## MB03VW

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

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

Purpose

      To reduce the generalized matrix product

                   S(1)           S(2)                 S(K)
           A(:,:,1)     * A(:,:,2)     * ... * A(:,:,K)

      to upper Hessenberg-triangular form, where A is N-by-N-by-K and S
      is the signature array with values 1 or -1. The H-th matrix of A
      is reduced to upper Hessenberg form while the other matrices are
      triangularized. Unblocked version.

      If COMPQ = 'U' or COMPZ = 'I', then the orthogonal factors are
      computed and stored in the array Q so that for S(I) = 1,
                                                               T
            Q(:,:,I)(in)   A(:,:,I)(in)   Q(:,:,MOD(I,K)+1)(in)
                                                                T    (1)
         =  Q(:,:,I)(out)  A(:,:,I)(out)  Q(:,:,MOD(I,K)+1)(out) ,

      and for S(I) = -1,
                                                               T
            Q(:,:,MOD(I,K)+1)(in)   A(:,:,I)(in)   Q(:,:,I)(in)
                                                                T    (2)
         =  Q(:,:,MOD(I,K)+1)(out)  A(:,:,I)(out)  Q(:,:,I)(out) .

      A partial generation of the orthogonal factors can be realized via
      the array QIND.

Specification

          SUBROUTINE MB03VW( COMPQ, QIND, TRIU, N, K, H, ILO, IHI, S, A,
         $                   LDA1, LDA2, Q, LDQ1, LDQ2, IWORK, LIWORK,
         $                   DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         COMPQ, TRIU
          INTEGER           H, IHI, ILO, INFO, K, LDA1, LDA2, LDQ1, LDQ2,
         $                  LDWORK, LIWORK, N
    C     .. Array Arguments ..
          INTEGER           IWORK(*), QIND(*), S(*)
          DOUBLE PRECISION  A(LDA1,LDA2,*), DWORK(LDWORK), Q(LDQ1,LDQ2,*)

Arguments

Mode Parameters

      COMPQ   CHARACTER*1
              Specifies whether or not the orthogonal transformations
              should be accumulated in the array Q, as follows:
              = 'N': do not modify Q;
              = 'U': modify (update) the array Q by the orthogonal
                     transformations that are applied to the matrices in
                     the array A to reduce them to periodic Hessenberg-
                     triangular form;
              = 'I': like COMPQ = 'U', except that each matrix in the
                     array Q will be first initialized to the identity
                     matrix;
              = 'P': use the parameters as encoded in QIND.

      QIND    INTEGER array, dimension (K)
              If COMPQ = 'P', then this array describes the generation
              of the orthogonal factors as follows:
                 If QIND(I) > 0, then the array Q(:,:,QIND(I)) is
              modified by the transformations corresponding to the
              i-th orthogonal factor in (1) and (2).
                 If QIND(I) < 0, then the array Q(:,:,-QIND(I)) is
              initialized to the identity and modified by the
              transformations corresponding to the i-th orthogonal
              factor in (1) and (2).
                 If QIND(I) = 0, then the transformations corresponding
              to the i-th orthogonal factor in (1), (2) are not applied.

      TRIU    CHARACTER*1
              Indicates how many matrices are reduced to upper
              triangular form in the first stage of the algorithm,
              as follows
              = 'N':  only matrices with negative signature;
              = 'A':  all possible N - 1 matrices.
              The first choice minimizes the computational costs of the
              algorithm, whereas the second is more cache efficient and
              therefore faster on modern architectures.

Input/Output Parameters

      N       (input)  INTEGER
              The order of each factor in the array A.  N >= 0.

      K       (input) INTEGER
              The number of factors.  K >= 0.

      H       (input/output) INTEGER
              On entry, if H is in the interval [1,K] then the H-th
              factor of A will be transformed to upper Hessenberg form.
              Otherwise the most efficient H is chosen.
              On exit, H indicates the factor of A which is in upper
              Hessenberg form.

      ILO     (input)  INTEGER
      IHI     (input)  INTEGER
              It is assumed that each factor in A is already upper
              triangular in rows and columns 1:ILO-1 and IHI+1:N.
              1 <= ILO <= IHI <= N, if N > 0;
              ILO = 1 and IHI  = 0, if N = 0.
              If ILO = IHI, all factors are upper triangular.

      S       (input)  INTEGER array, dimension (K)
              The leading K elements of this array must contain the
              signatures of the factors. Each entry in S must be either
              1 or -1.

      A       (input/output) DOUBLE PRECISION array, dimension
                             (LDA1,LDA2,K)
              On entry, the leading N-by-N-by-K part of this array must
              contain the factors of the general product to be reduced.
              On exit, A(:,:,H) is overwritten by an upper Hessenberg
              matrix and each A(:,:,I), for I not equal to H, is
              overwritten by an upper triangular matrix.

      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).

      Q       (input/output) DOUBLE PRECISION array, dimension
                             (LDQ1,LDQ2,K)
              On entry, if COMPQ = 'U', the leading N-by-N-by-K part
              of this array must contain the initial orthogonal factors
              as described in (1) and (2).
              On entry, if COMPQ = 'P', only parts of the leading
              N-by-N-by-K part of this array must contain some
              orthogonal factors as described by the parameters QIND.
              If COMPQ = 'I', this array should not be set on entry.
              On exit, if COMPQ = 'U' or COMPQ = 'I', the leading
              N-by-N-by-K part of this array contains the modified
              orthogonal factors as described in (1) and (2).
              On exit, if COMPQ = 'P', only parts of the leading
              N-by-N-by-K part contain some modified orthogonal factors
              as described by the parameters QIND.
              This array is not referenced if COMPQ = 'N'.

      LDQ1    INTEGER
              The first leading dimension of the array Q.  LDQ1 >= 1,
              and, if COMPQ <> 'N', LDQ1 >= MAX(1,N).

      LDQ2    INTEGER
              The second leading dimension of the array Q.  LDQ2 >= 1,
              and, if COMPQ <> 'N', LDQ2 >= MAX(1,N).

Workspace

      IWORK   INTEGER array, dimension (LIWORK)
              On exit, if  INFO = -17,  IWORK(1)  returns the needed
              value of LIWORK.

      LIWORK  INTEGER
              The length of the array IWORK.  LIWORK >= MAX(1,3*K).

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal value
              of LDWORK.
              On exit, if  INFO = -19,  DWORK(1)  returns the minimum
              value of LIWORK.

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= 1, if MIN(N,K) = 0, or N = 1 or ILO = IHI;
              LDWORK >= M+MAX(IHI,N-ILO+1)), otherwise, where
                        M = IHI-ILO+1.
              For optimum performance LDWORK should be 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.

Error Indicator

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

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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