MB03WD

Schur decomposition and eigenvalues of a product of matrices in periodic Hessenberg form

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

Purpose

  To compute the Schur decomposition and the eigenvalues of a
  product of matrices, H = H_1*H_2*...*H_p, with H_1 an upper
  Hessenberg matrix and H_2, ..., H_p upper triangular matrices,
  without evaluating the product. Specifically, the matrices Z_i
  are computed, such that

          Z_1' * H_1 * Z_2 = T_1,
          Z_2' * H_2 * Z_3 = T_2,
                 ...
          Z_p' * H_p * Z_1 = T_p,

  where T_1 is in real Schur form, and T_2, ..., T_p are upper
  triangular.

  The routine works primarily with the Hessenberg and triangular
  submatrices in rows and columns ILO to IHI, but optionally applies
  the transformations to all the rows and columns of the matrices
  H_i, i = 1,...,p. The transformations can be optionally
  accumulated.

Specification
      SUBROUTINE MB03WD( JOB, COMPZ, N, P, ILO, IHI, ILOZ, IHIZ, H,
     $                   LDH1, LDH2, Z, LDZ1, LDZ2, WR, WI, DWORK,
     $                   LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER         COMPZ, JOB
      INTEGER           IHI, IHIZ, ILO, ILOZ, INFO, LDH1, LDH2, LDWORK,
     $                  LDZ1, LDZ2, N, P
C     .. Array Arguments ..
      DOUBLE PRECISION  DWORK( * ), H( LDH1, LDH2, * ), WI( * ),
     $                  WR( * ), Z( LDZ1, LDZ2, * )

Arguments

Mode Parameters

  JOB     CHARACTER*1
          Indicates whether the user wishes to compute the full
          Schur form or the eigenvalues only, as follows:
          = 'E':  Compute the eigenvalues only;
          = 'S':  Compute the factors T_1, ..., T_p of the full
                  Schur form, T = T_1*T_2*...*T_p.

  COMPZ   CHARACTER*1
          Indicates whether or not the user wishes to accumulate
          the matrices Z_1, ..., Z_p, as follows:
          = 'N':  The matrices Z_1, ..., Z_p are not required;
          = 'I':  Z_i is initialized to the unit matrix and the
                  orthogonal transformation matrix Z_i is returned,
                  i = 1, ..., p;
          = 'V':  Z_i must contain an orthogonal matrix Q_i on
                  entry, and the product Q_i*Z_i is returned,
                  i = 1, ..., p.

Input/Output Parameters
  N       (input) INTEGER
          The order of the matrix H.  N >= 0.

  P       (input) INTEGER
          The number of matrices in the product H_1*H_2*...*H_p.
          P >= 1.

  ILO     (input) INTEGER
  IHI     (input) INTEGER
          It is assumed that all matrices H_j, j = 2, ..., p, are
          already upper triangular in rows and columns 1:ILO-1 and
          IHI+1:N, and H_1 is upper quasi-triangular in rows and
          columns 1:ILO-1 and IHI+1:N, with H_1(ILO,ILO-1) = 0
          (unless ILO = 1), and H_1(IHI+1,IHI) = 0 (unless IHI = N).
          The routine works primarily with the Hessenberg submatrix
          in rows and columns ILO to IHI, but applies the
          transformations to all the rows and columns of the
          matrices H_i, i = 1,...,p, if JOB = 'S'.
          1 <= ILO <= max(1,N); min(ILO,N) <= IHI <= N.

  ILOZ    (input) INTEGER
  IHIZ    (input) INTEGER
          Specify the rows of Z to which the transformations must be
          applied if COMPZ = 'I' or COMPZ = 'V'.
          1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

  H       (input/output) DOUBLE PRECISION array, dimension
          (LDH1,LDH2,P)
          On entry, the leading N-by-N part of H(*,*,1) must contain
          the upper Hessenberg matrix H_1 and the leading N-by-N
          part of H(*,*,j) for j > 1 must contain the upper
          triangular matrix H_j, j = 2, ..., p.
          On exit, if JOB = 'S', the leading N-by-N part of H(*,*,1)
          is upper quasi-triangular in rows and columns ILO:IHI,
          with any 2-by-2 diagonal blocks corresponding to a pair of
          complex conjugated eigenvalues, and the leading N-by-N
          part of H(*,*,j) for j > 1 contains the resulting upper
          triangular matrix T_j.
          If JOB = 'E', the contents of H are unspecified on exit.

