naginterfaces.library.lapacklin.zgetrf

naginterfaces.library.lapacklin.zgetrf(a)

zgetrf computes the $LU$ factorization of a complex $m\times n$ matrix.

For full information please refer to the NAG Library document for f07ar

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07arf.html

Parameters

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

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

Returns

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

The factors $L$ and $U$ from the factorization $A=PLU$; the unit diagonal elements of $L$ are not stored.

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

The pivot indices that define the permutation matrix. At the $\text{i}$th step, if $ipiv[\text{i}-1]>\text{i}$ then row $\text{i}$ of the matrix $A$ was interchanged with row $ipiv[\text{i}-1]$, for $\text{i}=1,2,\dots ,min(m,n)$. $ipiv[i-1]\le i$ indicates that, at the $i$th step, a row interchange was not required.

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: $n\ge 0$.

Warns

NagAlgorithmicWarning

(errno $i>0$)

Element $⟨value⟩$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, and division by zero will occur if it is used to solve a system of equations.

Notes

zgetrf forms the $LU$ factorization of a complex $m\times n$ matrix $A$ as $A=PLU$, where $P$ is a permutation matrix, $L$ is lower triangular with unit diagonal elements (lower trapezoidal if $m>n$) and $U$ is upper triangular (upper trapezoidal if $m<n$). Usually $A$ is square $(m=n)$, and both $L$ and $U$ are triangular. The function uses partial pivoting, with row interchanges.

References

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