naginterfaces.library.lapacklin.zhetrf

naginterfaces.library.lapacklin.zhetrf(uplo, n, a)

zhetrf computes the Bunch–Kaufman factorization of a complex Hermitian indefinite matrix.

For full information please refer to the NAG Library document for f07mr

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/f07/f07mrf.html

Parameters

uplostr, length 1

Specifies whether the upper or lower triangular part of $A$ is stored and how $A$ is to be factorized.

$uplo=\text{'U'}$

The upper triangular part of $A$ is stored and $A$ is factorized as $PUD{U}^H{P}^T$, where $U$ is upper triangular.

$uplo=\text{'L'}$

The lower triangular part of $A$ is stored and $A$ is factorized as $PLD{L}^H{P}^T$, where $L$ is lower triangular.

nint

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

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

The $n\times n$ Hermitian indefinite matrix $A$.

Returns

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

The upper or lower triangle of $A$ is overwritten by details of the block diagonal matrix $D$ and the multipliers used to obtain the factor $U$ or $L$ as specified by $uplo$.

ipivint, ndarray, shape $\left(n\right)$

Details of the interchanges and the block structure of $D$. More precisely,

if $ipiv[i-1]=k>0$, $d_{ii}$ is a $1\times 1$ pivot block and the $i$th row and column of $A$ were interchanged with the $k$th row and column;

if $uplo=\text{'U'}$ and $ipiv[i-2]=ipiv[i-1]=-l<0$, $\left(\begin{array}{cc}{d}_{i-1,i-1}& {\overline{d}}_{i,i-1}\\ {\overline{d}}_{i,i-1}& d_{ii}\end{array}\right)$ is a $2\times 2$ pivot block and the $(i-1)$th row and column of $A$ were interchanged with the $l$th row and column;

if $uplo=\text{'L'}$ and $ipiv[i-1]=ipiv\left[i\right]=-m<0$, $\left(\begin{array}{cc}{d}_{ii}& d_{i+1,i}\\ d_{i+1,i}& d_{i+1,i+1}\end{array}\right)$ is a $2\times 2$ pivot block and the $(i+1)$th row and column of $A$ were interchanged with the $m$th row and column.

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. The factorization has been completed, but the block diagonal matrix $D$ is exactly singular, and division by zero will occur if it is used to solve a system of equations.

Notes

zhetrf factorizes a complex Hermitian matrix $A$, using the Bunch–Kaufman diagonal pivoting method. $A$ is factorized either as $A=PUD{U}^H{P}^T$ if $uplo=\text{'U'}$ or $A=PLD{L}^H{P}^T$ if $uplo=\text{'L'}$, where $P$ is a permutation matrix, $U$ (or $L$) is a unit upper (or lower) triangular matrix and $D$ is an Hermitian block diagonal matrix with $1\times 1$ and $2\times 2$ diagonal blocks; $U$ (or $L$) has $2\times 2$ unit diagonal blocks corresponding to the $2\times 2$ blocks of $D$. Row and column interchanges are performed to ensure numerical stability while keeping the matrix Hermitian.

This method is suitable for Hermitian matrices which are not known to be positive definite. If $A$ is in fact positive definite, no interchanges are performed and no $2\times 2$ blocks occur in $D$.

References

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore