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<HTML>
<HEAD><TITLE>MB03KC - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB03KC">MB03KC</A></H2>
<H3>
Reducing a 2-by-2 formal matrix product to periodic Hessenberg-triangular form
</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 a 2-by-2 general, formal matrix product A of length K,
A_K^s(K) * A_K-1^s(K-1) * ... * A_1^s(1),
to the periodic Hessenberg-triangular form using a K-periodic
sequence of elementary reflectors (Householder matrices). The
matrices A_k, k = 1, ..., K, are stored in the N-by-N-by-K array A
starting in the R-th row and column, and N can be 3 or 4.
Each elementary reflector H_k is represented as
H_k = I - tau_k * v_k * v_k', (1)
where I is the 2-by-2 identity, tau_k is a real scalar, and v_k is
a vector of length 2, k = 1,...,K, and it is constructed such that
the following holds for k = 1,...,K:
H_{k+1} * A_k * H_k = T_k, if s(k) = 1,
(2)
H_k * A_k * H_{k+1} = T_k, if s(k) = -1,
with H_{K+1} = H_1 and all T_k upper triangular except for
T_{khess} which is full. Clearly,
T_K^s(K) *...* T_1^s(1) = H_1 * A_K^s(K) *...* A_1^s(1) * H_1.
The reflectors are suitably applied to the whole, extended N-by-N
matrices Ae_k, not only to the submatrices A_k, k = 1, ..., K.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB03KC( K, KHESS, N, R, S, A, LDA, V, TAU )
C .. Scalar Arguments ..
INTEGER K, KHESS, LDA, N, R
C .. Array Arguments ..
INTEGER S( * )
DOUBLE PRECISION A( * ), TAU( * ), V( * )
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
</PRE>
<B>Input/Output Parameters</B>
<PRE>
K (input) INTEGER
The number of matrices in the sequence A_k. K >= 2.
KHESS (input) INTEGER
The index for which the returned matrix A_khess should be
in the Hessenberg form on output. 1 <= KHESS <= K.
N (input) INTEGER
The order of the extended matrices. N = 3 or N = 4.
R (input) INTEGER
The starting row and column index for the
2-by-2 submatrices. R = 1, or R = N-1.
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; the value S(k) = -1 corresponds to using the
inverse of the factor A_k.
A (input/output) DOUBLE PRECISION array, dimension (*)
On entry, this array must contain at position IXA(k) =
(k-1)*N*LDA+1 the N-by-N matrix Ae_k stored with leading
dimension LDA.
On exit, this array contains at position IXA(k) the
N-by-N matrix Te_k stored with leading dimension LDA.
LDA INTEGER
Leading dimension of the matrices Ae_k and Te_k in the
one-dimensional array A. LDA >= N.
V (output) DOUBLE PRECISION array, dimension (2*K)
On exit, this array contains the K vectors v_k,
k = 1,...,K, defining the elementary reflectors H_k as
in (1). The k-th reflector is stored in V(2*k-1:2*k).
TAU (output) DOUBLE PRECISION array, dimension (K)
On exit, this array contains the K values of tau_k,
k = 1,...,K, defining the elementary reflectors H_k
as in (1).
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
A K-periodic sequence of elementary reflectors (Householder
matrices) is used. The computations start for k = khess with the
left reflector in (1), which is the identity matrix.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The implemented method is numerically backward stable.
</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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