MB03LD

Eigenvalues and right deflating subspace of a real skew-Hamiltonian/Hamiltonian pencil

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

Purpose

  To compute the relevant eigenvalues of a real N-by-N skew-
  Hamiltonian/Hamiltonian pencil aS - bH, with

        (  A  D  )         (  B  F  )
    S = (        ) and H = (        ),                           (1)
        (  E  A' )         (  G -B' )

  where the notation M' denotes the transpose of the matrix M.
  Optionally, if COMPQ = 'C', an orthogonal basis of the right
  deflating subspace of aS - bH corresponding to the eigenvalues
  with strictly negative real part is computed.

Specification
      SUBROUTINE MB03LD( COMPQ, ORTH, N, A, LDA, DE, LDDE, B, LDB, FG,
     $                   LDFG, NEIG, Q, LDQ, ALPHAR, ALPHAI, BETA,
     $                   IWORK, LIWORK, DWORK, LDWORK, BWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER          COMPQ, ORTH
      INTEGER            INFO, LDA, LDB, LDDE, LDFG, LDQ, LDWORK,
     $                   LIWORK, N, NEIG
C     .. Array Arguments ..
      LOGICAL            BWORK( * )
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
     $                   B( LDB, * ), BETA( * ), DE( LDDE, * ),
     $                   DWORK( * ), FG( LDFG, * ), Q( LDQ, * )

Arguments

Mode Parameters

  COMPQ   CHARACTER*1
          Specifies whether to compute the right deflating subspace
          corresponding to the eigenvalues of aS - bH with strictly
          negative real part.
          = 'N':  do not compute the deflating subspace;
          = 'C':  compute the deflating subspace and store it in the
                  leading subarray of Q.

  ORTH    CHARACTER*1
          If COMPQ = 'C', specifies the technique for computing the
          orthogonal basis of the deflating subspace, as follows:
          = 'P':  QR factorization with column pivoting;
          = 'S':  singular value decomposition.
          If COMPQ = 'N', the ORTH value is not used.

Input/Output Parameters
  N       (input) INTEGER
          The order of the pencil aS - bH.  N >= 0, even.

  A       (input/output) DOUBLE PRECISION array, dimension
                         (LDA, N/2)
          On entry, the leading N/2-by-N/2 part of this array must
          contain the matrix A.
          On exit, if COMPQ = 'C', the leading N/2-by-N/2 part of
          this array contains the upper triangular matrix Aout
          (see METHOD); otherwise, it contains the upper triangular
          matrix A obtained just before the application of the
          periodic QZ algorithm (see SLICOT Library routine MB04BD).

  LDA     INTEGER
          The leading dimension of the array A.  LDA >= MAX(1, N/2).

  DE      (input/output) DOUBLE PRECISION array, dimension
                         (LDDE, N/2+1)
          On entry, the leading N/2-by-N/2 lower triangular part of
          this array must contain the lower triangular part of the
          skew-symmetric matrix E, and the N/2-by-N/2 upper
          triangular part of the submatrix in the columns 2 to N/2+1
          of this array must contain the upper triangular part of the
          skew-symmetric matrix D.
          The entries on the diagonal and the first superdiagonal of
          this array need not be set, but are assumed to be zero.
          On exit, if COMPQ = 'C', the leading N/2-by-N/2 lower
          triangular part and the first superdiagonal contain the
          transpose of the upper quasi-triangular matrix C2out (see
          METHOD), and the (N/2-1)-by-(N/2-1) upper triangular part
          of the submatrix in the columns 3 to N/2+1 of this array
          contains the strictly upper triangular part of the
          skew-symmetric matrix Dout (see METHOD), without the main
          diagonal, which is zero.
          On exit, if COMPQ = 'N', the leading N/2-by-N/2 lower
          triangular part and the first superdiagonal contain the
          transpose of the upper Hessenberg matrix C2, and the
          (N/2-1)-by-(N/2-1) upper triangular part of the submatrix
          in the columns 3 to N/2+1 of this array contains the
          strictly upper triangular part of the skew-symmetric
          matrix D (without the main diagonal) just before the
          application of the periodic QZ algorithm.

