naginterfaces.library.lapackeig.dtpqrt

naginterfaces.library.lapackeig.dtpqrt(l, nb, a, b)

dtpqrt computes the $QR$ factorization of a real $(m+n)\times n$ triangular-pentagonal matrix.

For full information please refer to the NAG Library document for f08bb

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f08/f08bbf.html

Parameters

lint

$l$, the number of rows of the trapezoidal part of $B$ (i.e., $B_2$).

nbint

The explicitly chosen block-size to be used in the algorithm for computing the $QR$ factorization. See Further Comments for details.

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

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

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

The $m\times n$ pentagonal matrix $B$ composed of an $(m-l)\times n$ rectangular matrix $B_1$ above an $l\times n$ upper trapezoidal matrix $B_2$.

Returns

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

The upper triangle is overwritten by the corresponding elements of the $n\times n$ upper triangular matrix $R$.

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

Details of the orthogonal matrix $Q$.

tfloat, ndarray, shape $(nb,n)$

Further details of the orthogonal matrix $Q$. The number of blocks is $b=\lceil \frac{k}{nb}\rceil$, where $k=min(m,n)$ and each block is of order $nb$ except for the last block, which is of order $k-(b-1)\times nb$. For each of the blocks, an upper triangular block reflector factor is computed: $T_1,T_2,\dots ,T_b$. These are stored in the $nb\times n$ matrix $T$ as $T=[T_1|T_2|\cdots |T_b]$.

Raises

NagValueError

(errno $-1$)

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

Constraint: $m\ge 0$.

(errno $-2$)

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

Constraint: $n\ge 0$.

(errno $-3$)

On entry, error in parameter $l$.

Constraint: $0\le l\le min(m,n)$.

(errno $-4$)

On entry, error in parameter $nb$.

Constraint: $nb\ge 1$.

(errno $-4$)

On entry, error in parameter $nb$.

Constraint: $nb\le n$.

Notes

dtpqrt forms the $QR$ factorization of a real $(m+n)\times n$ triangular-pentagonal matrix $C$,

$$\begin{array}{c}C=\left(\begin{array}{c}A\\ B\end{array}\right)\end{array}$$

where $A$ is an upper triangular $n\times n$ matrix and $B$ is an $m\times n$ pentagonal matrix consisting of an $(m-l)\times n$ rectangular matrix $B_1$ on top of an $l\times n$ upper trapezoidal matrix $B_2$:

$$\begin{array}{c}B=\left(\begin{array}{c}{B}_1\\ B_2\end{array}\right)\text{.}\end{array}$$

The upper trapezoidal matrix $B_2$ consists of the first $l$ rows of an $n\times n$ upper triangular matrix, where $0\le l\le min(m,n)$. If $l=0$, $B$ is $m\times n$ rectangular; if $l=n$ and $m=n$, $B$ is upper triangular.

A recursive, explicitly blocked, $QR$ factorization (see dgeqrt()) is performed on the matrix $C$. The upper triangular matrix $R$, details of the orthogonal matrix $Q$, and further details (the block reflector factors) of $Q$ are returned.

Typically the matrix $A$ or $B_2$ contains the matrix $R$ from the $QR$ factorization of a subproblem and dtpqrt performs the $QR$ update operation from the inclusion of matrix $B_1$.

For example, consider the $QR$ factorization of an $l\times n$ matrix $\hat{B}$ with $l<n$: $\hat{B}=\hat{Q}\hat{R}$, $\hat{R}=\left(\begin{array}{cc}\widehat{R_1}& \widehat{R_2}\end{array}\right)$, where $\widehat{R_1}$ is $l\times l$ upper triangular and $\widehat{R_2}$ is $(n-l)\times n$ rectangular (this can be performed by dgeqrt()). Given an initial least squares problem $\hat{B}\hat{X}=\hat{Y}$ where $X$ and $Y$ are $l\times \text{nrhs}$ matrices, we have $\hat{R}\hat{X}={\hat{Q}}^T\hat{Y}$.

Now, adding an additional $m-l$ rows to the original system gives the augmented least squares problem

$$BX=Y$$

where $B$ is an $m\times n$ matrix formed by adding $m-l$ rows on top of $\hat{R}$ and $Y$ is an $m\times \text{nrhs}$ matrix formed by adding $m-l$ rows on top of ${\hat{Q}}^T\hat{Y}$.

dtpqrt can then be used to perform the $QR$ factorization of the pentagonal matrix $B$; the $n\times n$ matrix $A$ will be zero on input and contain $R$ on output.

In the case where $\hat{B}$ is $r\times n$, $r\ge n$, $\hat{R}$ is $n\times n$ upper triangular (forming $A$) on top of $r-n$ rows of zeros (forming first $r-n$ rows of $B$). Augmentation is then performed by adding rows to the bottom of $B$ with $l=0$.

References

Elmroth, E and Gustavson, F, 2000, Applying Recursion to Serial and Parallel $QR$ Factorization Leads to Better Performance, IBM Journal of Research and Development. (Volume 44) (4), 605–624

Golub, G H and Van Loan, C F, 2012, Matrix Computations, (4th Edition), Johns Hopkins University Press, Baltimore