naginterfaces.library.sparse.direct_​real_​gen_​refine

naginterfaces.library.sparse.direct_real_gen_refine(trans, icolzp, irowix, a, iprm, il, lval, iu, uval, nrhs, b, x)

direct_real_gen_refine returns error bounds for the solution of a real sparse system of linear equations with multiple right-hand sides, $AX=B$ or $A^{T}X=B$. It improves the solution by iterative refinement in standard precision, in order to reduce the backward error as much as possible.

For full information please refer to the NAG Library document for f11mh

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/f11/f11mhf.html

Parameters

transstr, length 1

Specifies whether $AX=B$ or $A^{T}X=B$ is solved.

$trans=\text{'N'}$

$AX=B$ is solved.

$trans=\text{'T'}$

$A^{T}X=B$ is solved.

icolzpint, array-like, shape $(n+1)$

The new column index array of sparse matrix $A$. See the F11 Introduction.

irowixint, array-like, shape $\left(\text{asize}\right)$

The row index array of sparse matrix $A$. See the F11 Introduction.

afloat, array-like, shape $\left(\text{asize}\right)$

The array of nonzero values in the sparse matrix $A$.

iprmint, array-like, shape $(7\times n)$

The column permutation which defines $P_c$, the row permutation which defines $P_r$, plus associated data structures as computed by direct_real_gen_lu().

ilint, array-like, shape $(:)$

Records the sparsity pattern of matrix $L$ as computed by direct_real_gen_lu().

lvalfloat, array-like, shape $(:)$

Records the nonzero values of matrix $L$ and some nonzero values of matrix $U$ as computed by direct_real_gen_lu().

iuint, array-like, shape $(:)$

Records the sparsity pattern of matrix $U$ as computed by direct_real_gen_lu().

uvalfloat, array-like, shape $(:)$

Records some nonzero values of matrix $U$ as computed by direct_real_gen_lu().

nrhsint

$\text{nrhs}$, the number of right-hand sides in $B$.

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

The $n\times \text{nrhs}$ right-hand side matrix $B$.

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

The $n\times \text{nrhs}$ solution matrix $X$, as returned by direct_real_gen_solve().

Returns

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

The $n\times \text{nrhs}$ 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 $\text{j}=1,2,\dots ,\text{nrhs}$.

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 $\text{j}=1,2,\dots ,\text{nrhs}$.

Raises

NagValueError

(errno $1$)

On entry, $nrhs=⟨value⟩$.

Constraint: $nrhs\ge 0$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $1$)

On entry, $trans=⟨value⟩$.

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

(errno $2$)

Incorrect row permutations in array $iprm$.

(errno $3$)

Incorrect column permutations in array $iprm$.

Notes

direct_real_gen_refine returns the backward errors and estimated bounds on the forward errors for the solution of a real 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 direct_real_gen_refine 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 if $x$ is the exact solution of a perturbed system:

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

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.

The function uses the $LU$ factorization $P_{r}A{P}_c=LU$ computed by direct_real_gen_lu() and the solution computed by direct_real_gen_solve().

References

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