  LDDE    INTEGER
          The leading dimension of the array DE.
          LDDE >= MAX(1, N/2).

  B       (input/output) DOUBLE PRECISION array, dimension
                         (LDB, N/2)
          On entry, the leading N/2-by-N/2 part of this array must
          contain the matrix B.
          On exit, if COMPQ = 'C', the leading N/2-by-N/2 part of
          this array contains the upper triangular matrix C1out
          (see METHOD); otherwise, it contains the upper triangular
          matrix C1 obtained just before the application of the
          periodic QZ algorithm.

  LDB     INTEGER
          The leading dimension of the array B.  LDB >= MAX(1, N/2).

  FG      (input/output) DOUBLE PRECISION array, dimension
                         (LDFG, N/2+1)
          On entry, the leading N/2-by-N/2 lower triangular part of
          this array must contain the lower triangular part of the
          symmetric matrix G, and the N/2-by-N/2 upper triangular
          part of the submatrix in the columns 2 to N/2+1 of this
          array must contain the upper triangular part of the
          symmetric matrix F.
          On exit, if COMPQ = 'C', the leading N/2-by-N/2 part of
          the submatrix in the columns 2 to N/2+1 of this array
          contains the matrix Vout (see METHOD); otherwise, it
          contains the matrix V obtained just before the application
          of the periodic QZ algorithm.

  LDFG    INTEGER
          The leading dimension of the array FG.
          LDFG >= MAX(1, N/2).

  NEIG    (output) INTEGER
          If COMPQ = 'C', the number of eigenvalues in aS - bH with
          strictly negative real part.

  Q       (output) DOUBLE PRECISION array, dimension (LDQ, 2*N)
          On exit, if COMPQ = 'C', the leading N-by-NEIG part of
          this array contains an orthogonal basis of the right
          deflating subspace corresponding to the eigenvalues of
          aA - bB with strictly negative real part. The remaining
          part of this array is used as workspace.
          If COMPQ = 'N', this array is not referenced.

  LDQ     INTEGER
          The leading dimension of the array Q.
          LDQ >= 1,           if COMPQ = 'N';
          LDQ >= MAX(1, 2*N), if COMPQ = 'C'.

  ALPHAR  (output) DOUBLE PRECISION array, dimension (N/2)
          The real parts of each scalar alpha defining an eigenvalue
          of the pencil aS - bH.

  ALPHAI  (output) DOUBLE PRECISION array, dimension (N/2)
          The imaginary parts of each scalar alpha defining an
          eigenvalue of the pencil aS - bH.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real.

  BETA    (output) DOUBLE PRECISION array, dimension (N/2)
          The scalars beta that define the eigenvalues of the pencil
          aS - bH.
          Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
          beta = BETA(j) represent the j-th eigenvalue of the pencil
          aS - bH, in the form lambda = alpha/beta. Since lambda may
          overflow, the ratios should not, in general, be computed.
          Due to the skew-Hamiltonian/Hamiltonian structure of the
          pencil, for every eigenvalue lambda, -lambda is also an
          eigenvalue, and thus it has only to be saved once in
          ALPHAR, ALPHAI and BETA.
          Specifically, only eigenvalues with imaginary parts
          greater than or equal to zero are stored; their conjugate
          eigenvalues are not stored. If imaginary parts are zero
          (i.e., for real eigenvalues), only positive eigenvalues
          are stored. The remaining eigenvalues have opposite signs.

Workspace
  IWORK   INTEGER array, dimension (LIWORK)
          On exit, if INFO = -19, IWORK(1) returns the minimum value
          of LIWORK.

  LIWORK  INTEGER
          The dimension of the array IWORK.  LIWORK = 1, if N = 0,
          LIWORK >= MAX( N + 12, 2*N + 3 ),     if COMPQ = 'N',
          LIWORK >= MAX( 32, N + 12, 2*N + 3 ), if COMPQ = 'Q'.

