# NAG FL Interfacef06qpf (dutr1)

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## 1Purpose

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.

## 2Specification

Fortran Interface
 Subroutine f06qpf ( n, 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)
#include <nag.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[])
The routine may be called by the names f06qpf or nagf_blas_dutr1.

## 3Description

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 $n×n$ real upper triangular matrices, $x$ and $y$ are $n$-element real vectors, $\alpha$ 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:
 $QT = Qn-1 ⋯ Q2 Q1 P1 P2 ⋯ Pn-1$
where
• ${P}_{k}$ is a rotation in the $\left(k,n\right)$ plane, chosen to annihilate ${x}_{k}$: thus $Px=\beta {e}_{n}$, where $P={P}_{1}{P}_{2}\cdots {P}_{n-1}$ and ${e}_{n}$ is the last column of the unit matrix;
• ${Q}_{k}$ is a rotation in the $\left(k,n\right)$ plane, chosen to annihilate the $\left(n,k\right)$ element of $\left(\alpha \beta {e}_{n}{y}^{\mathrm{T}}+PU\right)$, and thus restore it to upper triangular form.
The $2×2$ plane rotation part of ${P}_{k}$ or ${Q}_{k}$ has the form
 $( ck sk -sk ck ) .$
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.
None.

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrices $U$ and $R$.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{alpha}$Real (Kind=nag_wp) Input
On entry: the scalar $\alpha$.
3: $\mathbf{x}\left(*\right)$Real (Kind=nag_wp) array Input/Output
Note: the dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{n}}-1\right)×{\mathbf{incx}}\right)$.
On entry: the $n$-element vector $x$. ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left(1+\left(\mathit{i}-1\right)×{\mathbf{incx}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Intermediate elements of x are not referenced.
On exit: the referenced elements are overwritten by details of the sequence of plane rotations.
4: $\mathbf{incx}$Integer Input
On entry: the increment in the subscripts of x between successive elements of $x$.
Constraint: ${\mathbf{incx}}>0$.
5: $\mathbf{y}\left(*\right)$Real (Kind=nag_wp) array Input
Note: the dimension of the array y must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{n}}-1\right)×{\mathbf{incy}}\right)$.
On entry: the $n$-element vector $y$. ${y}_{\mathit{i}}$ must be stored in ${\mathbf{y}}\left(1+\left(\mathit{i}-1\right)×{\mathbf{incy}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Intermediate elements of y are not referenced.
6: $\mathbf{incy}$Integer Input
On entry: the increment in the subscripts of y between successive elements of $y$.
Constraint: ${\mathbf{incy}}>0$.
7: $\mathbf{a}\left({\mathbf{lda}},*\right)$Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least ${\mathbf{n}}$.
On entry: the $n×n$ upper triangular matrix $U$.
On exit: the upper triangular matrix $R$.
8: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f06qpf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9: $\mathbf{c}\left({\mathbf{n}}-1\right)$Real (Kind=nag_wp) array Output
On exit: the cosines of the rotations ${Q}_{\mathit{k}}$, for $\mathit{k}=1,2,\dots ,n-1$.
10: $\mathbf{s}\left({\mathbf{n}}-1\right)$Real (Kind=nag_wp) array Output
On exit: the sines of the rotations ${Q}_{\mathit{k}}$, for $\mathit{k}=1,2,\dots ,n-1$.

None.

Not applicable.