## MB03KD

### Reordering the diagonal blocks of a formal matrix product using periodic QZ algorithm

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

Purpose

      To reorder the diagonal blocks of the formal matrix product

         T22_K^S(K) * T22_K-1^S(K-1) * ... * T22_1^S(1),             (1)

      of length K, in the generalized periodic Schur form,

               [  T11_k  T12_k  T13_k  ]
         T_k = [    0    T22_k  T23_k  ],    k = 1, ..., K,          (2)
               [    0      0    T33_k  ]

      where

      - the submatrices T11_k are NI(k+1)-by-NI(k), if S(k) = 1, or
        NI(k)-by-NI(k+1), if S(k) = -1, and contain dimension-induced
        infinite eigenvalues,
      - the submatrices T22_k are NC-by-NC and contain core eigenvalues,
        which are generically neither zero nor infinite,
      - the submatrices T33_k contain dimension-induced zero
        eigenvalues,

      such that the M selected eigenvalues pointed to by the logical
      vector SELECT end up in the leading part of the matrix sequence
      T22_k.

      Given that N(k) = N(k+1) for all k where S(k) = -1, the T11_k are
      void and the first M columns of the updated orthogonal
      transformation matrix sequence Q_1, ..., Q_K span a periodic
      deflating subspace corresponding to the same eigenvalues.

Specification

          SUBROUTINE MB03KD( COMPQ, WHICHQ, STRONG, K, NC, KSCHUR, N, NI, S,
         $                   SELECT, T, LDT, IXT, Q, LDQ, IXQ, M, TOL,
         $                   IWORK, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          COMPQ, STRONG
          INTEGER            INFO, K, KSCHUR, LDWORK, M, NC
          DOUBLE PRECISION   TOL
    C     .. Array Arguments ..
          LOGICAL            SELECT( * )
          INTEGER            IWORK( * ), IXQ( * ), IXT( * ), LDQ( * ),
         $                   LDT( * ), N( * ), NI( * ), S( * ), WHICHQ( * )
          DOUBLE PRECISION   DWORK( * ), Q( * ), T( * )

Arguments

Mode Parameters

      COMPQ   CHARACTER*1
              Specifies whether to compute the orthogonal transformation
              matrices Q_k, as follows:
              = 'N': do not compute any of the matrices Q_k;
              = 'I': each coefficient of Q is initialized internally to
                     the identity matrix, and the orthogonal matrices
                     Q_k are returned, where Q_k, k = 1, ..., K,
                     performed the reordering;
              = 'U': each coefficient of Q must contain an orthogonal
                     matrix Q1_k on entry, and the products Q1_k*Q_k are
                     returned;
              = 'W': the computation of each Q_k is specified
                     individually in the array WHICHQ.

      WHICHQ  INTEGER array, dimension (K)
              If COMPQ = 'W', WHICHQ(k) specifies the computation of Q_k
              as follows:
              = 0:   do not compute Q_k;
              = 1:   the kth coefficient of Q is initialized to the
                     identity matrix, and the orthogonal matrix Q_k is
                     returned;
              = 2:   the kth coefficient of Q must contain an orthogonal
                     matrix Q1_k on entry, and the product Q1_k*Q_k is
                     returned.
              This array is not referenced if COMPQ <> 'W'.

      STRONG  CHARACTER*1
              Specifies whether to perform the strong stability tests,
              as follows:
              = 'N': do not perform the strong stability tests;
              = 'S': perform the strong stability tests; often, this is
                     not needed, and omitting them can save some
                     computations.

Input/Output Parameters

      K       (input) INTEGER
              The period of the periodic matrix sequences T and Q (the
              number of factors in the matrix product).  K >= 2.
              (For K = 1, a standard eigenvalue reordering problem is
              obtained.)

      NC      (input) INTEGER
              The number of core eigenvalues.  0 <= NC <= min(N).

      KSCHUR  (input) INTEGER
              The index for which the matrix T22_kschur is upper quasi-
              triangular. All other T22 matrices are upper triangular.

      N       (input) INTEGER array, dimension (K)
              The leading K elements of this array must contain the
              dimensions of the factors of the formal matrix product T,
              such that the k-th coefficient T_k is an N(k+1)-by-N(k)
              matrix, if S(k) = 1, or an N(k)-by-N(k+1) matrix,
              if S(k) = -1, k = 1, ..., K, where N(K+1) = N(1).

