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<HTML>
<HEAD><TITLE>MB04OW - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB04OW">MB04OW</A></H2>
<H3>
Rank-one update of a Cholesky factorization (variant)
</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 perform the QR factorization
( U ) = Q*( R ), where U = ( U1 U2 ), R = ( R1 R2 ),
( x' ) ( 0 ) ( 0 T ) ( 0 R3 )
where U and R are (m+n)-by-(m+n) upper triangular matrices, x is
an m+n element vector, U1 is m-by-m, T is n-by-n, stored
separately, and Q is an (m+n+1)-by-(m+n+1) orthogonal matrix.
The matrix ( U1 U2 ) must be supplied in the m-by-(m+n) upper
trapezoidal part of the array A and this is overwritten by the
corresponding part ( R1 R2 ) of R. The remaining upper triangular
part of R, R3, is overwritten on the array T.
The transformations performed are also applied to the (m+n+1)-by-p
matrix ( B' C' d )' (' denotes transposition), where B, C, and d'
are m-by-p, n-by-p, and 1-by-p matrices, respectively.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB04OW( M, N, P, A, LDA, T, LDT, X, INCX, B, LDB,
$ C, LDC, D, INCD )
C .. Scalar Arguments ..
INTEGER INCD, INCX, LDA, LDB, LDC, LDT, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(*), T(LDT,*),
$ X(*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
</PRE>
<B>Input/Output Parameters</B>
<PRE>
M (input) INTEGER
The number of rows of the matrix ( U1 U2 ). M >= 0.
N (input) INTEGER
The order of the matrix T. N >= 0.
P (input) INTEGER
The number of columns of the matrices B and C. P >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading M-by-(M+N) upper trapezoidal part of
this array must contain the upper trapezoidal matrix
( U1 U2 ).
On exit, the leading M-by-(M+N) upper trapezoidal part of
this array contains the upper trapezoidal matrix ( R1 R2 ).
The strict lower triangle of A is not referenced.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
T (input/output) DOUBLE PRECISION array, dimension (LDT,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix T.
On exit, the leading N-by-N upper triangular part of this
array contains the upper triangular matrix R3.
The strict lower triangle of T is not referenced.
LDT INTEGER
The leading dimension of the array T. LDT >= max(1,N).
X (input/output) DOUBLE PRECISION array, dimension
(1+(M+N-1)*INCX), if M+N > 0, or dimension (0), if M+N = 0.
On entry, the incremented array X must contain the
vector x. On exit, the content of X is changed.
INCX (input) INTEGER
Specifies the increment for the elements of X. INCX > 0.
B (input/output) DOUBLE PRECISION array, dimension (LDB,P)
On entry, the leading M-by-P part of this array must
contain the matrix B.
On exit, the leading M-by-P part of this array contains
the transformed matrix B.
If M = 0 or P = 0, this array is not referenced.
LDB INTEGER
The leading dimension of the array B.
LDB >= max(1,M), if P > 0;
LDB >= 1, if P = 0.
C (input/output) DOUBLE PRECISION array, dimension (LDC,P)
On entry, the leading N-by-P part of this array must
contain the matrix C.
On exit, the leading N-by-P part of this array contains
the transformed matrix C.
If N = 0 or P = 0, this array is not referenced.
LDC INTEGER
The leading dimension of the array C.
LDC >= max(1,N), if P > 0;
LDC >= 1, if P = 0.
D (input/output) DOUBLE PRECISION array, dimension
(1+(P-1)*INCD), if P > 0, or dimension (0), if P = 0.
On entry, the incremented array D must contain the
vector d.
On exit, this incremented array contains the transformed
vector d.
If P = 0, this array is not referenced.
INCD (input) INTEGER
Specifies the increment for the elements of D. INCD > 0.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
Let q = m+n. The matrix Q is formed as a sequence of plane
rotations in planes (1, q+1), (2, q+1), ..., (q, q+1), the
rotation in the (j, q+1)th plane, Q(j), being chosen to
annihilate the jth element of x.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The algorithm requires 0((M+N)*(M+N+P)) operations and is backward
stable.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
For P = 0, this routine produces the same result as SLICOT Library
routine MB04OX, but matrix T may not be stored in the array A.
</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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