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<HTML>
<HEAD><TITLE>MB04KD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB04KD">MB04KD</A></H2>
<H3>
QR factorization of a special structured block matrix
</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 calculate a QR factorization of the first block column and
apply the orthogonal transformations (from the left) also to the
second block column of a structured matrix, as follows
_
[ R 0 ] [ R C ]
Q' * [ ] = [ ]
[ A B ] [ 0 D ]
_
where R and R are upper triangular. The matrix A can be full or
upper trapezoidal/triangular. The problem structure is exploited.
This computation is useful, for instance, in combined measurement
and time update of one iteration of the Kalman filter (square
root information filter).
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB04KD( UPLO, N, M, P, R, LDR, A, LDA, B, LDB, C, LDC,
$ TAU, DWORK )
C .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, LDB, LDC, LDR, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
$ R(LDR,*), TAU(*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
UPLO CHARACTER*1
Indicates if the matrix A is or not triangular as follows:
= 'U': Matrix A is upper trapezoidal/triangular;
= 'F': Matrix A is full.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER _
The order of the matrices R and R. N >= 0.
M (input) INTEGER
The number of columns of the matrices B, C and D. M >= 0.
P (input) INTEGER
The number of rows of the matrices A, B and D. P >= 0.
R (input/output) DOUBLE PRECISION array, dimension (LDR,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix R.
On exit, the leading N-by-N upper triangular part of this
_
array contains the upper triangular matrix R.
The strict lower triangular part of this array is not
referenced.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, if UPLO = 'F', the leading P-by-N part of this
array must contain the matrix A. If UPLO = 'U', the
leading MIN(P,N)-by-N part of this array must contain the
upper trapezoidal (upper triangular if P >= N) matrix A,
and the elements below the diagonal are not referenced.
On exit, the leading P-by-N part (upper trapezoidal or
triangular, if UPLO = 'U') of this array contains 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,P).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading P-by-M part of this array must
contain the matrix B.
On exit, the leading P-by-M part of this array contains
the computed matrix D.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,P).
C (output) DOUBLE PRECISION array, dimension (LDC,M)
The leading N-by-M part of this array contains the
computed matrix C.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,N).
TAU (output) DOUBLE PRECISION array, dimension (N)
The scalar factors of the elementary reflectors used.
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension (N)
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The routine uses N Householder transformations exploiting the zero
pattern of the block matrix. A Householder matrix has the form
( 1 ),
H = I - tau *u *u', u = ( v )
i i i i i ( i)
where v is a P-vector, if UPLO = 'F', or an min(i,P)-vector, if
i
UPLO = 'U'. The components of v are stored in the i-th column
i
of A, and tau is stored in TAU(i).
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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