naginterfaces.library.blas.dutr1

naginterfaces.library.blas.dutr1(alpha, x, y, a)

dutr1 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.

For full information please refer to the NAG Library document for f06qp

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f06/f06qpf.html

Parameters

alphafloat

The scalar $\alpha$.

xfloat, array-like, shape $\left(n\right)$

The $n$-element vector $x$.

yfloat, array-like, shape $\left(n\right)$

The $n$-element vector $y$.

afloat, array-like, shape $(n,n)$

The $n\times n$ upper triangular matrix $U$.

Returns

xfloat, ndarray, shape $\left(n\right)$

The referenced elements are overwritten by details of the sequence of plane rotations.

afloat, ndarray, shape $(n,n)$

The upper triangular matrix $R$.

cfloat, ndarray, shape $(n-1)$

The cosines of the rotations $Q_{\text{k}}$, for $\text{k}=1,2,\dots ,n-1$.

sfloat, ndarray, shape $(n-1)$

The sines of the rotations $Q_{\text{k}}$, for $\text{k}=1,2,\dots ,n-1$.

Raises

NagValueError

(errno $1$)

On entry, error in parameter $\text{n}$.

Constraint: $n\ge 0$.

(errno $4$)

On entry, error in parameter $\text{incx}$.

Constraint: $\text{incx}>0$.

(errno $6$)

On entry, error in parameter $\text{incy}$.

Constraint: $\text{incy}>0$.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

dutr1 performs a $QR$ factorization of an upper triangular matrix which has been modified by a rank-1 update:

$$\alpha x{y}^T+U=QR$$

where $U$ and $R$ are $n\times n$ real upper triangular matrices, $x$ and $y$ are $n$-element real vectors, $\alpha$ is a real scalar, and $Q$ is an $n\times n$ real orthogonal matrix.

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

$$Q^T=Q_{n-1}\cdots Q_2{Q}_1{P}_1{P}_2\cdots P_{n-1}$$

where

$P_k$ is a rotation in the $(k,n)$ 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 $(k,n)$ plane, chosen to annihilate the $(n,k)$ element of $(\alpha \beta e_n{y}^T+PU)$, and thus restore it to upper triangular form.

The $2\times 2$ plane rotation part of $P_k$ or $Q_k$ has the form

$$\begin{array}{c}\left(\begin{array}{cc}{c}_k& s_k\\ -s_k& c_k\end{array}\right)\text{.}\end{array}$$

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 dcsg(). The cosines and sines of the rotations $Q_k$ are returned directly in the arrays $c$ and $s$.