MB04WR

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

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 ]

      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(*)

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).

  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.

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

  [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.

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