naginterfaces.library.lapacklin.dpprfs

naginterfaces.library.lapacklin.dpprfs(uplo, n, ap, afp, b, x)

dpprfs returns error bounds for the solution of a real symmetric positive definite system of linear equations with multiple right-hand sides, $AX=B$, using packed storage. 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 f07gh

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07ghf.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.

nint

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

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

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

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

The Cholesky factor of $A$ stored in packed form, as returned by dpptrf().

bfloat, array-like, shape $(n,\text{nrhs})$

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

xfloat, array-like, shape $(n,\text{nrhs})$

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

Returns

xfloat, ndarray, shape $(n,\text{nrhs})$

The improved solution matrix $X$.

ferrfloat, ndarray, shape $\left(\text{nrhs}\right)$

$ferr[\text{j}-1]$ 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 $\left(\text{nrhs}\right)$

$berr[\text{j}-1]$ 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 $n$.

Constraint: $n\ge 0$.

(errno $-3$)

On entry, error in parameter $\text{nrhs}$.

Constraint: $\text{nrhs}\ge 0$.

Notes

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