NAG Library Routine Document

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

Q R

factorization of an upper triangular matrix which has been modified by a rank-1 update:

α x y^{T} + U = Q R

where

U

and

R

are

n

n

real upper triangular matrices,

x

and

y

are

n

-element real vectors,

α

is a real scalar, and

Q

is an

n

n

real orthogonal matrix.

Q

is formed as the product of two sequences of plane rotations:

Q^{T} = Q_{n - 1} \dots Q_{2} Q_{1} P_{1} P_{2} \dots P_{n - 1}

where

$P_{k}$ is a rotation in the $(k, n)$ plane, chosen to annihilate $x_{k}$ : thus $P x = β e_{n}$ , where $P = P_{1} P_{2} \dots P_{n - 1}$ and $e_{n}$ is the last column of the unit matrix;
$Q_{k}$ is a rotation in the $(k, n)$ plane, chosen to annihilate the $(n, k)$ element of $(α β e_{n} y^{T} + P U)$ , and thus restore it to upper triangular form.

The

2

2

plane rotation part of

P_{k}

Q_{k}

has the form

(\begin{array}{r} c_{k} & s_{k} \\ - s_{k} & c_{k} \end{array}) .

The tangents of the rotations

P_{k}

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

Q_{k}

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: $n \geq 0$ .
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 $\max (1, 1 + (n - 1) \times incx)$ .

On entry: the $n$ -element vector $x$ . $x_{i}$ must be stored in $x (1 + (i - 1) \times incx)$ , for $i = 1, 2, \dots, n$ .
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 $\max (1, 1 + (n - 1) \times incy)$ .

On entry: the $n$ -element vector $y$ . $y_{i}$ must be stored in $y (1 + (i - 1) \times incy)$ , for $i = 1, 2, \dots, n$ .
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: $a (lda *)$ – 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: $lda \geq \max (1, n)$ .
9: $c (n - 1)$ – Real (Kind=nag_wp) arrayOutput: On exit: the cosines of the rotations $Q_{k}$ , for $k = 1, 2, \dots, n - 1$ .
10: $s (n - 1)$ – Real (Kind=nag_wp) arrayOutput: On exit: the sines of the rotations $Q_{k}$ , for $k = 1, 2, \dots, n - 1$ .

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.