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
          On exit, if INFO = -21, DWORK(1) returns the minimum value
          of LDWORK.

  LDWORK  INTEGER
          The dimension of the array DWORK.  LDWORK = 1, if N = 0,
          LDWORK >= 3*(N/2)**2 + N**2 + MAX( MAX( N, 32 ) + 4,
                                          4*(N+1) ), if COMPQ = 'N',
            where the last term can be reduced to 4*N if N/2 is odd;
          LDWORK >= 8*N**2 + MAX( 8*N + 32, N/2 + 168, 272 ),
                                                     if COMPQ = 'C'.
          For good performance LDWORK should be generally larger.

          If LDWORK = -1  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 is issued by XERBLA.

  BWORK   LOGICAL array, dimension (N/2)

Error Indicator
  INFO    INTEGER
          = 0: succesful exit;
          < 0: if INFO = -i, the i-th argument had an illegal value;
          = 1: periodic QZ iteration failed in the SLICOT Library
               routines MB04BD or MB04HD (QZ iteration did not
               converge or computation of the shifts failed);
          = 2: standard QZ iteration failed in the SLICOT Library
               routines MB04HD or MB03DD (called by MB03JD);
          = 3: a numerically singular matrix was found in the SLICOT
               Library routine MB03HD (called by MB03JD);
          = 4: the singular value decomposition failed in the LAPACK
               routine DGESVD (for ORTH = 'S');
          = 5: some eigenvalues might be inaccurate. This is a
               warning.

Method
  First, the decompositions of S and H are computed via orthogonal
  transformations Q1 and Q2 as follows:

                    (  Aout  Dout  )
    Q1' S J Q1 J' = (              ),
                    (   0    Aout' )

                    (  Bout  Fout  )
    J' Q2' J S Q2 = (              ) =: T,                       (2)
                    (   0    Bout' )

               (  C1out  Vout  )            (  0  I  )
    Q1' H Q2 = (               ), where J = (        ),
               (  0     C2out' )            ( -I  0  )

  and Aout, Bout, C1out are upper triangular, C2out is upper quasi-
  triangular and Dout and Fout are skew-symmetric.

  Then, orthogonal matrices Q3 and Q4 are found, for the extended
  matrices

         (  Aout   0  )          (    0   C1out )
    Se = (            ) and He = (              ),
         (   0   Bout )          ( -C2out   0   )

  such that S11 := Q4' Se Q3 is upper triangular and
  H11 := Q4' He Q3 is upper quasi-triangular. The following matrices
  are computed:

               (  Dout   0  )                   (   0   Vout )
    S12 := Q4' (            ) Q4 and H12 := Q4' (            ) Q4.
               (   0   Fout )                   ( Vout'   0  )

  Then, an orthogonal matrix Q is found such that the eigenvalues
  with strictly negative real parts of the pencil

      (  S11  S12  )     (  H11  H12  )
    a (            ) - b (            )
      (   0   S11' )     (   0  -H11' )

  are moved to the top of this pencil.

  Finally, an orthogonal basis of the right deflating subspace
  corresponding to the eigenvalues with strictly negative real part
  is computed. See also page 12 in [1] for more details.

References
  [1] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
      Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
      Eigenproblems.
      Tech. Rep., Technical University Chemnitz, Germany,
      Nov. 2007.

Numerical Aspects
                                                            3
  The algorithm is numerically backward stable and needs O(N )
  floating point operations.

Further Comments
  This routine does not perform any scaling of the matrices. Scaling
  might sometimes be useful, and it should be done externally.

