naginterfaces.library.lapacklin.dposv

naginterfaces.library.lapacklin.dposv(uplo, a, b)

dposv computes the solution to a real system of linear equations

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

where $A$ is an $n\times n$ symmetric positive definite matrix and $X$ and $B$ are $n\times r$ matrices.

For full information please refer to the NAG Library document for f07fa

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07faf.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.

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

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

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

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

Returns

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

If no exception or warning is raised, the factor $U$ or $L$ from the Cholesky factorization $A=U^{T}U$ or $A=L{L}^T$.

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 $\text{nrhs}$.

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

dposv 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 matrix and $L$ is a lower triangular matrix. 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