MB04OW

Rank-one update of a Cholesky factorization (variant)

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

Purpose

```  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.

```
Specification
```      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(*)

```
Arguments

Input/Output Parameters

```  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.

```
Method
```  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.

```
Numerical Aspects
```  The algorithm requires 0((M+N)*(M+N+P)) operations and is backward
stable.

```
```  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.

```
Example

Program Text

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