Example

Program Text

*     MB03LD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
      INTEGER            NIN, NOUT
      PARAMETER          ( NIN = 5, NOUT = 6 )
      INTEGER            NMAX
      PARAMETER          ( NMAX = 50 )
      INTEGER            LDA, LDB, LDDE, LDFG, LDQ, LDWORK, LIWORK
      PARAMETER          (  LDA = NMAX/2, LDB = NMAX/2, LDDE = NMAX/2,
     $                     LDFG = NMAX/2, LDQ = 2*NMAX,
     $                     LDWORK = 8*NMAX*NMAX +
     $                              MAX( 8*NMAX + 32, NMAX/2 + 168,
     $                                   272 ),
     $                     LIWORK = MAX( 32, NMAX + 12, NMAX*2 + 3 ) )
*
*     .. Local Scalars ..
      CHARACTER          COMPQ, ORTH
      INTEGER            I, INFO, J, M, N, NEIG
*
*     .. Local Arrays ..
      LOGICAL            BWORK( NMAX/2 )
      INTEGER            IWORK( LIWORK )
      DOUBLE PRECISION   A( LDA, NMAX/2 ),  ALPHAI( NMAX/2 ),
     $                   ALPHAR( NMAX/2 ),  B( LDB, NMAX/2 ),
     $                   BETA( NMAX/2 ),  DE( LDDE, NMAX/2+1 ),
     $                   DWORK( LDWORK ), FG( LDFG, NMAX/2+1 ),
     $                   Q( LDQ, 2*NMAX )
*
*     .. External Subroutines ..
      EXTERNAL           MB03LD
*
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*
*     .. Executable Statements ..
*
      WRITE( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read in the data.
      READ( NIN, FMT = * )
      READ( NIN, FMT = * ) COMPQ, ORTH, N
      IF( N.LT.0 .OR. N.GT.NMAX ) THEN
         WRITE( NOUT, FMT = 99998 ) N
      ELSE
         M = N/2
         READ( NIN, FMT = * ) ( (  A( I, J ), J = 1, M   ), I = 1, M )
         READ( NIN, FMT = * ) ( ( DE( I, J ), J = 1, M+1 ), I = 1, M )
         READ( NIN, FMT = * ) ( (  B( I, J ), J = 1, M   ), I = 1, M )
         READ( NIN, FMT = * ) ( ( FG( I, J ), J = 1, M+1 ), I = 1, M )
*        Compute the eigenvalues and an orthogonal basis of the right
*        deflating subspace of a real skew-Hamiltonian/Hamiltonian
*        pencil, corresponding to the eigenvalues with strictly negative
*        real part.
         CALL MB03LD( COMPQ, ORTH, N, A, LDA, DE, LDDE, B, LDB, FG,
     $                LDFG, NEIG, Q, LDQ, ALPHAR, ALPHAI, BETA, IWORK,
     $                LIWORK, DWORK, LDWORK, BWORK, INFO )
*
         IF( INFO.NE.0 ) THEN
            WRITE( NOUT, FMT = 99997 ) INFO
         ELSE
            WRITE( NOUT, FMT = 99996 )
            DO 10 I = 1, M
               WRITE( NOUT, FMT = 99995 ) ( A( I, J ), J = 1, M )
   10       CONTINUE
            WRITE( NOUT, FMT = 99994 )
            DO 20 I = 1, M
               WRITE( NOUT, FMT = 99995 ) ( DE( I, J ), J = 1, M+1 )
   20       CONTINUE
            WRITE( NOUT, FMT = 99993 )
            DO 30 I = 1, M
               WRITE( NOUT, FMT = 99995 ) ( B( I, J ), J = 1, M )
   30       CONTINUE
            WRITE( NOUT, FMT = 99992 )
            DO 40 I = 1, M
               WRITE( NOUT, FMT = 99995 ) ( FG( I, J ), J = 2, M+1 )
   40       CONTINUE
            WRITE( NOUT, FMT = 99991 )
            WRITE( NOUT, FMT = 99995 ) ( ALPHAR( I ), I = 1, M )
            WRITE( NOUT, FMT = 99990 )
            WRITE( NOUT, FMT = 99995 ) ( ALPHAI( I ), I = 1, M )
            WRITE( NOUT, FMT = 99989 )
            WRITE( NOUT, FMT = 99995 ) (   BETA( I ), I = 1, M )
            IF( LSAME( COMPQ, 'C' ) .AND. NEIG.GT.0 ) THEN
               WRITE( NOUT, FMT = 99988 )
               DO 50 I = 1, N
                  WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, NEIG )
   50          CONTINUE
            END IF
         END IF
      END IF
      STOP
*
99999 FORMAT( 'MB03LD EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT( 'INFO on exit from MB03LD = ', I2 )
99996 FORMAT( 'The matrix A on exit is ' )
99995 FORMAT( 50( 1X, F8.4 ) )
99994 FORMAT( 'The matrix DE on exit is ' )
99993 FORMAT( 'The matrix C1 on exit is ' )
99992 FORMAT( 'The matrix V on exit is ' )
99991 FORMAT( 'The vector ALPHAR is ' )
99990 FORMAT( 'The vector ALPHAI is ' )
99989 FORMAT( 'The vector BETA is ' )
99988 FORMAT( 'The matrix Q is ' )
      END
Program Data
MB03LD EXAMPLE PROGRAM DATA
   C   P   8
   3.1472   1.3236   4.5751   4.5717
   4.0579  -4.0246   4.6489  -0.1462
  -3.7301  -2.2150  -3.4239   3.0028
   4.1338   0.4688   4.7059  -3.5811
   0.0000   0.0000  -1.5510  -4.5974  -2.5127
   3.5071   0.0000   0.0000   1.5961   2.4490  
  -3.1428   2.5648   0.0000   0.0000  -0.0596 
   3.0340   2.4892  -1.1604   0.0000   0.0000
   0.6882  -3.3782  -3.3435   1.8921
  -0.3061   2.9428   1.0198   2.4815
  -4.8810  -1.8878  -2.3703  -0.4946
  -1.6288   0.2853   1.5408  -4.1618
  -2.4013  -2.7102   0.3834  -3.9335   3.1730
  -3.1815  -2.3620   4.9613   4.6190   3.6869
   3.6929   0.7970   0.4986  -4.9537  -4.1556
   3.5303   1.2206  -1.4905   0.1325  -1.0022

