## MB04RU

### Reduction of a skew-Hamiltonian matrix to Paige/Van Loan (PVL) form (unblocked version)

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

Purpose

      To reduce a skew-Hamiltonian matrix,

                    [  A   G  ]
              W  =  [       T ] ,
                    [  Q   A  ]

      where A is an N-by-N matrix and G, Q are N-by-N skew-symmetric
      matrices, to Paige/Van Loan (PVL) form. That is, an orthogonal
      symplectic matrix U is computed so that

                T       [  Aout  Gout  ]
               U W U =  [            T ] ,
                        [    0   Aout  ]

      where Aout is in upper Hessenberg form.
      Unblocked version.

Specification

          SUBROUTINE MB04RU( N, ILO, A, LDA, QG, LDQG, CS, TAU, DWORK,
         $                   LDWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER           ILO, INFO, LDA, LDQG, LDWORK, N
    C     .. Array Arguments ..
          DOUBLE PRECISION  A(LDA,*), CS(*), DWORK(*), QG(LDQG,*), TAU(*)

Arguments

Input/Output Parameters

      N       (input) INTEGER
              The order of the matrix A.  N >= 0.

      ILO     (input) INTEGER
              It is assumed that A is already upper triangular and Q is
              zero in rows and columns 1:ILO-1. ILO is normally set by a
              previous call to the SLICOT Library routine MB04DS;
              otherwise it should be set to 1.
              1 <= ILO <= N+1, if N > 0; ILO = 1, if N = 0.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the matrix A.
              On exit, the leading N-by-N part of this array contains
              the matrix Aout and, in the zero part of Aout, information
              about the elementary reflectors used to compute the
              PVL factorization.

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

      QG      (input/output) DOUBLE PRECISION array, dimension
                             (LDQG,N+1)
              On entry, the leading N-by-N+1 part of this array must
              contain in columns 1:N the strictly lower triangular part
              of the matrix Q and in columns 2:N+1 the strictly upper
              triangular part of the matrix G. The parts containing the
              diagonal and the first superdiagonal of this array are not
              referenced.
              On exit, the leading N-by-N+1 part of this array contains
              in its first N-1 columns information about the elementary
              reflectors used to compute the PVL factorization and in
              its last N columns the strictly upper triangular part of
              the matrix Gout.

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

      CS      (output) DOUBLE PRECISION array, dimension (2N-2)
              On exit, the first 2N-2 elements of this array contain the
              cosines and sines of the symplectic Givens rotations used
              to compute the PVL factorization.

      TAU     (output) DOUBLE PRECISION array, dimension (N-1)
              On exit, the first N-1 elements of this array contain the
              scalar factors of some of the elementary reflectors.

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,N-1).

Error Indicator

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

Method

      The matrix U is represented 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) ).

      Each H(i) has the form

            H(i) = I - tau * v * v'

      where tau is a real scalar, and v is a real vector with v(1:i) = 0
      and v(i+1) = 1; v(i+2:n) is stored on exit in QG(i+2:n,i), and
      tau in QG(i+1,i).

      Each F(i) has the form

            F(i) = I - nu * w * w'

      where nu is a real scalar, and w is a real vector with w(1:i) = 0
      and w(i+1) = 1; w(i+2:n) is stored on exit in A(i+2:n,i), and
      nu in TAU(i).

      Each G(i) is a Givens rotation acting on rows i+1 and n+i+1, where
      the cosine is stored in CS(2*i-1) and the sine in CS(2*i).

Numerical Aspects

      The algorithm requires 40/3 N**3 + O(N) floating point operations
      and is strongly backward stable.

References

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

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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