      NI      (input) INTEGER array, dimension (K)
              The leading K elements of this array must contain the
              dimensions of the factors of the matrix sequence T11_k.
              N(k) >= NI(k) + NC >= 0.

      S       (input) INTEGER array, dimension (K)
              The leading K elements of this array must contain the
              signatures (exponents) of the factors in the K-periodic
              matrix sequence. Each entry in S must be either 1 or -1;
              the value S(k) = -1 corresponds to using the inverse of
              the factor T_k.

      SELECT  (input) LOGICAL array, dimension (NC)
              SELECT specifies the eigenvalues in the selected cluster.
              To select a real eigenvalue w(j), SELECT(j) must be set to
              .TRUE.. To select a complex conjugate pair of eigenvalues
              w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
              either SELECT(j) or SELECT(j+1) or both must be set to
              .TRUE.; a complex conjugate pair of eigenvalues must be
              either both included in the cluster or both excluded.

      T       (input/output) DOUBLE PRECISION array, dimension (*)
              On entry, this array must contain at position IXT(k) the
              matrix T_k, which is at least N(k+1)-by-N(k), if S(k) = 1,
              or at least N(k)-by-N(k+1), if S(k) = -1, in periodic
              Schur form.
              On exit, the matrices T_k are overwritten by the reordered
              periodic Schur form.

      LDT     INTEGER array, dimension (K)
              The leading dimensions of the matrices T_k in the one-
              dimensional array T.
              LDT(k) >= max(1,N(k+1)),  if S(k) =  1,
              LDT(k) >= max(1,N(k)),    if S(k) = -1.

      IXT     INTEGER array, dimension (K)
              Start indices of the matrices T_k in the one-dimensional
              array T.

      Q       (input/output) DOUBLE PRECISION array, dimension (*)
              On entry, this array must contain at position IXQ(k) a
              matrix Q_k of size at least N(k)-by-N(k), provided that
              COMPQ = 'U', or COMPQ = 'W' and WHICHQ(k) = 2.
              On exit, if COMPQ = 'I' or COMPQ = 'W' and WHICHQ(k) = 1,
              Q_k contains the orthogonal matrix that performed the
              reordering. If COMPQ = 'U', or COMPQ = 'W' and
              WHICHQ(k) = 2, Q_k is post-multiplied with the orthogonal
              matrix that performed the reordering.
              This array is not referenced if COMPQ = 'N'.

      LDQ     INTEGER array, dimension (K)
              The leading dimensions of the matrices Q_k in the one-
              dimensional array Q.
              LDQ(k) >= max(1,N(k)), if COMPQ = 'I', or COMPQ = 'U', or
                                        COMPQ = 'W' and WHICHQ(k) > 0;
              This array is not referenced if COMPQ = 'N'.

      IXQ     INTEGER array, dimension (K)
              Start indices of the matrices Q_k in the one-dimensional
              array Q.
              This array is not referenced if COMPQ = 'N'.

      M       (output) INTEGER
              The number of selected core eigenvalues which were
              reordered to the top of T22_k.

Tolerances

      TOL     DOUBLE PRECISION
              The tolerance parameter c. The weak and strong stability
              tests performed for checking the reordering use a
              threshold computed by the formula  MAX(c*EPS*NRM, SMLNUM),
              where NRM is the varying Frobenius norm of the matrices
              formed by concatenating K pairs of adjacent diagonal
              blocks of sizes 1 and/or 2 in the T22_k submatrices from
              (2), which are swapped, and EPS and SMLNUM are the machine
              precision and safe minimum divided by EPS, respectively
              (see LAPACK Library routine DLAMCH). The value c should
              normally be at least 10.

Workspace

      IWORK   INTEGER array, dimension (4*K)

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

      LDWORK  INTEGER
              The dimension of the array DWORK.
              LDWORK >= 10*K + MN, if all blocks involved in reordering
                                   have order 1;
              LDWORK >= 25*K + MN, if there is at least a block of
                                   order 2, but no adjacent blocks of
                                   order 2 are involved in reordering;
              LDWORK >= MAX(42*K + MN, 80*K - 48), if there is at least
                                   a pair of adjacent blocks of order 2
                                   involved in reordering;
              where MN = MXN, if MXN > 10, and MN = 0, otherwise, with
              MXN = MAX(N(k),k=1,...,K).

