naginterfaces.library.lapacklin.dsptrf

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

dsptrf computes the Bunch–Kaufman factorization of a real symmetric indefinite matrix, using packed storage.

For full information please refer to the NAG Library document for f07pd

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07pdf.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 $PUD{U}^T{P}^T$, where $U$ is upper triangular.

$uplo=\text{'L'}$

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

nint

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

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

The $n\times n$ symmetric matrix $A$, packed by columns.

Returns

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

$A$ 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

dsptrf factorizes a real symmetric matrix $A$, using the Bunch–Kaufman diagonal pivoting method and packed storage. $A$ is factorized as either $A=PUD{U}^T{P}^T$ if $uplo=\text{'U'}$ or $A=PLD{L}^T{P}^T$ if $uplo=\text{'L'}$, where $P$ is a permutation matrix, $U$ (or $L$) is a unit upper (or lower) triangular matrix and $D$ is a symmetric 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 preserving symmetry.

This method is suitable for symmetric 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