naginterfaces.library.lapacklin.zgbsv

naginterfaces.library.lapacklin.zgbsv(kl, ku, ab, b)

zgbsv computes the solution to a complex system of linear equations

$$AX=B\text{,}$$

where $A$ is an $n\times n$ band matrix, with $k_l$ subdiagonals and $k_u$ superdiagonals, and $X$ and $B$ are $n\times r$ matrices.

For full information please refer to the NAG Library document for f07bn

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07bnf.html

Parameters

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 $n\times n$ coefficient matrix $A$.

See Further Comments for further details.

bcomplex, array-like, shape $(n,\text{nrhs})$

The $n\times r$ right-hand side matrix $B$.

Returns

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

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

ipivint, ndarray, shape $\left(n\right)$

If no constraints are violated, the pivot indices that define the permutation matrix $P$; at the $i$th step row $i$ of the matrix was interchanged with row $ipiv[i-1]$. $ipiv[i-1]=i$ indicates a row interchange was not required.

bcomplex, ndarray, shape $(n,\text{nrhs})$

If no exception or warning is raised, the $n\times r$ solution matrix $X$.

Raises

NagValueError

(errno $-1$)

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

Constraint: $n\ge 0$.

(errno $-2$)

On entry, error in parameter $kl$.

Constraint: $kl\ge 0$.

(errno $-3$)

On entry, error in parameter $ku$.

Constraint: $ku\ge 0$.

(errno $-4$)

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

Constraint: $\text{nrhs}\ge 0$.

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, so the solution could not be computed.

Notes

zgbsv uses the $LU$ decomposition with partial pivoting and row interchanges to factor $A$ as $A=PLU$, where $P$ is a permutation matrix, $L$ is a product of permutation and unit lower triangular matrices with $k_l$ subdiagonals, and $U$ is upper triangular with $(k_l+k_u)$ superdiagonals. The factored form of $A$ is then used to solve the system of equations $AX=B$.

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