naginterfaces.library.lapacklin.dgbrfs

naginterfaces.library.lapacklin.dgbrfs(trans, kl, ku, nrhs, ab, afb, ipiv, b, x)

dgbrfs returns error bounds for the solution of a real band system of linear equations with multiple right-hand sides, $AX=B$ or $A^{T}X=B$. 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 f07bh

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/f07/f07bhf.html

Parameters

transstr, length 1

Indicates the form of the linear equations for which $X$ is the computed solution.

$trans=\text{'N'}$

The linear equations are of the form $AX=B$.

$trans=\text{'T'}$ or $\text{'C'}$

The linear equations are of the form $A^{T}X=B$.

klint

$k_l$, the number of subdiagonals within the band of the matrix $A$.

kuint

$k_u$, the number of superdiagonals within the band of the matrix $A$.

nrhsint

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

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

The original $n\times n$ band matrix $A$ as supplied to dgbtrf() but with reduced requirements since the matrix is not factorized.

See Further Comments for further details.

afbfloat, array-like, shape $(2\times kl+ku+1,n)$

The $LU$ factorization of $A$, as returned by dgbtrf().

ipivint, array-like, shape $\left(n\right)$

The pivot indices, as returned by dgbtrf().

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 dgbtrs().

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 $trans$.

Constraint: $trans=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.

(errno $-2$)

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

Constraint: $n\ge 0$.

(errno $-3$)

On entry, error in parameter $kl$.

Constraint: $kl\ge 0$.

(errno $-4$)

On entry, error in parameter $ku$.

Constraint: $ku\ge 0$.

(errno $-5$)

On entry, error in parameter $nrhs$.

Constraint: $nrhs\ge 0$.

Notes

dgbrfs returns the backward errors and estimated bounds on the forward errors for the solution of a real band system of linear equations with multiple right-hand sides $AX=B$ or $A^{T}X=B$. The function handles each right-hand side vector (stored as a column of the matrix $B$) independently, so we describe the function of dgbrfs 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