naginterfaces.library.lapacklin.zhetri

naginterfaces.library.lapacklin.zhetri(uplo, n, a, ipiv)

zhetri computes the inverse of a complex Hermitian indefinite matrix $A$, where $A$ has been factorized by zhetrf().

For full information please refer to the NAG Library document for f07mw

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f07/f07mwf.html

Parameters

uplostr, length 1

Specifies how $A$ has been factorized.

$uplo=\text{'U'}$

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

$uplo=\text{'L'}$

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

nint

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

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

Details of the factorization of $A$, as returned by zhetrf().

ipivint, array-like, shape $\left(n\right)$

Details of the interchanges and the block structure of $D$, as returned by zhetrf().

Returns

acomplex, ndarray, shape $(n,n)$

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

If $uplo=\text{'U'}$, the upper triangle of $A^{-1}$ is stored in the upper triangular part of the array.

If $uplo=\text{'L'}$, the lower triangle of $A^{-1}$ is stored in the lower triangular part of the array.

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 $i>0$)

Element $⟨value⟩$ of the diagonal is exactly zero. $D$ is singular and the inverse of $A$ cannot be computed.

Notes

zhetri is used to compute the inverse of a complex Hermitian indefinite matrix $A$, the function must be preceded by a call to zhetrf(), which computes the Bunch–Kaufman factorization of $A$.

If $uplo=\text{'U'}$, $A=PUD{U}^H{P}^T$ and $A^{-1}$ is computed by solving $U^H{P}^{T}XPU=D^{-1}$ for $X$.

If $uplo=\text{'L'}$, $A=PLD{L}^H{P}^T$ and $A^{-1}$ is computed by solving $L^H{P}^{T}XPL=D^{-1}$ for $X$.

References

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