naginterfaces.library.lapacklin.dpbsv

naginterfaces.library.lapacklin.dpbsv(uplo, kd, ab, b)

dpbsv computes the solution to a real system of linear equations

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

where $A$ is an $n\times n$ symmetric positive definite band matrix of bandwidth $(2k_d+1)$ and $X$ and $B$ are $n\times r$ matrices.

For full information please refer to the NAG Library document for f07ha

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07haf.html

Parameters

uplostr, length 1

If $uplo=\text{'U'}$, the upper triangle of $A$ is stored.

If $uplo=\text{'L'}$, the lower triangle of $A$ is stored.

kdint

$k_d$, the number of superdiagonals of the matrix $A$ if $uplo=\text{'U'}$, or the number of subdiagonals if $uplo=\text{'L'}$.

abfloat, array-like, shape $(kd+1,n)$

The upper or lower triangle of the symmetric band matrix $A$.

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

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

Returns

abfloat, ndarray, shape $(kd+1,n)$

If no exception or warning is raised, the triangular factor $U$ or $L$ from the Cholesky factorization $A=U^{T}U$ or $A=L{L}^T$ of the band matrix $A$, in the same storage format as $A$.

bfloat, 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 $uplo$.

Constraint: $uplo=\text{'U'}$ or $\text{'L'}$.

(errno $-2$)

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

Constraint: $n\ge 0$.

(errno $-3$)

On entry, error in parameter $kd$.

Constraint: $kd\ge 0$.

(errno $-4$)

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

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

(errno $i>0$)

The leading minor of order $⟨value⟩$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has not been computed.

Notes

dpbsv uses the Cholesky decomposition to factor $A$ as $A=U^{T}U$ if $uplo=\text{'U'}$ or $A=L{L}^T$ if $uplo=\text{'L'}$, where $U$ is an upper triangular band matrix, and $L$ is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as $A$. 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