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<HTML>
<HEAD><TITLE>MB03BC - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB03BC">MB03BC</A></H2>
<H3>
Product singular value decomposition of K-1 triangular factors of order 2
</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 the product singular value decomposition of the K-1
triangular factors corresponding to a 2-by-2 product of K
factors in upper Hessenberg-triangular form.
For a general product of 2-by-2 triangular matrices
S(2) S(3) S(K)
A = A(:,:,2) A(:,:,3) ... A(:,:,K),
Givens rotations are computed so that
S(i)
[ CV(i-1) SV(i-1) ] [ A(1,1,i)(in) A(1,2,i)(in) ]
[ -SV(i-1) CV(i-1) ] [ 0 A(2,2,i)(in) ]
S(i)
[ A(1,1,i)(out) A(1,2,i)(out) ] [ CV(i) SV(i) ]
= [ 0 A(2,2,i)(out) ] [ -SV(i) CV(i) ]
stays upper triangular and
[ CV(1) SV(1) ] [ CV(K) -SV(K) ]
[ -SV(1) CV(1) ] * A * [ SV(K) CV(K) ]
is diagonal.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB03BC( K, AMAP, S, SINV, A, LDA1, LDA2, MACPAR, CV,
$ SV, DWORK )
C .. Scalar Arguments ..
INTEGER K, LDA1, LDA2, SINV
C .. Array Arguments ..
INTEGER AMAP(*), S(*)
DOUBLE PRECISION A(LDA1,LDA2,*), CV(*), DWORK(*), MACPAR(*),
$ SV(*)
</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 factors. K >= 1.
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.
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/output) DOUBLE PRECISION array, dimension
(LDA1,LDA2,K)
On entry, the leading 2-by-2-by-K part of this array must
contain a 2-by-2 product (implicitly represented by its K
factors) in upper Hessenberg-triangular form.
On exit, the leading 2-by-2-by-K part of this array
contains modified triangular factors such that their
product is diagonal.
LDA1 INTEGER
The first leading dimension of the array A. LDA1 >= 2.
LDA2 INTEGER
The second leading dimension of the array A. LDA2 >= 2.
MACPAR (input) DOUBLE PRECISION array, dimension (5)
Machine parameters:
MACPAR(1) overflow threshold, DLAMCH( 'O' );
MACPAR(2) underflow threshold, DLAMCH( 'U' );
MACPAR(3) safe minimum, DLAMCH( 'S' );
MACPAR(4) relative machine precision, DLAMCH( 'E' );
MACPAR(5) base of the machine, DLAMCH( 'B' ).
CV (output) DOUBLE PRECISION array, dimension (K)
On exit, the first K elements of this array contain the
cosines of the Givens rotations.
SV (output) DOUBLE PRECISION array, dimension (K)
On exit, the first K elements of this array contain the
sines of the Givens rotations.
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension (3*(K-1))
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The product singular value decomposition of the K-1
triangular factors are computed as described in [1].
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Bojanczyk, A. and Van Dooren, P.
On propagating orthogonal transformations in a product of 2x2
triangular matrices.
In Reichel, Ruttan and Varga: 'Numerical Linear Algebra',
pp. 1-9, 1993.
</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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