MB04WP - SLICOT Library Routine Documentation

## MB04WP

### Generating an orthogonal symplectic matrix which performed the reduction in MB04PU

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

Purpose

      To generate an orthogonal symplectic matrix U, which is defined as
      a product of symplectic reflectors and Givens rotations

      U = 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(n-1),H(n-1) ) G(n-1) diag( F(n-1),F(n-1) ).

      as returned by MB04PU. The matrix U is returned in terms of its
      first N rows

                       [  U1   U2 ]
                   U = [          ].
                       [ -U2   U1 ]

Specification

          SUBROUTINE MB04WP( N, ILO, U1, LDU1, U2, LDU2, CS, TAU, DWORK,
         $                   LDWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER           ILO, INFO, LDU1, LDU2, LDWORK, N
    C     .. Array Arguments ..
          DOUBLE PRECISION  CS(*), DWORK(*), U1(LDU1,*), U2(LDU2,*), TAU(*)

Arguments

Input/Output Parameters

      N       (input) INTEGER
              The order of the matrices U1 and U2.  N >= 0.

      ILO     (input) INTEGER
              ILO must have the same value as in the previous call of
              MB04PU. U is equal to the unit matrix except in the
              submatrix
              U([ilo+1:n n+ilo+1:2*n], [ilo+1:n n+ilo+1:2*n]).
              1 <= ILO <= N, if N > 0; ILO = 1, if N = 0.

      U1      (input/output) DOUBLE PRECISION array, dimension (LDU1,N)
              On entry, the leading N-by-N part of this array must
              contain in its i-th column the vector which defines the
              elementary reflector F(i).
              On exit, the leading N-by-N part of this array contains
              the matrix U1.

      LDU1    INTEGER
              The leading dimension of the array U1.  LDU1 >= MAX(1,N).

      U2      (input/output) DOUBLE PRECISION array, dimension (LDU2,N)
              On entry, the leading N-by-N part of this array must
              contain in its i-th column the vector which defines the
              elementary reflector H(i) and, on the subdiagonal, the
              scalar factor of H(i).
              On exit, the leading N-by-N part of this array contains
              the matrix U2.

      LDU2    INTEGER
              The leading dimension of the array U2.  LDU2 >= MAX(1,N).

      CS      (input) DOUBLE PRECISION array, dimension (2N-2)
              On entry, the first 2N-2 elements of this array must
              contain the cosines and sines of the symplectic Givens
              rotations G(i).

      TAU     (input) DOUBLE PRECISION array, dimension (N-1)
              On entry, the first N-1 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 = -10,  DWORK(1)  returns the minimum
              value of LDWORK.

      LDWORK  INTEGER
              The length of the array DWORK. LDWORK >= MAX(1,2*(N-ILO)).
              For optimum performance LDWORK should be larger. (See
              SLICOT Library routine MB04WD).

              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.

Numerical Aspects

      The algorithm requires O(N**3) floating point operations and is
      strongly backward stable.

References

      [1] C. F. VAN LOAN:
          A symplectic method for approximating all the eigenvalues of
          a Hamiltonian matrix.
          Linear Algebra and its Applications, 61, pp. 233-251, 1984.

      [2] D. KRESSNER:
          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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