naginterfaces.library.lapacklin.dpptri

naginterfaces.library.lapacklin.dpptri(uplo, n, ap)

dpptri computes the inverse of a real symmetric positive definite matrix $A$, where $A$ has been factorized by dpptrf(), using packed storage.

For full information please refer to the NAG Library document for f07gj

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f07/f07gjf.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.

nint

$n$, the order of the matrix $A$.

apfloat, array-like, shape $(n\times (n+1)/2)$

The Cholesky factor of $A$ stored in packed form, as returned by dpptrf().

Returns

apfloat, ndarray, shape $(n\times (n+1)/2)$

The factorization is overwritten by the $n\times n$ matrix $A^{-1}$.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $uplo$.

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

(errno $-2$)

On entry, error in parameter $n$.

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

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

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