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<HTML>
<HEAD><TITLE>MB03AI - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB03AI">MB03AI</A></H2>
<H3>
Reducing the first column of a real Wilkinson shift polynomial for a product of matrices to the first unit vector (variant with evaluation, Hessenberg factor is the last one)
</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 compute two Givens rotations (C1,S1) and (C2,S2)
such that the orthogonal matrix
[ Q 0 ] [ C1 S1 0 ] [ 1 0 0 ]
Z = [ ], Q := [ -S1 C1 0 ] * [ 0 C2 S2 ],
[ 0 I ] [ 0 0 1 ] [ 0 -S2 C2 ]
makes the first column of the real Wilkinson double shift
polynomial of the product of matrices in periodic upper Hessenberg
form, stored in the array A, parallel to the first unit vector.
Only the rotation defined by C1 and S1 is used for the real
Wilkinson single shift polynomial (see SLICOT Library routines
MB03BE or MB03BF). All factors whose exponents differ from that of
the Hessenberg factor are assumed nonsingular. The matrix product
is evaluated.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB03AI( SHFT, K, N, AMAP, S, SINV, A, LDA1, LDA2, C1,
$ S1, C2, S2, DWORK )
C .. Scalar Arguments ..
CHARACTER SHFT
INTEGER K, LDA1, LDA2, N, SINV
DOUBLE PRECISION C1, C2, S1, S2
C .. Array Arguments ..
INTEGER AMAP(*), S(*)
DOUBLE PRECISION A(LDA1,LDA2,*), DWORK(*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
SHFT CHARACTER*1
Specifies the number of shifts employed by the shift
polynomial, as follows:
= 'D': two shifts (assumes N > 2);
= 'S': one real shift.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
K (input) INTEGER
The number of factors. K >= 1.
N (input) INTEGER
The order of the factors. N >= 2.
AMAP (input) INTEGER array, dimension (K)
The map for accessing the factors, i.e., if AMAP(I) = J,
then the factor A_I is stored at the J-th position in A.
AMAP(K) is the pointer to the Hessenberg matrix.
S (input) INTEGER array, dimension (K)
The signature array. Each entry of S must be 1 or -1.
SINV (input) INTEGER
Signature multiplier. Entries of S are virtually
multiplied by SINV.
A (input) DOUBLE PRECISION array, dimension (LDA1,LDA2,K)
The leading N-by-N-by-K part of this array must contain
the product (implicitly represented by its K factors)
in periodic upper Hessenberg form.
LDA1 INTEGER
The first leading dimension of the array A. LDA1 >= N.
LDA2 INTEGER
The second leading dimension of the array A. LDA2 >= N.
C1 (output) DOUBLE PRECISION
S1 (output) DOUBLE PRECISION
On exit, C1 and S1 contain the parameters for the first
Givens rotation.
C2 (output) DOUBLE PRECISION
S2 (output) DOUBLE PRECISION
On exit, if SHFT = 'D', C2 and S2 contain the parameters
for the second Givens rotation. Otherwise, C2 = 1, S2 = 0.
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension (N*(N+2))
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The necessary elements of the real Wilkinson double shift
polynomial are computed, and suitable Givens rotations are
found. For numerical reasons, this routine should be called
when convergence difficulties are encountered for small order
matrices and small K, e.g., N, K <= 6.
</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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