              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.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  the reordering of T failed because some eigenvalues
                    are too close to separate (the problem is very ill-
                    conditioned); T may have been partially reordered.

Method

      An adaptation of the LAPACK Library routine DTGSEN is used.

Numerical Aspects

      The implemented method is numerically backward stable.

Further Comments

      None

Example

Program Text

    *     MB03KD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER            NIN, NOUT
          PARAMETER          ( NIN = 5, NOUT = 6 )
          INTEGER            KMAX, NMAX
          PARAMETER          ( KMAX = 6, NMAX = 50 )
          INTEGER            LDA1, LDA2, LDQ1, LDQ2, LDWORK, LIWORK
          PARAMETER          ( LDA1 = NMAX, LDA2 = NMAX, LDQ1 = NMAX,
         $                     LDQ2 = NMAX,
         $                     LDWORK = MAX( 42*KMAX + NMAX, 80*KMAX - 48,
         $                                   KMAX + MAX( 2*NMAX, 8*KMAX ) ),
         $                     LIWORK = MAX( 4*KMAX, 2*KMAX + NMAX ) )
          DOUBLE PRECISION   HUND, ZERO
          PARAMETER          ( HUND = 1.0D2, ZERO = 0.0D0 )
    *
    *     .. Local Scalars ..
          CHARACTER          COMPQ, DEFL, JOB, STRONG
          INTEGER            H, I, IHI, ILO, INFO, IWARN, J, K, L, M, N, P
          DOUBLE PRECISION   TOL
    *
    *     .. Local Arrays ..
          LOGICAL            SELECT( NMAX )
          INTEGER            IWORK( LIWORK ), IXQ( KMAX ), IXT( KMAX ),
         $                   LDQ( KMAX ), LDT( KMAX ), ND( KMAX ),
         $                   NI( KMAX ), QIND( KMAX ), S( KMAX ),
         $                   SCAL( NMAX )
          DOUBLE PRECISION   A( LDA1, LDA2, KMAX ), ALPHAI( NMAX ),
         $                   ALPHAR( NMAX ), BETA( NMAX ), DWORK( LDWORK),
         $                   Q( LDQ1, LDQ2, KMAX ), QK( NMAX*NMAX*KMAX ),
         $                   T( NMAX*NMAX*KMAX )
    *
    *     .. External Functions ..
          LOGICAL            LSAME
          EXTERNAL           LSAME
    *
    *     .. External Subroutines ..
          EXTERNAL           DLACPY, MB03BD, MB03KD
    *
    *     .. Intrinsic Functions ..
          INTRINSIC          INT, 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 = * ) JOB, DEFL, COMPQ, STRONG, K, N, H, ILO, IHI
          IF( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE( NOUT, FMT = 99998 ) N
          ELSE
             TOL = HUND
             READ( NIN, FMT = * ) ( S( I ), I = 1, K )
             READ( NIN, FMT = * ) ( ( ( A( I, J, L ), J = 1, N ),
         $                                I = 1, N ), L = 1, K )
             IF( LSAME( COMPQ, 'U' ) )
         $      READ( NIN, FMT = * ) ( ( ( Q( I, J, L ), J = 1, N ),
         $                                   I = 1, N ), L = 1, K )
             IF( LSAME( COMPQ, 'P' ) ) THEN
                READ( NIN, FMT = * ) ( QIND( I ), I = 1, K )
                DO 10 L = 1, K
                   IF( QIND( L ).GT.0 )
         $            READ( NIN, FMT = * ) ( ( Q( I, J, QIND( L ) ),
         $                                    J = 1, N ), I = 1, N )
       10       CONTINUE
             END IF
             IF( LSAME( JOB, 'E' ) )
         $      JOB = 'S'
    *        Compute the eigenvalues and the transformed matrices.
             CALL MB03BD( JOB, DEFL, COMPQ, QIND, K, N, H, ILO, IHI, S, A,
         $                LDA1, LDA2, Q, LDQ1, LDQ2, ALPHAR, ALPHAI, BETA,
         $                SCAL, IWORK, LIWORK, DWORK, LDWORK, IWARN, INFO )
    *
             IF( INFO.NE.0 ) THEN
                WRITE( NOUT, FMT = 99997 ) INFO
             ELSE IF( IWARN.EQ.0 ) THEN
    *           Prepare the data for calling MB03KD, which uses different
    *           data structures and reverse ordering of the factors.
                DO 20 L = 1, K
                   ND(  L ) = MAX( 1, N )
                   NI(  L ) = 0
                   LDT( L ) = MAX( 1, N )
                   IXT( L ) = ( L - 1 )*LDT( L )*N + 1
                   LDQ( L ) = MAX( 1, N )
                   IXQ( L ) = IXT( L )
                   IF( L.LE.INT( K/2 ) ) THEN
                      I = S( K - L + 1 )
                      S( K - L + 1 ) = S( L )
                      S( L ) = I
                   END IF
       20       CONTINUE
                DO 30 L = 1, K
                   CALL DLACPY( 'Full', N, N, A( 1, 1, K-L+1 ), LDA1,
