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NAG Toolbox: nag_roots_withdraw_contfn_cntin_start (c05aj)

Purpose

nag_roots_withdraw_contfn_cntin_start (c05aj) attempts to locate a zero of a continuous function using a continuation method based on a secant iteration.
Note: this function is scheduled to be withdrawn, please see c05aj in Advice on Replacement Calls for Withdrawn/Superseded Routines..

Syntax

[x, ifail] = c05aj(x, eps, eta, f, nfmax)
[x, ifail] = nag_roots_withdraw_contfn_cntin_start(x, eps, eta, f, nfmax)

Description

nag_roots_withdraw_contfn_cntin_start (c05aj) 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] = f(xx)

Input Parameters

1:     xx – double scalar
The point at which the function must be evaluated.

Output Parameters

1:     result – double scalar
The result of the function.
5:     nfmax – int64int32nag_int scalar
The maximum permitted number of calls to f from nag_roots_withdraw_contfn_cntin_start (c05aj). If f is inexpensive to evaluate, nfmax should be given a large value (say > 1000 >1000 ).
Constraint: nfmax > 0 nfmax>0 .

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

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:     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
On entry, eps0.0 eps0.0 ,
or nfmax0 nfmax0 .
  ifail = 2ifail=2
An internally calculated scale factor has the wrong order of magnitude for the problem. If this error exit occurs, you are advised to call nag_roots_contfn_cntin_rcomm (c05ax) instead where different scale values can be tried.
  ifail = 3ifail=3
Either the function f(x) f(x)  given by f has no zero near x or too much accuracy has been requested in calculating the zero. The first is a more likely cause of this error exit and you should check the coding of f and make an independent investigation of its behaviour near x. The second can be alleviated by increasing eps.
  ifail = 4ifail=4
More than nfmax calls have been made to f. This error exit can occur because nfmax is too small for the problem (essentially because x is too far away from the zero) or for either of the reasons given under ifail = 3ifail=3 above. If nfmax is increased considerably and this error exit occurs again at approximately the same final value of x, then it is likely that one of the reasons given under ifail = 3ifail=3 is the cause.
   ifail = 5 ifail=5  (nag_roots_contfn_cntin_rcomm (c05ax))
A serious error has occurred in the specified function. Check all function calls. Seek expert help.

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_withdraw_contfn_cntin_start (c05aj) 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_withdraw_contfn_cntin_start (c05aj).

Example

function nag_roots_withdraw_contfn_cntin_start_example
x = 1;
eta = 0;
nfmax = int64(200);
f = @(x) exp(-x)-x;
for k=3:4
  [xOut, ifail] = nag_roots_withdraw_contfn_cntin_start(x, 10^-k, eta, f, nfmax);

  if ifail == 3 || ifail == 4
    fprintf('With epsilon = %10.2e, final value = %14.5f\n', 10^-k, xOut);
  else
    fprintf('With epsilon = %10.2e, root = %14.5f\n', 10^-k, xOut);
  end
end
 
With epsilon =   1.00e-03, root =        0.56715
With epsilon =   1.00e-04, root =        0.56715

function c05aj_example
x = 1;
eta = 0;
nfmax = int64(200);
f = @(x) exp(-x)-x;
for k=3:4
  [xOut, ifail] = c05aj(x, 10^-k, eta, f, nfmax);

  if ifail == 3 || ifail == 4
    fprintf('With epsilon = %10.2e, final value = %14.5f\n', 10^-k, xOut);
  else
    fprintf('With epsilon = %10.2e, root = %14.5f\n', 10^-k, xOut);
  end
end
 
With epsilon =   1.00e-03, root =        0.56715
With epsilon =   1.00e-04, root =        0.56715


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