification

          SUBROUTINE MB04WR( JOB, TRANS, N, ILO, Q1, LDQ1, Q2, LDQ2, CS,
         $                   TAU, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         JOB, TRANS
          INTEGER           ILO, INFO, LDQ1, LDQ2, LDWORK, N
    C     .. Array Arguments ..
          DOUBLE PRECISION  CS(*), DWORK(*), Q1(LDQ1,*), Q2(LDQ2,*), TAU(*)

Arguments

Input/Output Parameters

      JOB     CHARACTER*1
              Specifies whether the matrix U or the matrix V is
              required:
              = 'U':  generate U;
              = 'V':  generate V.

      TRANS   CHARACTER*1
              If  JOB = 'U'  then TRANS must have the same value as
              the argument TRANA in the previous call of MB04TS or
              MB04TB.
              If  JOB = 'V'  then TRANS must have the same value as
              the argument TRANB in the previous call of MB04TS or
              MB04TB.

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

      ILO     (input) INTEGER
              ILO must have the same value as in the previous call of
              MB04TS or MB04TB. U and V are equal to the unit matrix
              except in the submatrices
              U([ilo:n n+ilo:2*n], [ilo:n n+ilo:2*n]) and
              V([ilo+1:n n+ilo+1:2*n], [ilo+1:n n+ilo+1:2*n]),
              respectively.
              1 <= ILO <= N, if N > 0; ILO = 1, if N = 0.

      Q1      (input/output) DOUBLE PRECISION array, dimension (LDQ1,N)
              On entry, if  JOB = 'U'  and  TRANS = 'N'  then the
              leading N-by-N part of this array must contain in its i-th
              column the vector which defines the elementary reflector
              FU(i).
              If  JOB = 'U'  and  TRANS = 'T'  or  TRANS = 'C' then the
              leading N-by-N part of this array must contain in its i-th
              row the vector which defines the elementary reflector
              FU(i).
              If  JOB = 'V'  and  TRANS = 'N'  then the leading N-by-N
              part of this array must contain in its i-th row the vector
              which defines the elementary reflector FV(i).
              If  JOB = 'V'  and  TRANS = 'T'  or  TRANS = 'C' then the
              leading N-by-N part of this array must contain in its i-th
              column the vector which defines the elementary reflector
              FV(i).
              On exit, if  JOB = 'U'  and  TRANS = 'N'  then the leading
              N-by-N part of this array contains the matrix U1.
              If  JOB = 'U'  and  TRANS = 'T'  or  TRANS = 'C' then the
              leading N-by-N part of this array contains the matrix
              U1**T.
              If  JOB = 'V'  and  TRANS = 'N'  then the leading N-by-N
              part of this array contains the matrix V1**T.
              If  JOB = 'V'  and  TRANS = 'T'  or  TRANS = 'C' then the
              leading N-by-N part of this array contains the matrix V1.

      LDQ1    INTEGER
              The leading dimension of the array Q1.  LDQ1 >= MAX(1,N).

      Q2      (input/output) DOUBLE PRECISION array, dimension (LDQ2,N)
              On entry, if  JOB = 'U'  then the leading N-by-N part of
              this array must contain in its i-th column the vector
              which defines the elementary reflector HU(i).
              If  JOB = 'V'  then the leading N-by-N part of this array
              must contain in its i-th row the vector which defines the
              elementary reflector HV(i).
              On exit, if  JOB = 'U'  then the leading N-by-N part of
              this array contains the matrix U2.
              If  JOB = 'V'  then the leading N-by-N part of this array
              contains the matrix V2**T.

      LDQ2    INTEGER
              The leading dimension of the array Q2.  LDQ2 >= MAX(1,N).

      CS      (input) DOUBLE PRECISION array, dimension (2N)
              On entry, if  JOB = 'U'  then the first 2N elements of
              this array must contain the cosines and sines of the
              symplectic Givens rotations GU(i).
              If  JOB = 'V'  then the first 2N-2 elements of this array
              must contain the cosines and sines of the symplectic
              Givens rotations GV(i).

      TAU     (input) DOUBLE PRECISION array, dimension (N)
              On entry, if  JOB = 'U'  then the first N elements of
              this array must contain the scalar factors of the
              elementary reflectors FU(i).
              If  JOB = 'V'  then the first N-1 elements of this array
              must contain the scalar factors of the elementary
              reflectors FV(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 = -12,  DWORK(1)  returns the minimum
              value of LDWORK.

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= MAX(1,2*(N-ILO+1)).

              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] Benner, P., Mehrmann, V., and Xu, H.
          A numerically stable, structure preserving method for
          computing the eigenvalues of real Hamiltonian or symplectic
          pencils. Numer. Math., Vol 78 (3), pp. 329-358, 1998.

      [2] 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

Return to Supporting Routines index