## MB03VD

### Periodic Hessenberg form of a product of p matrices using orthogonal similarity transformations

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

Purpose

      To reduce a product of p real general matrices A = A_1*A_2*...*A_p
      to upper Hessenberg form, H = H_1*H_2*...*H_p, where H_1 is
      upper Hessenberg, and H_2, ..., H_p are upper triangular, by using
      orthogonal similarity transformations on A,

              Q_1' * A_1 * Q_2 = H_1,
              Q_2' * A_2 * Q_3 = H_2,
                     ...
              Q_p' * A_p * Q_1 = H_p.

Specification

          SUBROUTINE MB03VD( N, P, ILO, IHI, A, LDA1, LDA2, TAU, LDTAU,
         $                   DWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER           IHI, ILO, INFO, LDA1, LDA2, LDTAU, N, P
    C     .. Array Arguments ..
          DOUBLE PRECISION  A( LDA1, LDA2, * ), DWORK( * ), TAU( LDTAU, * )

Arguments

Input/Output Parameters

      N       (input) INTEGER
              The order of the square matrices A_1, A_2, ..., A_p.
              N >= 0.

      P       (input) INTEGER
              The number of matrices in the product A_1*A_2*...*A_p.
              P >= 1.

      ILO     (input) INTEGER
      IHI     (input) INTEGER
              It is assumed that all matrices A_j, j = 2, ..., p, are
              already upper triangular in rows and columns 1:ILO-1 and
              IHI+1:N, and A_1 is upper Hessenberg in rows and columns
              1:ILO-1 and IHI+1:N, with A_1(ILO,ILO-1) = 0 (unless
              ILO = 1), and A_1(IHI+1,IHI) = 0 (unless IHI = N).
              If this is not the case, ILO and IHI should be set to 1
              and N, respectively.
              1 <= ILO <= max(1,N); min(ILO,N) <= IHI <= N.

      A       (input/output) DOUBLE PRECISION array, dimension
              (LDA1,LDA2,P)
              On entry, the leading N-by-N-by-P part of this array must
              contain the matrices of factors to be reduced;
              specifically, A(*,*,j) must contain A_j, j = 1, ..., p.
              On exit, the leading N-by-N upper triangle and the first
              subdiagonal of A(*,*,1) contain the upper Hessenberg
              matrix H_1, and the elements below the first subdiagonal,
              with the first column of the array TAU represent the
              orthogonal matrix Q_1 as a product of elementary
              reflectors. See FURTHER COMMENTS.
              For j > 1, the leading N-by-N upper triangle of A(*,*,j)
              contains the upper triangular matrix H_j, and the elements
              below the diagonal, with the j-th column of the array TAU
              represent the orthogonal matrix Q_j as a product of
              elementary reflectors. See FURTHER COMMENTS.

      LDA1    INTEGER
              The first leading dimension of the array A.
              LDA1 >= max(1,N).

      LDA2    INTEGER
              The second leading dimension of the array A.
              LDA2 >= max(1,N).

      TAU     (output) DOUBLE PRECISION array, dimension (LDTAU,P)
              The leading N-1 elements in the j-th column contain the
              scalar factors of the elementary reflectors used to form
              the matrix Q_j, j = 1, ..., P. See FURTHER COMMENTS.

      LDTAU   INTEGER
              The leading dimension of the array TAU.
              LDTAU >= max(1,N-1).

Workspace

      DWORK   DOUBLE PRECISION array, dimension (N)

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value.

Method

      The algorithm consists in ihi-ilo major steps. In each such
      step i, ilo <= i <= ihi-1, the subdiagonal elements in the i-th
      column of A_j are annihilated using a Householder transformation
      from the left, which is also applied to A_(j-1) from the right,
      for j = p:-1:2. Then, the elements below the subdiagonal of the
      i-th column of A_1 are annihilated, and the Householder
      transformation is also applied to A_p from the right.
      See FURTHER COMMENTS.

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

      Each matrix Q_j is represented as a product of (ihi-ilo)
      elementary reflectors,

         Q_j = H_j(ilo) H_j(ilo+1) . . . H_j(ihi-1).

