naginterfaces.library.matop.real_​trapez_​rq

naginterfaces.library.matop.real_trapez_rq(m, a)

real_trapez_rq reduces the $m\times n$ ($m\le n$) real upper trapezoidal matrix $A$ to upper triangular form by means of orthogonal transformations.

For full information please refer to the NAG Library document for f01qg

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f01/f01qgf.html

Parameters

mint

$m$, the number of rows of the matrix $A$.

When $m=0$ then an immediate return is effected.

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

The leading $m\times n$ upper trapezoidal part of the array $a$ must contain the matrix to be factorized.

Returns

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

The $m\times m$ upper triangular part of $a$ will contain the upper triangular matrix $R$, and the $m\times (n-m)$ upper trapezoidal part of $a$ will contain details of the factorization as described in Notes.

zetafloat, ndarray, shape $\left(m\right)$

$zeta[k-1]$ contains the scalar ${\zeta }_k$ for the $(m-k+1)$th transformation. If $T_k=I$ then $zeta[k-1]=0.0$, otherwise $zeta[k-1]$ contains ${\zeta }_k$ as described in Notes and ${\zeta }_k$ is always in the range $(1.0,\sqrt{2.0})$.

Raises

NagValueError

(errno $-1$)

On entry, $n=⟨value⟩$ and $m=⟨value⟩$.

Constraint: $n\ge m$.

(errno $-1$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 0$.

Notes

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

The $m\times n$ ($m\le n$) real upper trapezoidal matrix $A$ given by

$$A=\left(\begin{array}{cc}U& X\end{array}\right)\text{,}$$

where $U$ is an $m\times m$ upper triangular matrix, is factorized as

$$A=\left(\begin{array}{cc}R& 0\end{array}\right)P^T\text{,}$$

where $P$ is an $n\times n$ orthogonal matrix and $R$ is an $m\times m$ upper triangular matrix.

$P$ is given as a sequence of Householder transformation matrices

$$P=P_m\cdots P_2{P}_1\text{,}$$

the $(m-k+1)$th transformation matrix, $P_k$, being used to introduce zeros into the $k$th row of $A$. $P_k$ has the form

$$\begin{array}{c}{P}_k=\left(\begin{array}{cc}I& 0\\ 0& T_k\end{array}\right)\text{,}\end{array}$$

where

$$\begin{array}{c}\begin{array}{c}{T}_k=I-u_k{u}_k^T\text{,}\\ \\ u_k=\left(\begin{array}{c}{\zeta }_k\\ 0\\ z_k\end{array}\right)\text{,}\end{array}\end{array}$$

${\zeta }_k$ is a scalar and $z_k$ is an ($n-m$) element vector. ${\zeta }_k$ and $z_k$ are chosen to annihilate the elements of the $k$th row of $X$.

The vector $u_k$ is returned in the $k$th element of the array $zeta$ and in the $k$th row of $a$, such that ${\zeta }_k$ is in $zeta[k-1]$ and the elements of $z_k$ are in $a[k-1,m],\dots ,a[k-1,n-1]$. The elements of $R$ are returned in the upper triangular part of $a$.

For further information on this factorization and its use see Section 6.5 of Golub and Van Loan (1996).

References

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore

Wilkinson, J H, 1965, The Algebraic Eigenvalue Problem, Oxford University Press, Oxford