naginterfaces.library.lapackeig.dggrqf

naginterfaces.library.lapackeig.dggrqf(a, b)

dggrqf computes a generalized $RQ$ factorization of a real matrix pair $(A,B)$, where $A$ is an $m\times n$ matrix and $B$ is a $p\times n$ matrix.

For full information please refer to the NAG Library document for f08zf

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f08/f08zff.html

Parameters

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

The $m\times n$ matrix $A$.

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

The $p\times n$ matrix $B$.

Returns

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

If $m\le n$, the upper triangle of the subarray $a[0:m,n-m:n]$ contains the $m\times m$ upper triangular matrix $R_{12}$.

If $m\ge n$, the elements on and above the $(m-n)$th subdiagonal contain the $m\times n$ upper trapezoidal matrix $R$; the remaining elements, with the array $taua$, represent the orthogonal matrix $Q$ as a product of $min(m,n)$ elementary reflectors (see the F08 Introduction).

tauafloat, ndarray, shape $(min(m,n))$

The scalar factors of the elementary reflectors which represent the orthogonal matrix $Q$.

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

The elements on and above the diagonal of the array contain the $min(p,n)\times n$ upper trapezoidal matrix $T$ ($T$ is upper triangular if $p\ge n$); the elements below the diagonal, with the array $taub$, represent the orthogonal matrix $Z$ as a product of elementary reflectors (see the F08 Introduction).

taubfloat, ndarray, shape $(min(p,n))$

The scalar factors of the elementary reflectors which represent the orthogonal matrix $Z$.

Raises

NagValueError

(errno $-1$)

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

Constraint: $m\ge 0$.

(errno $-2$)

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

Constraint: $p\ge 0$.

(errno $-3$)

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

Constraint: $n\ge 0$.

Notes

dggrqf forms the generalized $RQ$ factorization of an $m\times n$ matrix $A$ and a $p\times n$ matrix $B$

$$A=RQ\text{,}B=ZTQ\text{,}$$

where $Q$ is an $n\times n$ orthogonal matrix, $Z$ is a $p\times p$ orthogonal matrix and $R$ and $T$ are of the form

$$\begin{array}{c}R=\{\begin{array}{c}\begin{array}{c}\\ m\end{array}\left(\begin{array}{cc}n-m& m\\ 0& R_{12}\end{array}\right)\text{; if}m\le n,\\ \begin{array}{c}\\ m-n\\ n\end{array}\left(\begin{array}{c}n\\ R_{11}\\ R_{21}\end{array}\right)\text{; if}m>n,\end{array}\end{array}$$

with $R_{12}$ or $R_{21}$ upper triangular,

$$\begin{array}{c}T=\{\begin{array}{c}\begin{array}{c}\\ n\\ p-n\end{array}\left(\begin{array}{c}n\\ T_{11}\\ 0\end{array}\right)\text{; if}p\ge n,\\ \begin{array}{c}\\ p\end{array}\left(\begin{array}{cc}p& n-p\\ T_{11}& T_{12}\end{array}\right)\text{; if}p<n,\end{array}\end{array}$$

with $T_{11}$ upper triangular.

In particular, if $B$ is square and nonsingular, the generalized $RQ$ factorization of $A$ and $B$ implicitly gives the $RQ$ factorization of $A{B}^{-1}$ as

$$A{B}^{-1}=\left(R{T}^{-1}\right)Z^T\text{.}$$

References

Anderson, E, Bai, Z, Bischof, C, Blackford, S, Demmel, J, Dongarra, J J, Du Croz, J J, Greenbaum, A, Hammarling, S, McKenney, A and Sorensen, D, 1999, LAPACK Users’ Guide, (3rd Edition), SIAM, Philadelphia, https://www.netlib.org/lapack/lug

Anderson, E, Bai, Z and Dongarra, J, 1992, Generalized $QR$ factorization and its applications, Linear Algebra Appl. (Volume 162–164), 243–271

Hammarling, S, 1987, The numerical solution of the general Gauss-Markov linear model, Mathematics in Signal Processing, (eds T S Durrani, J B Abbiss, J E Hudson, R N Madan, J G McWhirter and T A Moore), 441–456, Oxford University Press

Paige, C C, 1990, Some aspects of generalized $QR$ factorizations, . In Reliable Numerical Computation, (eds M G Cox and S Hammarling), 73–91, Oxford University Press