MB04RU

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

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.

      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(*)

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.

  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).

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

  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).

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

  [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.

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