naginterfaces.library.lapackeig.dgelqf

naginterfaces.library.lapackeig.dgelqf(a)

dgelqf computes the $LQ$ factorization of a real $m\times n$ matrix.

For full information please refer to the NAG Library document for f08ah

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f08/f08ahf.html

Parameters

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

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

Returns

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

If $m\le n$, the elements above the diagonal are overwritten by details of the orthogonal matrix $Q$ and the lower triangle is overwritten by the corresponding elements of the $m\times m$ lower triangular matrix $L$.

If $m>n$, the strictly upper triangular part is overwritten by details of the orthogonal matrix $Q$ and the remaining elements are overwritten by the corresponding elements of the $m\times n$ lower trapezoidal matrix $L$.

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

Further details of the orthogonal matrix $Q$.

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

dgelqf forms the $LQ$ factorization of an arbitrary rectangular real $m\times n$ matrix. No pivoting is performed.

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

$$A=\left(\begin{array}{cc}L& 0\end{array}\right)Q$$

where $L$ is an $m\times m$ lower triangular matrix and $Q$ is an $n\times n$ orthogonal matrix. It is sometimes more convenient to write the factorization as

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

which reduces to

$$A=L{Q}_1\text{,}$$

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

If $m>n$, $L$ is trapezoidal, and the factorization can be written

$$\begin{array}{c}A=\left(\begin{array}{c}{L}_1\\ L_2\end{array}\right)Q\end{array}$$

where $L_1$ is lower triangular and $L_2$ is rectangular.

The $LQ$ factorization of $A$ is essentially the same as the $QR$ factorization of $A^T$, since

$$\begin{array}{c}A=\left(\begin{array}{cc}L& 0\end{array}\right)Q\iff A^T=Q^T\left(\begin{array}{c}{L}^T\\ 0\end{array}\right)\text{.}\end{array}$$

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 $k<m$, the information returned in the first $k$ rows of the array $a$ represents an $LQ$ factorization of the first $k$ rows of the original matrix $A$.