## MB04ID

### QR factorization of a matrix with a lower left-hand side zero triangle

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

```  To compute a QR factorization of an n-by-m matrix A (A = Q * R),
having a p-by-min(p,m) zero triangle in the lower left-hand side
corner, as shown below, for n = 8, m = 7, and p = 2:

[ 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 ]
A = [ x x x x x x x ],
[ x x x x x x x ]
[ 0 x x x x x x ]
[ 0 0 x x x x x ]

and optionally apply the transformations to an n-by-l matrix B
(from the left). 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 information filter).

```
Specification
```      SUBROUTINE MB04ID( 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(*)

```
Arguments

Input/Output Parameters

```  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 columns 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 P-by-MIN(P,M) lower trapezoidal/triangular part
(if P > 0) are not referenced.
On exit, the elements on and above the diagonal of this
array contain the MIN(N,M)-by-M upper trapezoidal matrix
R (R is upper triangular, if N >= M) of the QR
factorization, and the relevant elements below 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,L)
On entry, the leading N-by-L part of this array must
contain the matrix B.
On exit, the leading N-by-L 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,N) if L > 0;
LDB >= 1        if L = 0.

TAU     (output) DOUBLE PRECISION array, dimension MIN(N,M)
The scalar factors of the elementary reflectors used.

```
Workspace
```  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,M-1,M-P,L).
For optimum performance LDWORK should be larger.

If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value.

```
Method
```  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 (N-P+I-2)-vector.  The components of v  are stored
i                                             i
in the i-th column of A, beginning from the location i+1, and
tau  is stored in TAU(i).
i

```
Numerical Aspects
```  The algorithm is backward stable.

```
```  None
```
Example

Program Text

```  None
```
Program Data
```  None
```
Program Results
```  None
```