naginterfaces.library.matop.complex_​trapez_​rq

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

complex_trapez_rq reduces the complex $m\times n$ ($m\le n$) upper trapezoidal matrix $A$ to upper triangular form by means of unitary transformations.

For full information please refer to the NAG Library document for f01rg

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/f01/f01rgf.html

Parameters

mint

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

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

acomplex, 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

acomplex, 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.

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

$theta[k-1]$ contains the scalar ${\theta }_k$ for the $(m-k+1)$th transformation. If $T_k=I$ then $theta[k-1]=0.0$; if

$$\begin{array}{c}{T}_k=\left(\begin{array}{cc}\alpha & 0\\ 0& I\end{array}\right)\text{,}Re\left(\alpha \right)<0.0\end{array}$$

then $theta[k-1]=\alpha$, otherwise $theta[k-1]$ contains ${\theta }_k$ as described in Notes and $Re\left({\theta }_k\right)$ 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 $m\times n(m\le n)$ 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

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

where $P$ is an $n\times n$ unitary 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

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

${\gamma }_k$ is a scalar for which $Re\left({\gamma }_k\right)=1.0$, ${\zeta }_k$ is a real scalar and $z_k$ is an $(n-m)$ element vector. ${\gamma }_k$, ${\zeta }_k$ and $z_k$ are chosen to annihilate the elements of the $k$th row of $X$ and to make the diagonal elements of $R$ real.

The scalar ${\gamma }_k$ and the vector $u_k$ are returned in the $k$th element of the array $theta$ and in the $k$th row of $a$, such that ${\theta }_k$, given by

$${\theta }_k=({\zeta }_k,Im\left({\gamma }_k\right))\text{,}$$

is in $theta[k-1]$ 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