      Each H_j(i), i = ilo, ..., ihi-1, has the form

         H_j(i) = I - tau_j * v_j * v_j',

      where tau_j is a real scalar, and v_j is a real vector with
      v_j(1:i) = 0, v_j(i+1) = 1 and v_j(ihi+1:n) = 0; v_j(i+2:ihi)
      is stored on exit in A_j(i+2:ihi,i), and tau_j in TAU(i,j).

      The contents of A_1 are illustrated by the following example
      for n = 7, ilo = 2, and ihi = 6:

      on entry                         on exit

      ( a   a   a   a   a   a   a )    ( a   h   h   h   h   h   a )
      ( 0   a   a   a   a   a   a )    ( 0   h   h   h   h   h   a )
      ( 0   a   a   a   a   a   a )    ( 0   h   h   h   h   h   h )
      ( 0   a   a   a   a   a   a )    ( 0   v2  h   h   h   h   h )
      ( 0   a   a   a   a   a   a )    ( 0   v2  v3  h   h   h   h )
      ( 0   a   a   a   a   a   a )    ( 0   v2  v3  v4  h   h   h )
      ( 0   0   0   0   0   0   a )    ( 0   0   0   0   0   0   a )

      where a denotes an element of the original matrix A_1, h denotes
      a modified element of the upper Hessenberg matrix H_1, and vi
      denotes an element of the vector defining H_1(i).

      The contents of A_j, j > 1, are illustrated by the following
      example for n = 7, ilo = 2, and ihi = 6:

      on entry                         on exit

      ( a   a   a   a   a   a   a )    ( a   h   h   h   h   h   a )
      ( 0   a   a   a   a   a   a )    ( 0   h   h   h   h   h   h )
      ( 0   a   a   a   a   a   a )    ( 0   v2  h   h   h   h   h )
      ( 0   a   a   a   a   a   a )    ( 0   v2  v3  h   h   h   h )
      ( 0   a   a   a   a   a   a )    ( 0   v2  v3  v4  h   h   h )
      ( 0   a   a   a   a   a   a )    ( 0   v2  v3  v4  v5  h   h )
      ( 0   0   0   0   0   0   a )    ( 0   0   0   0   0   0   a )

      where a denotes an element of the original matrix A_j, h denotes
      a modified element of the upper triangular matrix H_j, and vi
      denotes an element of the vector defining H_j(i). (The element
      (1,2) in A_p is also unchanged for this example.)

      Note that for P = 1, the LAPACK Library routine DGEHRD could be
      more efficient on some computer architectures than this routine
      (a BLAS 2 version).

