NAG Library Routine Document

f06qpf (dutr1)

1
Purpose

f06qpf performs a QR factorization (as a sequence of plane rotations) of a real upper triangular matrix that has been modified by a rank-1 update.

2
Specification

Fortran Interface
Subroutine f06qpf ( n, alpha, x, incx, y, incy, a, lda, c, s)
Integer, Intent (In):: n, incx, incy, lda
Real (Kind=nag_wp), Intent (In):: alpha, y(*)
Real (Kind=nag_wp), Intent (Inout):: x(*), a(lda,*)
Real (Kind=nag_wp), Intent (Out):: c(n-1), s(n-1)
C Header Interface
#include <nagmk26.h>
void  f06qpf_ (const Integer *n, const double *alpha, double x[], const Integer *incx, const double y[], const Integer *incy, double a[], const Integer *lda, double c[], double s[])

3
Description

f06qpf performs a QR factorization of an upper triangular matrix which has been modified by a rank-1 update:
αxyT + U=QR  
where U and R are n by n real upper triangular matrices, x and y are n-element real vectors, α is a real scalar, and Q is an n by n real orthogonal matrix.
Q is formed as the product of two sequences of plane rotations:
QT = Qn-1 Q2 Q1 P1 P2 Pn-1  
where
The 2 by 2 plane rotation part of Pk or Qk has the form
ck sk -sk ck .  
The tangents of the rotations Pk are returned in the array x; the cosines and sines of these rotations can be recovered by calling f06bcf. The cosines and sines of the rotations Qk are returned directly in the arrays c and s.

4
References

None.

5
Arguments

1:     n – IntegerInput
On entry: n, the order of the matrices U and R.
Constraint: n0.
2:     alpha – Real (Kind=nag_wp)Input
On entry: the scalar α.
3:     x* – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array x must be at least max1, 1+n-1 ×incx .
On entry: the n-element vector x. xi must be stored in x1+i-1×incx, for i=1,2,,n.
Intermediate elements of x are not referenced.
On exit: the referenced elements are overwritten by details of the sequence of plane rotations.
4:     incx – IntegerInput
On entry: the increment in the subscripts of x between successive elements of x.
Constraint: incx>0.
5:     y* – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array y must be at least max1, 1+n-1 ×incy .
On entry: the n-element vector y. yi must be stored in y1+i-1×incy, for i=1,2,,n.
Intermediate elements of y are not referenced.
6:     incy – IntegerInput
On entry: the increment in the subscripts of y between successive elements of y.
Constraint: incy>0.
7:     alda* – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a must be at least n.
On entry: the n by n upper triangular matrix U.
On exit: the upper triangular matrix R.
8:     lda – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f06qpf is called.
Constraint: lda max1,n .
9:     cn-1 – Real (Kind=nag_wp) arrayOutput
On exit: the cosines of the rotations Qk, for k=1,2,,n-1.
10:   sn-1 – Real (Kind=nag_wp) arrayOutput
On exit: the sines of the rotations Qk, for k=1,2,,n-1.

6
Error Indicators and Warnings

None.

7
Accuracy

Not applicable.

8
Parallelism and Performance

f06qpf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9
Further Comments

None.

10
Example

None.