## MB04ND

### RQ factorization of a special structured block matrix

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

Purpose

```  To calculate an RQ 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
_
[ A   R ]        [ 0   R ]
[       ] * Q' = [ _   _ ]
[ C   B ]        [ C   B ]
_
where R and R are upper triangular. The matrix A can be full or
upper trapezoidal/triangular. The problem structure is exploited.

```
Specification
```      SUBROUTINE MB04ND( 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(*)

```
Arguments

Mode Parameters

```  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.

```
Input/Output Parameters
```  N       (input) INTEGER                 _
The order of the matrices R and R.  N >= 0.

M       (input) INTEGER
The number of rows of the matrices B and C.  M >= 0.

P       (input) INTEGER
The number of columns of the matrices A and C.  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,P)
On entry, if UPLO = 'F', the leading N-by-P part of this
array must contain the matrix A. For UPLO = 'U', if
N <= P, the upper triangle of the subarray A(1:N,P-N+1:P)
must contain the N-by-N upper triangular matrix A, and if
N >= P, the elements on and above the (N-P)-th subdiagonal
must contain the N-by-P upper trapezoidal matrix A.
On exit, if UPLO = 'F', the leading N-by-P part of this
array contains the trailing components (the vectors v, see
METHOD) of the elementary reflectors used in the
factorization. If UPLO = 'U', the upper triangle of the
subarray A(1:N,P-N+1:P) (if N <= P), or the elements on
and above the (N-P)-th subdiagonal (if N >= P), contain
the trailing components (the vectors v, see METHOD) of the
elementary reflectors used in the factorization.
The remaining elements are not referenced.

LDA     INTEGER
The leading dimension of array A.  LDA >= MAX(1,N).

B       (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading M-by-N part of this array must
contain the matrix B.
On exit, the leading M-by-N part of this array contains
_
the computed matrix B.

LDB     INTEGER
The leading dimension of array B.  LDB >= MAX(1,M).

C       (input/output) DOUBLE PRECISION array, dimension (LDC,P)
On entry, the leading M-by-P part of this array must
contain the matrix C.
On exit, the leading M-by-P part of this array contains
_
the computed matrix C.

LDC     INTEGER
The leading dimension of array C.  LDC >= MAX(1,M).

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

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (MAX(N-1,M))

```
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 a P-vector, if UPLO = 'F', or a min(N-i+1,P)-vector,
i
if UPLO = 'U'.  The components of v  are stored in the i-th row
i
of A, and tau  is stored in TAU(i), i = N,N-1,...,1.
i
In-line code for applying Householder transformations is used
whenever possible (see MB04NY routine).

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

```
```  None
```
Example

Program Text

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