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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_dgerfs (f07ah)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_lapack_dgerfs (f07ah) returns error bounds for the solution of a real system of linear equations with multiple right-hand sides,

A X = B

A^{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

A X = B

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 nag_lapack_dgerfs (f07ah) 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

β

. 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{matrix} (A + δ A) x = b + δ b \\ |δ a_{i j}| \leq β |a_{i j}| and |δ b_{i}| \leq β |b_{i}| . \end{matrix}

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} |x_{i}|

where

\hat{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

is the computed solution.

$trans ='N'$: The linear equations are of the form $A X = B$ .
$trans ='T'$ or $'C'$: The linear equations are of the form $A^{T} X = B$ .

Constraint:

trans ='N'

'T'

'C'

2: $a (lda :)$ – double array

The first dimension of the array a must be at least

\max (1, n)

The second dimension of the array a must be at least

\max (1, n)

The

n

n

original matrix

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)

The second dimension of the array af must be at least

\max (1, n)

The

L U

factorization of

A

, as returned by nag_lapack_dgetrf (f07ad).

4: $ipiv (:)$ – int64int32nag_int array

The dimension of the array ipiv must be at least

\max (1, n)

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)

The second dimension of the array b must be at least

\max (1, nrhs_p)

The

n

r

right-hand side matrix

B

6: $x (ldx :)$ – double array

The first dimension of the array x must be at least

\max (1, n)

The second dimension of the array x must be at least

\max (1, nrhs_p)

The

n

r

solution matrix

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 and the second dimension of the arrays a, af, ipiv.
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .
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.)
$r$ , the number of right-hand sides.

Constraint: $nrhs_p \geq 0$ .

Output Parameters

1: $x (ldx :)$ – double array: The first dimension of the array x will be $\max (1, n)$ .
The second dimension of the array x will be $\max (1, nrhs_p)$ .

The improved solution matrix $X$ .
2: $ferr (nrhs_p)$ – double array: $ferr (j)$ contains an estimated error bound for the $j$ th solution vector, that is, the $j$ th column of $X$ , for $j = 1, 2, \dots, r$ .
3: $berr (nrhs_p)$ – double array: $berr (j)$ contains the component-wise backward error bound $β$ for the $j$ th solution vector, that is, the $j$ th column of $X$ , for $j = 1, 2, \dots, r$ .
4: $info$ – int64int32nag_int scalar: $info = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

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.

Further Comments

For each right-hand side, computation of the backward error involves a minimum of

4 n^{2}

floating-point operations. Each step of iterative refinement involves an additional

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

A x = b

A^{T} x = b

; the number is usually

4

5

and never more than

11

. Each solution involves approximately

2 n^{2}

operations.

The complex analogue of this function is nag_lapack_zgerfs (f07av).

Example

This example solves the system of equations

A X = B

using iterative refinement and to compute the forward and backward error bounds, where

A = (\begin{matrix} 1.80 & 2.88 & 2.05 & - 0.89 \\ 5.25 & - 2.95 & - 0.95 & - 3.80 \\ 1.58 & - 2.69 & - 2.90 & - 1.04 \\ - 1.11 & - 0.66 & - 0.59 & 0.80 \end{matrix}) and B = (\begin{matrix} 9.52 & 18.47 \\ 24.35 & 2.25 \\ 0.77 & - 13.28 \\ - 6.22 & - 6.21 \end{matrix}) .

Here

A

is nonsymmetric and must first be factorized by nag_lapack_dgetrf (f07ad).

Open in the MATLAB editor: f07ah_example

function f07ah_example


fprintf('f07ah example results\n\n');

a = [ 1.80,  2.88,  2.05, -0.89;
      5.25, -2.95, -0.95, -3.80;
      1.58, -2.69, -2.90, -1.04;
     -1.11, -0.66, -0.59,  0.80];
b = [ 9.52, 18.47;
     24.35,  2.25;
      0.77,-13.28;
     -6.22, -6.21];

% Factorize a
[af, ipiv, info] = f07ad(a);

% Compute solution x
trans = 'N';
[x, ferr, berr, info] = f07ah( ...
                               trans, a, af, ipiv, b, b);

[ifail] = x04ca( ...
                 'General', ' ', x, 'Solution(s)');

fprintf('\nBackward errors (machine-dependent)\n   ')
fprintf('%11.1e', berr);
fprintf('\nEstimated forward error bounds (machine-dependent)\n   ')
fprintf('%11.1e', ferr);
fprintf('\n');

f07ah example results

 Solution(s)
             1          2
 1      1.0000     3.0000
 2     -1.0000     2.0000
 3      3.0000     4.0000
 4     -5.0000     1.0000

Backward errors (machine-dependent)
       6.5e-17    8.9e-17
Estimated forward error bounds (machine-dependent)
       2.5e-14    3.5e-14

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox