naginterfaces.library.lapacklin.dpotri

naginterfaces.library.lapacklin.dpotri(uplo, a)

dpotri computes the inverse of a real symmetric positive definite matrix $A$, where $A$ has been factorized by dpotrf().

For full information please refer to the NAG Library document for f07fj

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/f07/f07fjf.html

Parameters

uplostr, length 1

Specifies how $A$ has been factorized.

$uplo=\text{'U'}$

$A=U^{T}U$, where $U$ is upper triangular.

$uplo=\text{'L'}$

$A=L{L}^T$, where $L$ is lower triangular.

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

The upper triangular matrix $U$ if $uplo=\text{'U'}$ or the lower triangular matrix $L$ if $uplo=\text{'L'}$, as returned by dpotrf().

Returns

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

$U$ is overwritten by the upper triangle of $A^{-1}$ if $uplo=\text{'U'}$; $L$ is overwritten by the lower triangle of $A^{-1}$ if $uplo=\text{'L'}$.

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: $n\ge 0$.

Warns

NagAlgorithmicWarning

(errno $i>0$)

Diagonal element $⟨value⟩$ of the Cholesky factor is zero; the Cholesky factor is singular and the inverse of $A$ cannot be computed.

Notes

dpotri is used to compute the inverse of a real symmetric positive definite matrix $A$, the function must be preceded by a call to dpotrf(), which computes the Cholesky factorization of $A$.

If $uplo=\text{'U'}$, $A=U^{T}U$ and $A^{-1}$ is computed by first inverting $U$ and then forming $\left(U^{-1}\right)U^{-T}$.

If $uplo=\text{'L'}$, $A=L{L}^T$ and $A^{-1}$ is computed by first inverting $L$ and then forming $L^{-T}\left(L^{-1}\right)$.

References

Du Croz, J J and Higham, N J, 1992, Stability of methods for matrix inversion, IMA J. Numer. Anal. (12), 1–19