MB03XS

Eigenvalues and real skew-Hamiltonian Schur form of a skew-Hamiltonian matrix

Purpose

  To compute the eigenvalues and real skew-Hamiltonian Schur form of
  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. Specifically, an orthogonal symplectic matrix U is
  computed so that

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

  where Aout is in Schur canonical form (as returned by the LAPACK
  routine DHSEQR). That is, Aout is block upper triangular with
  1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has
  its diagonal elements equal and its off-diagonal elements of
  opposite sign.

  Optionally, the matrix U is returned in terms of its first N/2
  rows

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

      SUBROUTINE MB03XS( JOBU, N, A, LDA, QG, LDQG, U1, LDU1, U2, LDU2,
     $                   WR, WI, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER         JOBU
      INTEGER           INFO, LDA, LDQG, LDU1, LDU2, LDWORK, N
C     .. Array Arguments ..
      DOUBLE PRECISION  A(LDA,*), DWORK(*), QG(LDQG,*), U1(LDU1,*),
     $                  U2(LDU2,*), WI(*), WR(*)

Mode Parameters

  JOBU    CHARACTER*1
          Specifies whether matrix U is computed or not, as follows:
          = 'N': transformation matrix U is not computed;
          = 'U': transformation matrix U is computed.

  N       (input) INTEGER
          The order of the matrix A.  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 in Schur canonical form.

  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.
          On exit, the leading N-by-N+1 part of this array contains
          in columns 2:N+1 the strictly upper triangular part of the
          skew-symmetric matrix Gout. The part which contained the
          matrix Q is set to zero.
          Note that the parts containing the diagonal and the first
          superdiagonal of this array are not overwritten by zeros
          only if JOBU = 'U' or LDWORK >= 2*N*N - N.

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

  U1      (output) DOUBLE PRECISION array, dimension (LDU1,N)
          On exit, if JOBU = 'U', the leading N-by-N part of this
          array contains the matrix U1.
          If JOBU = 'N', this array is not referenced.

  LDU1    INTEGER
          The leading dimension of the array U1.
          LDU1 >= MAX(1,N),  if JOBU = 'U';
          LDU1 >= 1,         if JOBU = 'N'.

  U2      (output) DOUBLE PRECISION array, dimension (LDU2,N)
          On exit, if JOBU = 'U', the leading N-by-N part of this
          array contains the matrix U2.
          If JOBU = 'N', this array is not referenced.

  LDU2    INTEGER
          The leading dimension of the array U2.
          LDU2 >= MAX(1,N),  if JOBU = 'U';
          LDU2 >= 1,         if JOBU = 'N'.

  WR      (output) DOUBLE PRECISION array, dimension (N)
  WI      (output) DOUBLE PRECISION array, dimension (N)
          The real and imaginary parts, respectively, of the
          eigenvalues of Aout, which are half of the eigenvalues
          of W. The eigenvalues are stored in the same order as on
          the diagonal of Aout, with WR(i) = Aout(i,i) and, if
          Aout(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0
          and WI(i+1) = -WI(i).

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0,  DWORK(1)  returns the optimal value
          of LDWORK.
          On exit, if  INFO = -14,  DWORK(1)  returns the minimum
          value of LDWORK.

  LDWORK  INTEGER
          The length of the array DWORK.
          LDWORK >= MAX(1,(N+5)*N),      if JOBU = 'U';
          LDWORK >= MAX(1,5*N,(N+1)*N),  if JOBU = 'N'.

          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;
          > 0:  if INFO = i, DHSEQR failed to compute all of the
                eigenvalues.  Elements 1:ILO-1 and i+1:N of WR
                and WI contain those eigenvalues which have been
                successfully computed. The matrix A (and QG) has
                been partially reduced; namely, A is upper
                Hessenberg in the rows and columns ILO through i.
                (See DHSEQR for details.)

  First, using the SLICOT Library routine MB04RB, an orthogonal
  symplectic matrix UP is computed so that

             T      [  AP  GP  ]
         UP W UP =  [        T ]
                    [  0   AP  ]

  is in Paige/Van Loan form. Next, the LAPACK routine DHSEQR is
  applied to the matrix AP to compute an orthogonal matrix V so
  that Aout = V'*AP*V is in Schur canonical form.
  Finally, the transformations

                    [ V  0 ]
          U =  UP * [      ],     Gout = V'*G*V,
                    [ 0  V ]

  using the SLICOT Library routine MB01LD for the latter, are
  performed.

  [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.

  [2] Kressner, D.
      Block algorithms for orthogonal symplectic factorizations.
      BIT, 43 (4), pp. 775-790, 2003.

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