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NAG Toolbox: nag_roots_contfn_brent (c05ay)

Purpose

nag_roots_contfn_brent (c05ay) locates a simple zero of a continuous function in a given interval using Brent's method, which is a combination of nonlinear interpolation, linear extrapolation and bisection.

Syntax

[x, user, ifail] = c05ay(a, b, eps, eta, f, 'user', user)
[x, user, ifail] = nag_roots_contfn_brent(a, b, eps, eta, f, 'user', user)

Description

nag_roots_contfn_brent (c05ay) attempts to obtain an approximation to a simple zero of the function f(x) f(x)  given an initial interval [a,b] [a,b]  such that f(a) × f(b) 0 f(a) × f(b) 0 . The same core algorithm is used by nag_roots_contfn_brent_rcomm (c05az) 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

Brent R P (1973) Algorithms for Minimization Without Derivatives Prentice–Hall

Parameters

Compulsory Input Parameters

1:     a – double scalar
aa, the lower bound of the interval.
2:     b – double scalar
bb, the upper bound of the interval.
Constraint: ba ba .
3:     eps – double scalar
The termination tolerance on xx (see Section [Description]).
Constraint: eps > 0.0 eps>0.0 .
4:     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]).
5:     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_brent (c05ay) with the object supplied to nag_roots_contfn_brent (c05ay).

Output Parameters

1:     result – double scalar
The result of the function.
2:     user – Any MATLAB object

Optional Input Parameters

1:     user – Any MATLAB object
user is not used by nag_roots_contfn_brent (c05ay), 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
If ifail = 0ifail=0 or 22, x is the final approximation to the zero. If ifail = 3ifail=3, x is likely to be a pole of f(x)f(x). Otherwise, x 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:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

  ifail = 1ifail=1
Constraint: abab.
Constraint: eps > 0.0eps>0.0.
On entry, f(a)f(a) and f(b)f(b) have the same sign with neither equalling 0.00.0.
W ifail = 2ifail=2
No further improvement in the solution is possible.
W ifail = 3ifail=3
The function values in the interval [a,b] [a,b]  might contain a pole rather than a zero. Reducing eps may help in distinguishing between a pole and a zero.

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 = 2ifail=2, 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_brent (c05ay) depends primarily on the time spent evaluating f (see Section [Parameters]).
If it is important to determine an interval of relative length less than 2 × eps2×eps containing the zero, or if f is expensive to evaluate and the number of calls to f is to be restricted, then use of nag_roots_contfn_brent_rcomm (c05az) is recommended. Use of nag_roots_contfn_brent_rcomm (c05az) is also recommended when the structure of the problem to be solved does not permit a simple f to be written: the reverse communication facilities of nag_roots_contfn_brent_rcomm (c05az) are more flexible than the direct communication of f required by nag_roots_contfn_brent (c05ay).

Example

function nag_roots_contfn_brent_example
a = 0;
b = 1;
eps = 1e-5;
eta = 0;
fprintf('\n');
[x, user, ifail] = nag_roots_contfn_brent(a, b, eps, eta, @f);
switch ifail
  case {0}
    fprintf('With eps = %10.2e, root = %14.5f\n', eps, x);
  case {2, 3}
    fprintf('With eps = %10.2e, final value = %14.5f\n', eps, x);
end


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

With eps =   1.00e-05, root =        0.56714

function c05ay_example
a = 0;
b = 1;
eps = 1e-5;
eta = 0;
fprintf('\n');
[x, user, ifail] = c05ay(a, b, eps, eta, @f);
switch ifail
  case {0}
    fprintf('With eps = %10.2e, root = %14.5f\n', eps, x);
  case {2, 3}
    fprintf('With eps = %10.2e, final value = %14.5f\n', eps, x);
end


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

With eps =   1.00e-05, root =        0.56714


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