  LDH1    INTEGER
          The first leading dimension of the array H.
          LDH1 >= max(1,N).

  LDH2    INTEGER
          The second leading dimension of the array H.
          LDH2 >= max(1,N).

  Z       (input/output) DOUBLE PRECISION array, dimension
          (LDZ1,LDZ2,P)
          On entry, if COMPZ = 'V', the leading N-by-N-by-P part of
          this array must contain the current matrix Q of
          transformations accumulated by SLICOT Library routine
          MB03VY.
          If COMPZ = 'I', Z need not be set on entry.
          On exit, if COMPZ = 'V', or COMPZ = 'I', the leading
          N-by-N-by-P part of this array contains the transformation
          matrices which produced the Schur form; the
          transformations are applied only to the submatrices
          Z_j(ILOZ:IHIZ,ILO:IHI), j = 1, ..., P.
          If COMPZ = 'N', Z is not referenced.

  LDZ1    INTEGER
          The first leading dimension of the array Z.
          LDZ1 >= 1,        if COMPZ = 'N';
          LDZ1 >= max(1,N), if COMPZ = 'I' or COMPZ = 'V'.

  LDZ2    INTEGER
          The second leading dimension of the array Z.
          LDZ2 >= 1,        if COMPZ = 'N';
          LDZ2 >= max(1,N), if COMPZ = 'I' or COMPZ = 'V'.

  WR      (output) DOUBLE PRECISION array, dimension (N)
  WI      (output) DOUBLE PRECISION array, dimension (N)
          The real and imaginary parts, respectively, of the
          computed eigenvalues ILO to IHI are stored in the
          corresponding elements of WR and WI. If two eigenvalues
          are computed as a complex conjugate pair, they are stored
          in consecutive elements of WR and WI, say the i-th and
          (i+1)th, with WI(i) > 0 and WI(i+1) < 0. If JOB = 'S', the
          eigenvalues are stored in the same order as on the
          diagonal of the Schur form returned in H.

Workspace
  DWORK   DOUBLE PRECISION array, dimension (LDWORK)

  LDWORK  INTEGER
          The length of the array DWORK.  LDWORK >= IHI-ILO+P-1.

Error Indicator
  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          > 0:  if INFO = i, ILO <= i <= IHI, the QR algorithm
                failed to compute all the eigenvalues ILO to IHI
                in a total of 30*(IHI-ILO+1) iterations;
                the elements i+1:IHI of WR and WI contain those
                eigenvalues which have been successfully computed.

Method
  A refined version of the QR algorithm proposed in [1] and [2] is
  used. The elements of the subdiagonal, diagonal, and the first
  supradiagonal of current principal submatrix of H are computed
  in the process.

References
  [1] Bojanczyk, A.W., Golub, G. and Van Dooren, P.
      The periodic Schur decomposition: algorithms and applications.
      Proc. of the SPIE Conference (F.T. Luk, Ed.), 1770, pp. 31-42,
      1992.

  [2] Sreedhar, J. and Van Dooren, P.
      Periodic Schur form and some matrix equations.
      Proc. of the Symposium on the Mathematical Theory of Networks
      and Systems (MTNS'93), Regensburg, Germany (U. Helmke,
      R. Mennicken and J. Saurer, Eds.), Vol. 1, pp. 339-362, 1994.

Numerical Aspects
  The algorithm is numerically stable.

Further Comments
  Note that for P = 1, the LAPACK Library routine DHSEQR could be
  more efficient on some computer architectures than this routine,
  because DHSEQR uses a block multishift QR algorithm.
  When P is large and JOB = 'S', it could be more efficient to
  compute the product matrix H, and use the LAPACK Library routines.

