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
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