Example

Program Text

    *     MB03VD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, PMAX
          PARAMETER        ( NMAX = 20, PMAX = 20 )
          INTEGER          LDA1, LDA2, LDQ1, LDQ2, LDTAU
          PARAMETER        ( LDA1 = NMAX, LDA2 = NMAX, LDQ1 = NMAX,
         $                   LDQ2 = NMAX, LDTAU = NMAX-1 )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = NMAX )
          DOUBLE PRECISION ZERO, ONE
          PARAMETER        ( ZERO = 0.0D0, ONE = 1.0D0 )
    *     .. Local Scalars ..
          DOUBLE PRECISION SSQ
          INTEGER          I, IHI, ILO, INFO, J, K, KP1, N, P
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA1,LDA2,PMAX), AS(LDA1,LDA2,PMAX),
         $                 DWORK(LDWORK), Q(LDQ1,LDQ2,PMAX),
         $                 QTA(LDQ1,NMAX), TAU(LDTAU,PMAX)
    *     .. External Functions ..
          DOUBLE PRECISION DLANGE, DLAPY2
          EXTERNAL         DLANGE, DLAPY2
    *     .. External Subroutines ..
          EXTERNAL         DGEMM, DLACPY, DLASET, MB03VD, MB03VY
    *     .. Intrinsic Functions ..
          INTRINSIC        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
          IF ( N.LT.0 .OR. N.GT.MIN( LDA1, LDA2 ) ) THEN
             WRITE ( NOUT, FMT = 99991 ) N
          ELSE
             IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
                WRITE ( NOUT, FMT = 99990 ) 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 = 99998 ) INFO
                ELSE
                   WRITE ( NOUT, FMT = 99996 )
                   DO 30 K = 1, P
                      CALL DLACPY( 'L', N, N, A(1,1,K), LDA1, Q(1,1,K),
         $                         LDQ1 )
                      IF ( N.GT.1 ) THEN
                         IF ( N.GT.2 .AND. K.EQ.1 ) THEN
                            CALL DLASET( 'L', N-2, N-2, ZERO, ZERO,
         $                               A(3,1,K), LDA1 )
                         ELSE IF ( K.GT.1 ) THEN
                            CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
         $                               A(2,1,K), LDA1 )
                         END IF
                      END IF
                      WRITE ( NOUT, FMT = 99995 ) K
                      DO 20 I = 1, N
                         WRITE ( NOUT, FMT = 99994 ) ( A(I,J,K), J = 1, N )
       20             CONTINUE
       30          CONTINUE
    *              Accumulate the transformations.
                   CALL MB03VY( N, P, ILO, IHI, Q, LDQ1, LDQ2, TAU, LDTAU,
         $                      DWORK, LDWORK, INFO )
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99997 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99993 )
                      DO 50 K = 1, P
                         WRITE ( NOUT, FMT = 99995 ) K
                         DO 40 I = 1, N
                            WRITE ( NOUT, FMT = 99994 )
         $                        ( Q(I,J,K), J = 1, N )
       40                CONTINUE
       50             CONTINUE
    *                 Compute error.
                      SSQ = ZERO
                      DO 60 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, Q(1,1,K), LDQ1,
         $                           AS(1,1,K), LDA1, ZERO, QTA, LDQ1 )
                         CALL DGEMM( 'N', 'N', N, N, N, ONE, QTA, LDQ1,
         $                           Q(1,1,KP1), LDQ1, -ONE, A(1,1,K),
         $                           LDA1 )
                         SSQ = DLAPY2( SSQ,
         $                             DLANGE( 'Frobenius', N, N, A(1,1,K),
         $                                     LDA1, DWORK ) )
       60             CONTINUE
                      WRITE ( NOUT, FMT = 99992 ) SSQ
                   END IF
                END IF
             END IF
          END IF
          STOP
    99999 FORMAT (' MB03VD EXAMPLE PROGRAM RESULTS', /1X)
    99998 FORMAT (' INFO on exit from MB03VD = ', I2)
    99997 FORMAT (' INFO on exit from MB03VY = ', I2)
    99996 FORMAT (' Reduced matrices')
    99995 FORMAT (/' K = ', I5)
    99994 FORMAT (8F8.4)
    99993 FORMAT (/' Transformation matrices')
    99992 FORMAT (/,' NORM (Q''*A*Q - Aout) = ', 1PD12.5)
    99991 FORMAT (/, ' N is out of range.',/' N = ', I5)
    99990 FORMAT (/, ' P is out of range.',/' P = ', I5)
          END

Program Data

    MB03VD EXAMPLE PROGRAM DATA
    4 2 1 4
    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

     MB03VD EXAMPLE PROGRAM RESULTS

     Reduced matrices

     K =     1
     -2.3926  2.7042 -0.9598 -1.2335
      4.1417 -1.7046  1.3001 -1.3120
      0.0000 -1.6247 -0.2534  1.6453
      0.0000  0.0000 -0.0169 -0.4451

     K =     2
     -2.5495  2.3402  4.7021  0.2329
      0.0000  1.9725 -0.2483 -2.3493
      0.0000  0.0000 -0.6290 -0.5975
      0.0000  0.0000  0.0000 -0.4426

     Transformation matrices

     K =     1
      1.0000  0.0000  0.0000  0.0000
      0.0000 -0.7103  0.5504 -0.4388
      0.0000 -0.4735 -0.8349 -0.2807
      0.0000 -0.5209  0.0084  0.8536

     K =     2
     -0.5883  0.2947  0.7528 -0.0145
     -0.3922 -0.8070  0.0009 -0.4415
     -0.5883  0.4292 -0.6329 -0.2630
     -0.3922 -0.2788 -0.1809  0.8577

     NORM (Q'*A*Q - Aout) =  2.93760D-15

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