Program Results
MB03LD EXAMPLE PROGRAM RESULTS
The matrix A on exit is 
  -4.7460   4.1855   3.2696  -0.2244
   0.0000   6.4157   2.8287   1.4553
   0.0000   0.0000   7.4626   1.5726
   0.0000   0.0000   0.0000   8.8702
The matrix DE on exit is 
  -5.4562   2.5550  -1.3137  -6.3615  -0.8940
  -2.1348  -7.9616   0.0000   1.0704  -0.0659
   4.9694   1.1516   4.8504   0.0000  -0.6922
  -2.2744   3.4912   0.5046   4.4394   0.0000
The matrix C1 on exit is 
   6.9525  -4.9881   2.3661   4.2188
   0.0000   8.5009   0.7182   5.5533
   0.0000   0.0000  -4.6650  -2.8177
   0.0000   0.0000   0.0000   1.5124
The matrix V on exit is 
   0.9136   4.1106  -0.0079   3.5789
  -1.1553  -1.4785  -1.5155  -0.8018
  -2.2167   4.8029   1.3645   2.5202
  -1.0994  -0.6144   0.3970   2.0730
The vector ALPHAR is 
   0.8314  -0.8314   0.8131   0.0000
The vector ALPHAI is 
   0.4372   0.4372   0.0000   0.9164
The vector BETA is 
   0.7071   0.7071   1.4142   2.8284
The matrix Q is 
  -0.5844  -0.2949   0.1692
   0.5470  -0.2324  -0.4524
   0.0382   0.4673  -0.3092
  -0.0378   0.0904  -0.4451
  -0.0255   0.1497  -0.1929
  -0.1286  -0.6067  -0.2275
  -0.2260   0.4901   0.0951
   0.5367  -0.0430   0.6123

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