## MB03XS

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

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

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 ]

Specification

          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(*)

Arguments

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.

Input/Output Parameters

      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).

Workspace

      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.

Error Indicator

      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.)

Method

      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.

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.

      [2] Kressner, D.
          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

Return to Supporting Routines index