An absolute tolerance to control the accuracy to which the zero is determined. In general, the smaller the value of eps the more accurate x will be as an approximation to

α

. Indeed, for very small positive values of eps, it is likely that the final approximation will satisfy

|x - α| < eps

. You are advised to call the function with more than one value for eps to check the accuracy obtained.

Constraint:

eps > 0.0

3: $eta$ – double scalar

A value such that if

|f (x)| < eta

x

is accepted as the zero. eta may be specified as

0.0

(see Accuracy).

4: $f$ – function handle or string containing name of m-file

f must evaluate the function

f

whose zero is to be determined.

[result, user] = f(x, user)

Input Parameters

1: $x$ – double scalar: The point at which the function must be evaluated.
2: $user$ – Any MATLAB object: f is called from nag_roots_contfn_cntin (c05aw) with the object supplied to nag_roots_contfn_cntin (c05aw).

Output Parameters

1: $result$ – double scalar: The value of $f$ evaluated at x.
2: $user$ – Any MATLAB object

5: $nfmax$ – int64int32nag_int scalar

The maximum permitted number of calls to f from nag_roots_contfn_cntin (c05aw). If f is inexpensive to evaluate, nfmax should be given a large value (say

> 1000

Constraint:

nfmax > 0

Optional Input Parameters

1: $user$ – Any MATLAB object: user is not used by nag_roots_contfn_cntin (c05aw), but is passed to f. Note that for large objects it may be more efficient to use a global variable which is accessible from the m-files than to use user.

Output Parameters

1: $x$ – double scalar: The final approximation to the zero, unless $ifail = 1$ , $2$ or $5$ , in which case it contains no useful information.
2: $user$ – Any MATLAB object
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: Constraint: $eps > 0.0$ .

Constraint: $nfmax > 0$ .

$ifail = 2$: Internal scale factor invalid for this problem. Consider using nag_roots_contfn_cntin_rcomm (c05ax) instead and setting scal.

$ifail = 3$: Either f has no zero near x or too much accuracy has been requested. Check the coding of f or increase eps.

$ifail = 4$: More than nfmax calls have been made to f.

nfmax may be too small for the problem (because x is too far away from the zero), or f has no zero near x, or too much accuracy has been requested in calculating the zero. Increase nfmax, check the coding of f or increase eps.

$ifail = 5$: A serious error occurred in an internal call to an auxiliary function.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The levels of accuracy depend on the values of eps and eta. If full machine accuracy is required, they may be set very small, resulting in an exit with

ifail = 3

4

, although this may involve many more iterations than a lesser accuracy. You are recommended to set

eta = 0.0

and to use eps to control the accuracy, unless you have considerable knowledge of the size of

f (x)

for values of

x

near the zero.

Further Comments

The time taken by nag_roots_contfn_cntin (c05aw) depends primarily on the time spent evaluating the function

f

(see Arguments) and on how close the initial value of x is to the zero.

If a more flexible way of specifying the function

f

is required or if you wish to have closer control of the calculation, then the reverse communication function nag_roots_contfn_cntin_rcomm (c05ax) is recommended instead of nag_roots_contfn_cntin (c05aw).

Example

This example calculates the zero of

f (x) = e^{- x} - x

from a starting value

x = 1.0

. Two calculations are made with

eps = 1.0e−3

and

1.0e−4

for comparison purposes, with

eta = 0.0

in both cases.

Open in the MATLAB editor: c05aw_example

function c05aw_example


fprintf('c05aw example results\n\n');

x = 1;
eta = 0;
nfmax = int64(200);
fprintf('\n');
% Repeat with tolerance eps set to varying powers of 10
for k=3:4
  [xOut, user, ifail] = c05aw(x, 10^-k, eta, @f, nfmax);
  switch ifail
    case {0}
      fprintf('With eps = %10.2e, root = %14.5f\n', 10^-k, xOut);
    case {3, 4}
      fprintf('With eps = %10.2e, final value = %14.5f\n', 10^-k, xOut);
    otherwise
      break;
  end
end



function [result, user] = f(x, user)
  result = x - exp(-x);

c05aw example results


With eps =   1.00e-03, root =        0.56715
With eps =   1.00e-04, root =        0.56715

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

Chapter Contents

Chapter Introduction

NAG Toolbox