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NAG Toolbox: nag_roots_contfn_cntin (c05aw)

Purpose

nag_roots_contfn_cntin (c05aw) attempts to locate a zero of a continuous function using a continuation method based on a secant iteration.

Syntax

[x, user, ifail] = c05aw(x, eps, eta, f, nfmax, 'user', user)
[x, user, ifail] = nag_roots_contfn_cntin(x, eps, eta, f, nfmax, 'user', user)

Description

nag_roots_contfn_cntin (c05aw) attempts to obtain an approximation to a simple zero αα of the function f(x) f(x)  given an initial approximation xx to αα. The zero is found by a call to nag_roots_contfn_cntin_rcomm (c05ax) whose specification should be consulted for details of the method used.
The approximation xx to the zero αα is determined so that at least one of the following criteria is satisfied:
(i) |xα|eps |x-α|eps ,
(ii) |f(x)| < eta |f(x)|<eta .

References

None.

Parameters

Compulsory Input Parameters

1:     x – double scalar
An initial approximation to the zero.
2:     eps – double scalar
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 αα. Indeed, for very small positive values of eps, it is likely that the final approximation will satisfy |xα| < eps |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 eps>0.0 .
3:     eta – double scalar
A value such that if |f(x)| < eta |f(x)|<eta , xx is accepted as the zero. eta may be specified as 0.00.0 (see Section [Accuracy]).
4:     f – function handle or string containing name of m-file
f must evaluate the function ff 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 result of the function.
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 >1000 ).
Constraint: nfmax > 0 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.

Input Parameters Omitted from the MATLAB Interface

iuser ruser

Output Parameters

1:     x – double scalar
The final approximation to the zero, unless ifail = 1ifail=1, 22 or 55, in which case it contains no useful information.
2:     user – Any MATLAB object
3:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
Constraint: eps > 0.0eps>0.0.
Constraint: nfmax > 0nfmax>0.
  ifail = 2ifail=2
Internal scale factor invalid for this problem. Consider using nag_roots_contfn_cntin_rcomm (c05ax) instead and setting scal.
  ifail = 3ifail=3
Either f has no zero near x or too much accuracy has been requested. Check the coding of f or increase eps.
  ifail = 4ifail=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 = 5ifail=5
A serious error occurred in an internal call to an auxiliary function.

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 = 3ifail=3 or 44, although this may involve many more iterations than a lesser accuracy. You are recommended to set eta = 0.0 eta=0.0  and to use eps to control the accuracy, unless you have considerable knowledge of the size of f(x) f(x)  for values of xx near the zero.

Further Comments

The time taken by nag_roots_contfn_cntin (c05aw) depends primarily on the time spent evaluating the function ff (see Section [Parameters]) and on how close the initial value of x is to the zero.
If a more flexible way of specifying the function ff 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

function nag_roots_contfn_cntin_example
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] = nag_roots_contfn_cntin(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);
 

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

function c05aw_example
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);
 

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


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Chapter Introduction
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