naginterfaces.library.lapackeig.dgeqlf

naginterfaces.library.lapackeig.dgeqlf(a)

dgeqlf computes a $QL$ factorization of a real $m\times n$ matrix $A$.

For full information please refer to the NAG Library document for f08ce

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f08/f08cef.html

Parameters

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

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

Returns

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

If $m\ge n$, the lower triangle of the subarray $a[m-n:m,0:n]$ contains the $n\times n$ lower triangular matrix $L$.

If $m\le n$, the elements on and below the $(n-m)$th superdiagonal contain the $m\times n$ lower trapezoidal matrix $L$.

The remaining elements, with the array $tau$, represent the orthogonal matrix $Q$ as a product of elementary reflectors (see the F08 Introduction).

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

The scalar factors of the elementary reflectors (see Further Comments).

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: $n\ge 0$.

Notes

dgeqlf forms the $QL$ factorization of an arbitrary rectangular real $m\times n$ matrix.

If $m\ge n$, the factorization is given by:

$$\begin{array}{c}A=Q\left(\begin{array}{c}0\\ L\end{array}\right)\text{,}\end{array}$$

where $L$ is an $n\times n$ lower triangular matrix and $Q$ is an $m\times m$ orthogonal matrix. If $m<n$ the factorization is given by

$$A=QL\text{,}$$

where $L$ is an $m\times n$ lower trapezoidal matrix and $Q$ is again an $m\times m$ orthogonal matrix. In the case where $m>n$ the factorization can be expressed as

$$\begin{array}{c}A=\left(\begin{array}{cc}{Q}_1& Q_2\end{array}\right)\left(\begin{array}{c}0\\ L\end{array}\right)=Q_{2}L\text{,}\end{array}$$

where $Q_1$ consists of the first $m-n$ columns of $Q$, and $Q_2$ the remaining $n$ columns.

The matrix $Q$ is not formed explicitly but is represented as a product of $min(m,n)$ elementary reflectors (see the F08 Introduction for details). Functions are provided to work with $Q$ in this representation (see Further Comments).

Note also that for any $k<n$, the information returned in the last $k$ columns of the array $a$ represents a $QL$ factorization of the last $\text{k}$ columns of the original matrix $A$.

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

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore