naginterfaces.library.lapackeig.dpbstf

naginterfaces.library.lapackeig.dpbstf(uplo, kb, bb)

dpbstf computes a split Cholesky factorization of a real symmetric positive definite band matrix.

For full information please refer to the NAG Library document for f08uf

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f08/f08uff.html

Parameters

uplostr, length 1

Indicates whether the upper or lower triangular part of $B$ is stored.

$uplo=\text{'U'}$

The upper triangular part of $B$ is stored.

$uplo=\text{'L'}$

The lower triangular part of $B$ is stored.

kbint

If $uplo=\text{'U'}$, the number of superdiagonals, $k_b$, of the matrix $B$.

If $uplo=\text{'L'}$, the number of subdiagonals, $k_b$, of the matrix $B$.

bbfloat, array-like, shape $(kb+1,n)$

The $n\times n$ symmetric positive definite band matrix $B$.

Returns

bbfloat, ndarray, shape $(kb+1,n)$

$B$ is overwritten by the elements of its split Cholesky factor $S$.

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: $n\ge 0$.

(errno $-3$)

On entry, error in parameter $kb$.

Constraint: $kb\ge 0$.

(errno $i>0$)

The factorization could not be completed, because the updated element $b(⟨value⟩,⟨value⟩)$ would be the square root of a negative number. Hence $B$ is not positive definite. This may indicate an error in forming the matrix $B$.

Notes

dpbstf computes a split Cholesky factorization of a real symmetric positive definite band matrix $B$. It is designed to be used in conjunction with dsbgst().

The factorization has the form $B=S^{T}S$, where $S$ is a band matrix of the same bandwidth as $B$ and the following structure: $S$ is upper triangular in the first $(n+k)/2$ rows, and transposed — hence, lower triangular — in the remaining rows. For example, if $n=9$ and $k=2$, then

$$\begin{array}{c}S=\left(\begin{array}{ccccccccc}{s}_{11}& s_{12}& s_{13}& & & & & & \\ & s_{22}& s_{23}& s_{24}& & & & & \\ & & s_{33}& s_{34}& s_{35}& & & & \\ & & & s_{44}& s_{45}& & & & \\ & & & & s_{55}& & & & \\ & & & s_{64}& s_{65}& s_{66}& & & \\ & & & & s_{75}& s_{76}& s_{77}& & \\ & & & & & s_{86}& s_{87}& s_{88}& \\ & & & & & & s_{97}& s_{98}& s_{99}\end{array}\right)\text{.}\end{array}$$