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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