Purpose
To calculate an LQ factorization of the first block row and apply
the orthogonal transformations (from the right) also to the second
block row of a structured matrix, as follows
_
[ L A ] [ L 0 ]
[ ]*Q = [ ]
[ 0 B ] [ C D ]
_
where L and L are lower triangular. The matrix A can be full or
lower 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 covariance filter).
Specification
SUBROUTINE MB04LD( UPLO, N, M, P, L, LDL, A, LDA, B, LDB, C, LDC,
$ TAU, DWORK )
C .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, LDB, LDC, LDL, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
$ L(LDL,*), TAU(*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Indicates if the matrix A is or not triangular as follows:
= 'L': Matrix A is lower trapezoidal/triangular;
= 'F': Matrix A is full.
Input/Output Parameters
N (input) INTEGER _
The order of the matrices L and L. N >= 0.
M (input) INTEGER
The number of columns of the matrices A, B and D. M >= 0.
P (input) INTEGER
The number of rows of the matrices B, C and D. P >= 0.
L (input/output) DOUBLE PRECISION array, dimension (LDL,N)
On entry, the leading N-by-N lower triangular part of this
array must contain the lower triangular matrix L.
On exit, the leading N-by-N lower triangular part of this
_
array contains the lower triangular matrix L.
The strict upper triangular part of this array is not
referenced.
LDL INTEGER
The leading dimension of array L. LDL >= MAX(1,N).
A (input/output) DOUBLE PRECISION array, dimension (LDA,M)
On entry, if UPLO = 'F', the leading N-by-M part of this
array must contain the matrix A. If UPLO = 'L', the
leading N-by-MIN(N,M) part of this array must contain the
lower trapezoidal (lower triangular if N <= M) matrix A,
and the elements above the diagonal are not referenced.
On exit, the leading N-by-M part (lower trapezoidal or
triangular, if UPLO = 'L') 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,N).
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,N)
The leading P-by-N part of this array contains the
computed matrix C.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
TAU (output) DOUBLE PRECISION array, dimension (N)
The scalar factors of the elementary reflectors used.
Workspace
DWORK DOUBLE PRECISION array, dimension (N)Method
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 an M-vector, if UPLO = 'F', or an min(i,M)-vector, if
i
UPLO = 'L'. The components of v are stored in the i-th row of A,
i
and tau is stored in TAU(i).
i
Numerical Aspects
The algorithm is backward stable.Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
None
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