         $                      T( IXT( L ) ), LDT( L ) )
       30       CONTINUE
                IF( LSAME( COMPQ, 'U' ) .OR. LSAME( COMPQ, 'I' ) ) THEN
                   COMPQ = 'U'
                   DO 40 L = 1, K
                      CALL DLACPY( 'Full', N, N, Q( 1, 1, K-L+1 ), LDQ1,
         $                         QK( IXQ( L ) ), LDQ( L ) )
       40          CONTINUE
                ELSE IF( LSAME( COMPQ, 'P' ) ) THEN
                   COMPQ = 'W'
                   DO 50 L = 1, K
                      IF( QIND( L ).LT.0 )
         $                QIND( L ) = 2
                      P = QIND( L )
                      IF( P.NE.0 )
         $               CALL DLACPY( 'Full', N, N, Q( 1, 1, K-P+1 ), LDQ1,
         $                            QK( IXQ( P ) ), LDQ( P ) )
       50          CONTINUE
                END IF
    *           Select eigenvalues with negative real part.
                DO 60 I = 1, N
                   SELECT( I ) = ALPHAR( I ).LT.ZERO
       60       CONTINUE
                WRITE( NOUT, FMT = 99996 )
                WRITE( NOUT, FMT = 99995 ) ( ALPHAR( I ), I = 1, N )
                WRITE( NOUT, FMT = 99994 )
                WRITE( NOUT, FMT = 99995 ) ( ALPHAI( I ), I = 1, N )
                WRITE( NOUT, FMT = 99993 )
                WRITE( NOUT, FMT = 99995 ) (   BETA( I ), I = 1, N )
                WRITE( NOUT, FMT = 99992 )
                WRITE( NOUT, FMT = 99991 ) (   SCAL( I ), I = 1, N )
    *           Compute the transformed matrices, after reordering the
    *           eigenvalues.
                CALL MB03KD( COMPQ, QIND, STRONG, K, N, H, ND, NI, S,
         $                   SELECT, T, LDT, IXT, QK, LDQ, IXQ, M, TOL,
         $                   IWORK, DWORK, LDWORK, INFO )
                IF( INFO.NE.0 ) THEN
                   WRITE( NOUT, FMT = 99990 ) INFO
                ELSE
                   WRITE( NOUT, FMT = 99989 )
                   DO 80 L = 1, K
                      P = K - L + 1
                      WRITE( NOUT, FMT = 99988 ) L
                      DO 70 I = 1, N
                         WRITE( NOUT, FMT = 99995 )
         $                    ( T( IXT( P ) + I - 1 + ( J - 1 )*LDT( P ) ),
         $                       J = 1, N )
       70             CONTINUE
       80          CONTINUE
                   IF( LSAME( COMPQ, 'U' ) .OR. LSAME( COMPQ, 'I' ) ) THEN
                      WRITE( NOUT, FMT = 99987 )
                      DO 100 L = 1, K
                         P = K - L + 1
                         WRITE( NOUT, FMT = 99988 ) L
                         DO 90 I = 1, N
                            WRITE( NOUT, FMT = 99995 )
         $                       ( QK( IXQ( P ) + I - 1 +
         $                           ( J - 1 )*LDQ( P ) ), J = 1, N )
       90                CONTINUE
      100             CONTINUE
                   ELSE IF( LSAME( COMPQ, 'W' ) ) THEN
                      WRITE( NOUT, FMT = 99987 )
                      DO 120 L = 1, K
                         IF( QIND( L ).GT.0 ) THEN
                            P = K - QIND( L ) + 1
                            WRITE( NOUT, FMT = 99988 ) QIND( L )
                            DO 110 I = 1, N
                               WRITE( NOUT, FMT = 99995 )
         $                          ( QK( IXQ( P ) + I - 1 +
         $                              ( J - 1 )*LDQ( P ) ), J = 1, N )
      110                   CONTINUE
                         END IF
      120             CONTINUE
                   END IF
                END IF
             ELSE
                WRITE( NOUT, FMT = 99979 ) IWARN
             END IF
          END IF
          STOP
    *
    99999 FORMAT( 'MB03KD EXAMPLE PROGRAM RESULTS', 1X )
    99998 FORMAT( 'N is out of range.', /, 'N = ', I5 )
    99997 FORMAT( 'INFO on exit from MB03BD = ', I2 )
    99996 FORMAT( 'The vector ALPHAR is ' )
    99995 FORMAT( 50( 1X, F8.4 ) )
    99994 FORMAT( 'The vector ALPHAI is ' )
    99993 FORMAT( 'The vector BETA is ' )
    99992 FORMAT( 'The vector SCAL is ' )
    99991 FORMAT( 50( 1X, I5 ) )
    99990 FORMAT( 'INFO on exit from MB03KD = ', I2 )
    99989 FORMAT( 'The matrix A on exit is ' )
    99988 FORMAT( 'The factor ', I2, ' is ' )
    99987 FORMAT( 'The matrix Q on exit is ' )
    99986 FORMAT( 'LDT', 3I5 )
    99985 FORMAT( 'IXT', 3I5 )
    99984 FORMAT( 'LDQ', 3I5 )
    99983 FORMAT( 'IXQ', 3I5 )
    99982 FORMAT( 'ND' , 3I5 )
    99981 FORMAT( 'NI' , 3I5)
    99980 FORMAT( 'SELECT', 3L5 )
    99979 FORMAT( 'IWARN on exit from MB03BD = ', I2 )
          END

