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<HTML>
<HEAD><TITLE>MB04QF - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB04QF">MB04QF</A></H2>
<H3>
Forming the triangular block factors of a symplectic block reflector
</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 form the triangular block factors R, S and T of a symplectic
block reflector SH, which is defined as a product of 2k
concatenated Householder reflectors and k Givens rotations,
SH = diag( H(1),H(1) ) G(1) diag( F(1),F(1) )
diag( H(2),H(2) ) G(2) diag( F(2),F(2) )
....
diag( H(k),H(k) ) G(k) diag( F(k),F(k) ).
The upper triangular blocks of the matrices
[ S1 ] [ T11 T12 T13 ]
R = [ R1 R2 R3 ], S = [ S2 ], T = [ T21 T22 T23 ],
[ S3 ] [ T31 T32 T33 ]
with R2 unit and S1, R3, T21, T31, T32 strictly upper triangular,
are stored rowwise in the arrays RS and T, respectively.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB04QF( DIRECT, STOREV, STOREW, N, K, V, LDV, W, LDW,
$ CS, TAU, RS, LDRS, T, LDT, DWORK )
C .. Scalar Arguments ..
CHARACTER DIRECT, STOREV, STOREW
INTEGER K, LDRS, LDT, LDV, LDW, N
C .. Array Arguments ..
DOUBLE PRECISION CS(*), DWORK(*), RS(LDRS,*), T(LDT,*),
$ TAU(*), V(LDV,*), W(LDW,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
DIRECT CHARACTER*1
This is a dummy argument, which is reserved for future
extensions of this subroutine. Not referenced.
STOREV CHARACTER*1
Specifies how the vectors which define the concatenated
Householder F(i) reflectors are stored:
= 'C': columnwise;
= 'R': rowwise.
STOREW CHARACTER*1
Specifies how the vectors which define the concatenated
Householder H(i) reflectors are stored:
= 'C': columnwise;
= 'R': rowwise.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the Householder reflectors F(i) and H(i).
N >= 0.
K (input) INTEGER
The number of Givens rotations. K >= 1.
V (input) DOUBLE PRECISION array, dimension
(LDV,K) if STOREV = 'C',
(LDV,N) if STOREV = 'R'
On entry with STOREV = 'C', the leading N-by-K part of
this array must contain in its i-th column the vector
which defines the elementary reflector F(i).
On entry with STOREV = 'R', the leading K-by-N part of
this array must contain in its i-th row the vector
which defines the elementary reflector F(i).
LDV INTEGER
The leading dimension of the array V.
LDV >= MAX(1,N), if STOREV = 'C';
LDV >= K, if STOREV = 'R'.
W (input) DOUBLE PRECISION array, dimension
(LDW,K) if STOREW = 'C',
(LDW,N) if STOREW = 'R'
On entry with STOREW = 'C', the leading N-by-K part of
this array must contain in its i-th column the vector
which defines the elementary reflector H(i).
On entry with STOREV = 'R', the leading K-by-N part of
this array must contain in its i-th row the vector
which defines the elementary reflector H(i).
LDW INTEGER
The leading dimension of the array W.
LDW >= MAX(1,N), if STOREW = 'C';
LDW >= K, if STOREW = 'R'.
CS (input) DOUBLE PRECISION array, dimension (2*K)
On entry, the first 2*K elements of this array must
contain the cosines and sines of the symplectic Givens
rotations G(i).
TAU (input) DOUBLE PRECISION array, dimension (K)
On entry, the first K elements of this array must
contain the scalar factors of the elementary reflectors
F(i).
RS (output) DOUBLE PRECISION array, dimension (K,6*K)
On exit, the leading K-by-6*K part of this array contains
the upper triangular matrices defining the factors R and
S of the symplectic block reflector SH. The (strictly)
lower portions of this array are not used.
LDRS INTEGER
The leading dimension of the array RS. LDRS >= K.
T (output) DOUBLE PRECISION array, dimension (K,9*K)
On exit, the leading K-by-9*K part of this array contains
the upper triangular matrices defining the factor T of the
symplectic block reflector SH. The (strictly) lower
portions of this array are not used.
LDT INTEGER
The leading dimension of the array T. LDT >= K.
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension (3*K)
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Kressner, D.
Block algorithms for orthogonal symplectic factorizations.
BIT, 43 (4), pp. 775-790, 2003.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The algorithm requires ( 4*K - 2 )*K*N + 19/3*K*K*K + 1/2*K*K
+ 43/6*K - 4 floating point operations.
</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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