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
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
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