Program Data

    MB03KD EXAMPLE PROGRAM DATA
       S   C   I   N   3   3   2   1   3
      -1     1    -1
       2.0   0.0   1.0
       0.0  -2.0  -1.0
       0.0   0.0   3.0
       1.0   2.0   0.0
       4.0  -1.0   3.0
       0.0   3.0   1.0
       1.0   0.0   1.0
       0.0   4.0  -1.0
       0.0   0.0  -2.0

Program Results

    MB03KD EXAMPLE PROGRAM RESULTS
    The vector ALPHAR is
       0.3230   0.3230  -0.8752
    The vector ALPHAI is
       0.5694  -0.5694   0.0000
    The vector BETA is
       1.0000   1.0000   1.0000
    The vector SCAL is
         0     0    -1
    The matrix A on exit is
    The factor  1 is
       2.5997  -0.0087   1.6898
       0.0000   1.9846   0.1942
       0.0000   0.0000   2.3259
    The factor  2 is
      -2.0990  -1.0831  -2.5601
       0.0000   3.4838   0.2950
       0.0000   3.4552  -2.1690
    The factor  3 is
       1.8451   0.9260   1.2717
       0.0000   1.3976  -2.3544
       0.0000   0.0000  -3.1023
    The matrix Q on exit is
    The factor  1 is
      -0.2052   0.4647  -0.8614
       0.2033   0.8811   0.4270
      -0.9574   0.0875   0.2753
    The factor  2 is
      -0.7743  -0.1384   0.6176
       0.6070  -0.4386   0.6627
       0.1791   0.8880   0.4236
    The factor  3 is
      -0.6714   0.7225  -0.1651
      -0.3658  -0.5168  -0.7740
      -0.6446  -0.4593   0.6112

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