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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgerfs (f07ah)

## Purpose

nag_lapack_dgerfs (f07ah) returns error bounds for the solution of a real system of linear equations with multiple right-hand sides, AX = B$AX=B$ or ATX = B${A}^{\mathrm{T}}X=B$. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.

## Syntax

[x, ferr, berr, info] = f07ah(trans, a, af, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)
[x, ferr, berr, info] = nag_lapack_dgerfs(trans, a, af, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dgerfs (f07ah) 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$AX=B$ or ATX = B${A}^{\mathrm{T}}X=B$. The function handles each right-hand side vector (stored as a column of the matrix B$B$) independently, so we describe the function of nag_lapack_dgerfs (f07ah) in terms of a single right-hand side b$b$ and solution x$x$.
Given a computed solution x$x$, the function computes the component-wise backward error β$\beta$. This is the size of the smallest relative perturbation in each element of A$A$ and b$b$ such that x$x$ is the exact solution of a perturbed system
 (A + δA)x = b + δb |δaij| ≤ β|aij|   and   |δbi| ≤ β|bi| .
$(A+δA)x=b+δb |δaij|≤β|aij| and |δbi|≤β|bi| .$
Then the function estimates a bound for the component-wise forward error in the computed solution, defined by:
 max |xi − x̂i| / max |xi| i i
$maxi | xi - x^i | / maxi |xi|$
where $\stackrel{^}{x}$ is the true solution.
For details of the method, see the F07 Chapter Introduction.

## References

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

## Parameters

### Compulsory Input Parameters

1:     trans – string (length ≥ 1)
Indicates the form of the linear equations for which X$X$ is the computed solution.
trans = 'N'${\mathbf{trans}}=\text{'N'}$
The linear equations are of the form AX = B$AX=B$.
trans = 'T'${\mathbf{trans}}=\text{'T'}$ or 'C'$\text{'C'}$
The linear equations are of the form ATX = B${A}^{\mathrm{T}}X=B$.
Constraint: trans = 'N'${\mathbf{trans}}=\text{'N'}$, 'T'$\text{'T'}$ or 'C'$\text{'C'}$.
2:     a(lda, : $:$) – double array
The first dimension of the array a must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The n$n$ by n$n$ original matrix A$A$ as supplied to nag_lapack_dgetrf (f07ad).
3:     af(ldaf, : $:$) – double array
The first dimension of the array af must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The LU$LU$ factorization of A$A$, as returned by nag_lapack_dgetrf (f07ad).
4:     ipiv( : $:$) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The pivot indices, as returned by nag_lapack_dgetrf (f07ad).
5:     b(ldb, : $:$) – double array
The first dimension of the array b must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
The n$n$ by r$r$ right-hand side matrix B$B$.
6:     x(ldx, : $:$) – double array
The first dimension of the array x must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
The n$n$ by r$r$ solution matrix X$X$, as returned by nag_lapack_dgetrs (f07ae).

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the arrays a, af, b, x The second dimension of the arrays a, af, ipiv.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the arrays b, x. (An error is raised if these dimensions are not equal.)
r$r$, the number of right-hand sides.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

### Input Parameters Omitted from the MATLAB Interface

lda ldaf ldb ldx work iwork

### Output Parameters

1:     x(ldx, : $:$) – double array
The first dimension of the array x will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
ldxmax (1,n)$\mathit{ldx}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The improved solution matrix X$X$.
2:     ferr(nrhs_p) – double array
ferr(j)${\mathbf{ferr}}\left(\mathit{j}\right)$ contains an estimated error bound for the j$\mathit{j}$th solution vector, that is, the j$\mathit{j}$th column of X$X$, for j = 1,2,,r$\mathit{j}=1,2,\dots ,r$.
3:     berr(nrhs_p) – double array
berr(j)${\mathbf{berr}}\left(\mathit{j}\right)$ contains the component-wise backward error bound β$\beta$ for the j$\mathit{j}$th solution vector, that is, the j$\mathit{j}$th column of X$X$, for j = 1,2,,r$\mathit{j}=1,2,\dots ,r$.
4:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: trans, 2: n, 3: nrhs_p, 4: a, 5: lda, 6: af, 7: ldaf, 8: ipiv, 9: b, 10: ldb, 11: x, 12: ldx, 13: ferr, 14: berr, 15: work, 16: iwork, 17: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

## Accuracy

The bounds returned in ferr are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.

For each right-hand side, computation of the backward error involves a minimum of 4n2$4{n}^{2}$ floating point operations. Each step of iterative refinement involves an additional 6n2$6{n}^{2}$ operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required.
Estimating the forward error involves solving a number of systems of linear equations of the form Ax = b$Ax=b$ or ATx = b${A}^{\mathrm{T}}x=b$; the number is usually 4$4$ or 5$5$ and never more than 11$11$. Each solution involves approximately 2n2$2{n}^{2}$ operations.
The complex analogue of this function is nag_lapack_zgerfs (f07av).

## Example

```function nag_lapack_dgerfs_example
trans = 'N';
a = [1.8, 2.88, 2.05, -0.89;
5.25, -2.95, -0.95, -3.8;
1.58, -2.69, -2.9, -1.04;
-1.11, -0.66, -0.59, 0.8];
b = [9.52, 18.47;
24.35, 2.25;
0.77, -13.28;
-6.22, -6.21];
% Factorize a and put results in the array af
[af, ipiv, info] = nag_lapack_dgetrf(a);

% Compute solution and place in x
[x, ferr, berr, info] = nag_lapack_dgerfs(trans, a, af, ipiv, b, b)
```
```

x =

1.0000    3.0000
-1.0000    2.0000
3.0000    4.0000
-5.0000    1.0000

ferr =

1.0e-13 *

0.2499
0.3471

berr =

1.0e-16 *

0.6455
0.8889

info =

0

```
```function f07ah_example
trans = 'N';
a = [1.8, 2.88, 2.05, -0.89;
5.25, -2.95, -0.95, -3.8;
1.58, -2.69, -2.9, -1.04;
-1.11, -0.66, -0.59, 0.8];
b = [9.52, 18.47;
24.35, 2.25;
0.77, -13.28;
-6.22, -6.21];
% Factorize a and put results in the array af

% Compute solution and place in x
[x, ferr, berr, info] = f07ah(trans, a, af, ipiv, b, b)
```
```

x =

1.0000    3.0000
-1.0000    2.0000
3.0000    4.0000
-5.0000    1.0000

ferr =

1.0e-13 *

0.2499
0.3471

berr =

1.0e-16 *

0.6455
0.8889

info =

0

```