## 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
    *
    *     .. 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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