Example

Program Text

*     MB03WD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
      INTEGER          NIN, NOUT
      PARAMETER        ( NIN = 5, NOUT = 6 )
      INTEGER          NMAX, PMAX
      PARAMETER        ( NMAX = 20, PMAX = 20 )
      INTEGER          LDA1, LDA2, LDTAU, LDZ1, LDZ2, LDZTA
      PARAMETER        ( LDA1 = NMAX, LDA2 = NMAX, LDTAU = NMAX-1,
     $                   LDZ1 = NMAX, LDZ2 = NMAX, LDZTA = NMAX )
      INTEGER          LDWORK
      PARAMETER        ( LDWORK = MAX( NMAX, NMAX + PMAX - 2 ) )
      DOUBLE PRECISION ZERO, ONE
      PARAMETER        ( ZERO = 0.0D0, ONE = 1.0D0 )
*     .. Local Scalars ..
      DOUBLE PRECISION SSQ
      INTEGER          I, IHI, IHIZ, ILO, ILOZ, INFO, J, K, KP1, N, P
      CHARACTER        COMPZ, JOB
*     .. Local Arrays ..
      DOUBLE PRECISION A(LDA1,LDA2,PMAX), AS(LDA1,LDA2,PMAX),
     $                 DWORK(LDWORK), TAU(LDTAU,PMAX), WI(NMAX),
     $                 WR(NMAX), Z(LDZ1,LDZ2,PMAX), ZTA(LDZTA,NMAX)
*     .. External Functions ..
      DOUBLE PRECISION DLANGE, DLAPY2
      LOGICAL          LSAME
      EXTERNAL         DLANGE, DLAPY2, LSAME
*     .. External Subroutines ..
      EXTERNAL         DGEMM, DLACPY, MB03VD, MB03VY, MB03WD, MB03WX
*     .. Intrinsic Functions ..
      INTRINSIC        MAX, MIN
*     .. Executable Statements ..
      WRITE (NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
      READ ( NIN, FMT = '()' )
      READ ( NIN, FMT = * ) N, P, ILO, IHI, ILOZ, IHIZ, JOB, COMPZ
      IF ( N.LT.0 .OR. N.GT.MIN( LDA1, LDA2 ) ) THEN
         WRITE ( NOUT, FMT = 99988 ) N
      ELSE
         IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
            WRITE ( NOUT, FMT = 99987 ) P
         ELSE
*           Read matrices A_1, ..., A_p from the input file.
            DO 10 K = 1, P
               READ ( NIN, FMT = * )
     $            ( ( A(I,J,K), J = 1, N ), I = 1, N )
               CALL DLACPY( 'F', N, N, A(1,1,K), LDA1, AS(1,1,K), LDA1 )
   10       CONTINUE
*           Reduce to the periodic Hessenberg form.
            CALL MB03VD( N, P, ILO, IHI, A, LDA1, LDA2, TAU, LDTAU,
     $                   DWORK, INFO )
            IF ( INFO.NE.0 ) THEN
               WRITE ( NOUT, FMT = 99997 ) INFO
            ELSE
               IF ( LSAME( COMPZ, 'V' ) ) THEN
                  DO 20 K = 1, P
                     CALL DLACPY( 'L', N, N, A(1,1,K), LDA1, Z(1,1,K),
     $                            LDZ1 )
   20             CONTINUE
*                 Accumulate the transformations.
                  CALL MB03VY( N, P, ILO, IHI, Z, LDZ1, LDZ2, TAU,
     $                         LDTAU, DWORK, LDWORK, INFO )
                  IF ( INFO.NE.0 ) THEN
                     WRITE ( NOUT, FMT = 99996 ) INFO
                     STOP
                  ELSE
*                    Reduce to the periodic Schur form.
                     CALL MB03WD( JOB, COMPZ, N, P, ILO, IHI, ILOZ,
     $                            IHIZ, A, LDA1, LDA2, Z, LDZ1, LDZ2,
     $                            WR, WI, DWORK, LDWORK, INFO )
                     IF ( INFO.GT.0 ) THEN
                        WRITE ( NOUT, FMT = 99998 ) INFO
                        WRITE ( NOUT, FMT = 99991 )
                        DO 30 I = MAX( ILO, INFO + 1 ), IHI
                           WRITE ( NOUT, FMT = 99990 ) WR(I), WI(I)
   30                   CONTINUE
                        STOP
                     END IF
                     IF ( INFO.LT.0 ) THEN
                        WRITE ( NOUT, FMT = 99998 ) INFO
                     ELSE
*                       Store the isolated eigenvalues.
                        CALL MB03WX( ILO-1, P, A, LDA1, LDA2, WR, WI,
     $                               INFO )
                        IF ( IHI.LT.N )
     $                     CALL MB03WX( N-IHI, P, A(IHI+1,IHI+1,1),
     $                                  LDA1, LDA2, WR(IHI+1),
