naginterfaces.library.lapackeig.dtzrzf

naginterfaces.library.lapackeig.dtzrzf(a)

dtzrzf reduces the $m\times n$ ($m\le n$) real upper trapezoidal matrix $A$ to upper triangular form by means of orthogonal transformations.

For full information please refer to the NAG Library document for f08bh

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f08/f08bhf.html

Parameters

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

The leading $m\times n$ upper trapezoidal part of the array $a$ must contain the matrix to be factorized.

Returns

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

The leading $m\times m$ upper triangular part of $a$ contains the upper triangular matrix $R$, and elements $m+1$ to $\text{n}$ of the first $m$ rows of $a$, with the array $tau$, represent the orthogonal matrix $Z$ as a product of $m$ elementary reflectors (see the F08 Introduction).

taufloat, ndarray, shape $\left(m\right)$

The scalar factors of the elementary reflectors.

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: $n\ge m$.

Notes

The $m\times n$ ($m\le n$) real upper trapezoidal matrix $A$ given by

$$A=\left(\begin{array}{cc}{R}_1& R_2\end{array}\right)\text{,}$$

where $R_1$ is an $m\times m$ upper triangular matrix and $R_2$ is an $m\times (n-m)$ matrix, is factorized as

$$A=\left(\begin{array}{cc}R& 0\end{array}\right)Z\text{,}$$

where $R$ is also an $m\times m$ upper triangular matrix and $Z$ is an $n\times n$ orthogonal matrix.

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