naginterfaces.library.lapacklin.dpbrfs

naginterfaces.library.lapacklin.dpbrfs(uplo, kd, nrhs, ab, afb, b, x)

dpbrfs returns error bounds for the solution of a real symmetric positive definite band system of linear equations with multiple right-hand sides, $AX=B$. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.

For full information please refer to the NAG Library document for f07hh

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f07/f07hhf.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 $U^{T}U$, where $U$ is upper triangular.

$uplo=\text{'L'}$

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

kdint

$k_d$, the number of superdiagonals or subdiagonals of the matrix $A$.

nrhsint

$r$, the number of right-hand sides.

abfloat, array-like, shape $(kd+1,n)$

The $n\times n$ original symmetric positive definite band matrix $A$ as supplied to dpbtrf().

afbfloat, array-like, shape $(kd+1,n)$

The Cholesky factor of $A$, as returned by dpbtrf().

bfloat, array-like, shape $(n,nrhs)$

The $n\times r$ right-hand side matrix $B$.

xfloat, array-like, shape $(n,nrhs)$

The $n\times r$ solution matrix $X$, as returned by dpbtrs().

Returns

xfloat, ndarray, shape $(n,nrhs)$

The improved solution matrix $X$.

ferrfloat, ndarray, shape $\left(nrhs\right)$

$ferr[\text{j}-1]$ contains an estimated error bound for the $\text{j}$th solution vector, that is, the $\text{j}$th column of $X$, for $\text{j}=1,2,\dots ,r$.

berrfloat, ndarray, shape $\left(nrhs\right)$

$berr[\text{j}-1]$ contains the component-wise backward error bound $\beta$ for the $\text{j}$th solution vector, that is, the $\text{j}$th column of $X$, for $\text{j}=1,2,\dots ,r$.

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 $kd$.

Constraint: $kd\ge 0$.

(errno $-4$)

On entry, error in parameter $nrhs$.

Constraint: $nrhs\ge 0$.

Notes

dpbrfs returns the backward errors and estimated bounds on the forward errors for the solution of a real symmetric positive definite band system of linear equations with multiple right-hand sides $AX=B$. The function handles each right-hand side vector (stored as a column of the matrix $B$) independently, so we describe the function of dpbrfs in terms of a single right-hand side $b$ and solution $x$.

Given a computed solution $x$, the function computes the component-wise backward error $\beta$. This is the size of the smallest relative perturbation in each element of $A$ and $b$ such that $x$ is the exact solution of a perturbed system

$$\begin{array}{c}\begin{array}{c}(A+\delta A)x=b+\delta b\\ \left|\delta a_{ij}\right|\le \beta \left|a_{ij}\right|\text{and}\left|\delta b_i\right|\le \beta \left|b_i\right|\text{.}\end{array}\end{array}$$

Then the function estimates a bound for the component-wise forward error in the computed solution, defined by:

$${max}_i|x_i-{\hat{x}}_i|/{max}_i\left|x_i\right|$$

where $\hat{x}$ is the true solution.

For details of the method, see the F07 Introduction.

References

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