naginterfaces.library.matop.complex_​gen_​rq_​formq

naginterfaces.library.matop.complex_gen_rq_formq(wheret, m, nrowp, a, theta)

complex_gen_rq_formq returns the first $\ell$ rows of the $n\times n$ unitary matrix $P^H$, where $P$ is given as the product of Householder transformation matrices.

This function is intended for use following complex_gen_rq().

For full information please refer to the NAG Library document for f01rk

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f01/f01rkf.html

Parameters

wheretstr, length 1

Indicates where the elements of $\theta$ are to be found.

$wheret=\text{'I'}$ (In $a$)

The elements of $\theta$ are in $a$.

$wheret=\text{'S'}$ (Separate)

The elements of $\theta$ are separate from $a$, in $theta$.

mint

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

nrowpint

$\ell$, the required number of rows of $P$.

If $nrowp=0$, an immediate return is effected.

acomplex, array-like, shape $(max(m,nrowp),n)$

The leading $m\times m$ strictly lower triangular part of the array $a$, and the $m\times (n-m)$ rectangular part of $a$ with top left-hand corner at element $a[0,m]$ must contain details of the matrix $P$. In addition, if $wheret=\text{'I'}$, the diagonal elements of $a$ must contain the elements of $\theta$.

thetacomplex, array-like, shape $(:)$

Note: the required length for this argument is determined as follows: if $wheret=\text{'S'}$: $m$; otherwise: $1$.

If $wheret=\text{'S'}$, the array $theta$ must contain the elements of $\theta$. If $theta[k-1]=0.0$, $P_k$ is assumed to be $I$, if $theta[k-1]=\alpha$ and $Re\left(\alpha \right)<0.0$, $P_k$ is assumed to be of the form

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

otherwise $theta[k-1]$ is assumed to contain ${\theta }_k$ given by

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

If $wheret=\text{'I'}$, the array $theta$ is not referenced.

Returns

acomplex, ndarray, shape $(max(m,nrowp),n)$

The first $nrowp$ rows of the array $a$ are overwritten by the first $nrowp$ rows of the $n\times n$ unitary matrix $P^H$.

Raises

NagValueError

(errno $-1$)

On entry, $\text{lda}=⟨value⟩$, $m=⟨value⟩$ and $nrowp=⟨value⟩$.

Constraint: $\text{lda}\ge max(m,nrowp)$.

(errno $-1$)

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

Constraint: $nrowp\ge 0$ and $nrowp\le n$.

(errno $-1$)

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

Constraint: $n\ge m$.

(errno $-1$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 0$.

(errno $-1$)

On entry, $wheret=⟨value⟩$.

Constraint: $wheret=\text{'I'}$ or $\text{'S'}$.

Notes

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

$P$ is assumed to be given by

$$P=P_m{P}_{m-1}\cdots P_1\text{,}$$

where

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

${\gamma }_k$ is a scalar for which $Re\left({\gamma }_k\right)=1.0$, ${\zeta }_k$ is a real scalar, $w_k$ is a $(k-1)$ element vector and $z_k$ is an $(n-m)$ element vector. $w_k$ must be supplied in the $k$th row of $a$ in elements $a[k-1,0],\dots ,a[k-1,k-2]$. $z_k$ must be supplied in the $k$th row of $a$ in elements $a[k-1,m],\dots ,a[k-1,n-1]$ and ${\theta }_k$, given by

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

must be supplied either in $a[k-1,k-1]$ or in $theta[k-1]$, depending upon the argument $wheret$.

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