naginterfaces.library.lapackeig.dggqrf

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

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

For full information please refer to the NAG Library document for f08ze

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f08/f08zef.html

Parameters

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

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

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

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

Returns

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

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

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

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

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

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

If $n>p$, the elements on and above the $(n-p)$th subdiagonal contain the $n\times p$ upper trapezoidal matrix $T$; the remaining elements, with the array $taub$, represent the orthogonal matrix $Z$ as a product of elementary reflectors (see the F08 Introduction).

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

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

Raises

NagValueError

(errno $-1$)

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

Constraint: $n\ge 0$.

(errno $-2$)

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

Constraint: $m\ge 0$.

(errno $-3$)

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

Constraint: $p\ge 0$.

Notes

dggqrf forms the generalized $QR$ factorization of an $n\times m$ matrix $A$ and an $n\times p$ matrix $B$

$$A=QR\text{,}B=QTZ\text{,}$$

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

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

with $R_{11}$ upper triangular,

$$\begin{array}{c}T=\{\begin{array}{c}\begin{array}{c}\\ n\end{array}\left(\begin{array}{cc}p-n& n\\ 0& T_{12}\end{array}\right)\text{, if}n\le p,\\ \begin{array}{c}\\ n-p\\ p\end{array}\left(\begin{array}{c}p\\ T_{11}\\ T_{21}\end{array}\right)\text{, if}n>p,\end{array}\end{array}$$

with $T_{12}$ or $T_{21}$ upper triangular.

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

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

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