naginterfaces.library.lapacklin.zgbtrf

naginterfaces.library.lapacklin.zgbtrf(m, kl, ku, ab)

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

For full information please refer to the NAG Library document for f07br

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07brf.html

Parameters

mint

$m$, the number of rows of the matrix $A$.

klint

$k_l$, the number of subdiagonals within the band of the matrix $A$.

kuint

$k_u$, the number of superdiagonals within the band of the matrix $A$.

abcomplex, array-like, shape $(2\times kl+ku+1,n)$

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

See Further Comments for further details.

Returns

abcomplex, ndarray, shape $(2\times kl+ku+1,n)$

If $errno$ >= 0, $ab$ is overwritten by details of the factorization.

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 $m$.

Constraint: $m\ge 0$.

(errno $-2$)

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

Constraint: $n\ge 0$.

(errno $-3$)

On entry, error in parameter $kl$.

Constraint: $kl\ge 0$.

(errno $-4$)

On entry, error in parameter $ku$.

Constraint: $ku\ge 0$.

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

zgbtrf forms the $LU$ factorization of a complex $m\times n$ band matrix $A$ using partial pivoting, with row interchanges. Usually $m=n$, and then, if $A$ has $k_l$ nonzero subdiagonals and $k_u$ nonzero superdiagonals, the factorization has the form $A=PLU$, where $P$ is a permutation matrix, $L$ is a lower triangular matrix with unit diagonal elements and at most $k_l$ nonzero elements in each column, and $U$ is an upper triangular band matrix with $k_l+k_u$ superdiagonals.

Note that $L$ is not a band matrix, but the nonzero elements of $L$ can be stored in the same space as the subdiagonal elements of $A$. $U$ is a band matrix but with $k_l$ additional superdiagonals compared with $A$. These additional superdiagonals are created by the row interchanges.

References

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