     $                                  WI(IHI+1), INFO )
                        WRITE ( NOUT, FMT = 99991 )
                        DO 40 I = 1, N
                           WRITE ( NOUT, FMT = 99990 ) WR(I), WI(I)
   40                   CONTINUE
                        WRITE ( NOUT, FMT = 99995 )
                        DO 60 K = 1, P
                           WRITE ( NOUT, FMT = 99994 ) K
                           DO 50 I = 1, N
                              WRITE ( NOUT, FMT = 99993 )
     $                              ( A(I,J,K), J = 1, N )
   50                      CONTINUE
   60                   CONTINUE
                        WRITE ( NOUT, FMT = 99992 )
                        DO 80 K = 1, P
                           WRITE ( NOUT, FMT = 99994 ) K
                           DO 70 I = 1, N
                              WRITE ( NOUT, FMT = 99993 )
     $                              ( Z(I,J,K), J = 1, N )
   70                      CONTINUE
   80                   CONTINUE
*                       Compute error.
                        SSQ = ZERO
                        DO 90 K = 1, P
                           KP1 = K+1
                           IF( KP1.GT.P ) KP1 = 1
*                          Compute NORM (Z' * A * Z - Aout)
                           CALL DGEMM( 'T', 'N', N, N, N, ONE, Z(1,1,K),
     $                                 LDZ1, AS(1,1,K), LDA1, ZERO, ZTA,
     $                                 LDZTA )
                           CALL DGEMM( 'N', 'N', N, N, N, ONE, ZTA,
     $                                 LDZTA, Z(1,1,KP1), LDZ1, -ONE,
     $                                 A(1,1,K), LDA1 )
                           SSQ = DLAPY2( SSQ,
     $                                   DLANGE( 'Frobenius', N, N,
     $                                           A(1,1,K), LDA1,
     $                                           DWORK ) )
   90                   CONTINUE
                        WRITE ( NOUT, FMT = 99989 ) SSQ
                     END IF
                  END IF
               END IF
            END IF
         END IF
      END IF
      STOP
99999 FORMAT (' MB03WD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT (' INFO on exit from MB03WD = ', I2)
99997 FORMAT (' INFO on exit from MB03VD = ', I2)
99996 FORMAT (' INFO on exit from MB03VY = ', I2)
99995 FORMAT (/' Reduced matrices')
99994 FORMAT (/' K = ', I5)
99993 FORMAT (8F8.4)
99992 FORMAT (/' Transformation matrices')
99991 FORMAT ( ' Computed eigenvalues'/)
99990 FORMAT (4X,'( ', F17.6,' ,', F17.6,' )')
99989 FORMAT (/,' NORM (Z''*A*Z - Aout) = ', 1PD12.5)
99988 FORMAT (/, ' N is out of range.',/' N = ', I5)
99987 FORMAT (/, ' P is out of range.',/' P = ', I5)
      END
Program Data
MB03WD EXAMPLE PROGRAM DATA
4 2 1 4 1 4 S V
1.5 -.7 3.5 -.7 
1.  0.  2.  3. 
1.5 -.7 2.5 -.3 
1.  0.  2.  1. 
1.5 -.7 3.5 -.7 
1.  0.  2.  3. 
1.5 -.7 2.5 -.3 
1.  0.  2.  1. 
Program Results
 MB03WD EXAMPLE PROGRAM RESULTS

 Computed eigenvalues

    (          6.449861 ,         7.817717 )
    (          6.449861 ,        -7.817717 )
    (          0.091315 ,         0.000000 )
    (          0.208964 ,         0.000000 )

 Reduced matrices

 K =     1
  2.2112  4.3718 -2.3362  0.8907
 -0.9179  2.7688 -0.6570 -2.2426
  0.0000  0.0000  0.3022  0.1932
  0.0000  0.0000  0.0000 -0.4571

 K =     2
  2.9169  3.4539  2.2016  1.2367
  0.0000  3.4745  1.0209 -2.0720
  0.0000  0.0000  0.3022 -0.1932
  0.0000  0.0000  0.0000 -0.4571

 Transformation matrices

 K =     1
  0.3493  0.6751 -0.6490  0.0327
  0.7483 -0.4863 -0.1249 -0.4336
  0.2939  0.5504  0.7148 -0.3158
  0.4813 -0.0700  0.2286  0.8433

 K =     2
  0.2372  0.7221  0.6490  0.0327
  0.8163 -0.3608  0.1249 -0.4336
  0.2025  0.5902 -0.7148 -0.3158
  0.4863  0.0076 -0.2286  0.8433

 NORM (Z'*A*Z - Aout) =  7.18432D-15

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