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<HTML>
<HEAD><TITLE>MB04JD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB04JD">MB04JD</A></H2>
<H3>
LQ factorization of a matrix with an upper right-hand side zero triangle
</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 an LQ factorization of an n-by-m matrix A (A = L * Q),
having a min(n,p)-by-p zero triangle in the upper right-hand side
corner, as shown below, for n = 8, m = 7, and p = 2:
[ x x x x x 0 0 ]
[ x x x x x x 0 ]
[ x x x x x x x ]
[ x x x x x x x ]
A = [ x x x x x x x ],
[ x x x x x x x ]
[ x x x x x x x ]
[ x x x x x x x ]
and optionally apply the transformations to an l-by-m matrix B
(from the right). The problem structure is exploited. This
computation is useful, for instance, in combined measurement and
time update of one iteration of the time-invariant Kalman filter
(square root covariance filter).
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB04JD( N, M, P, L, A, LDA, B, LDB, TAU, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
INTEGER INFO, L, LDA, LDB, LDWORK, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), TAU(*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The number of rows of the matrix A. N >= 0.
M (input) INTEGER
The number of columns of the matrix A. M >= 0.
P (input) INTEGER
The order of the zero triagle. P >= 0.
L (input) INTEGER
The number of rows of the matrix B. L >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,M)
On entry, the leading N-by-M part of this array must
contain the matrix A. The elements corresponding to the
zero MIN(N,P)-by-P upper trapezoidal/triangular part
(if P > 0) are not referenced.
On exit, the elements on and below the diagonal of this
array contain the N-by-MIN(N,M) lower trapezoidal matrix
L (L is lower triangular, if N <= M) of the LQ
factorization, and the relevant elements above the
diagonal contain the trailing components (the vectors v,
see Method) of the elementary reflectors used in the
factorization.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading L-by-M part of this array must
contain the matrix B.
On exit, the leading L-by-M part of this array contains
the updated matrix B.
If L = 0, this array is not referenced.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,L).
TAU (output) DOUBLE PRECISION array, dimension MIN(N,M)
The scalar factors of the elementary reflectors used.
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK The length of the array DWORK.
LDWORK >= MAX(1,N-1,N-P,L).
For optimum performance LDWORK should be larger.
</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="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The routine uses min(N,M) Householder transformations exploiting
the zero pattern of the matrix. A Householder matrix has the form
( 1 ),
H = I - tau *u *u', u = ( v )
i i i i i ( i)
where v is an (M-P+I-2)-vector. The components of v are stored
i i
in the i-th row of A, beginning from the location i+1, and tau
i
is stored in TAU(i).
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The algorithm is 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>
<HR>
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