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<HTML>
<HEAD><TITLE>MB03VW - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB03VW">MB03VW</A></H2>
<H3>
Periodic Hessenberg form of a formal product of p matrices using orthogonal similarity transformations
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
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.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
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,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
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.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
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).
</PRE>
<B>Workspace</B>
<PRE>
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.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
None
</PRE>
<B>Program Data</B>
<PRE>
None
</PRE>
<B>Program Results</B>
<PRE>
None
</PRE>
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