Steffensen's method: Difference between revisions

Revision as of 20:01, 29 October 2008

Revision as of 19:55, 7 November 2008

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Similar to [[Newton's method]] and most other quadratically convergent methods, the crucial weakness with the method is the choice of the starting value <math>x_0</math>&nbsp;. If the value of <math>x_0</math> is not "close enough" to the actual solution, the method will fail and the sequence of values <math>x_0, x_1, x_2, x_3,\dots</math> will either flip flop between two extremes, or diverge to infinity (possibly both!).

==Implementation in Matlab==

Here is the source for an implementation of Steffensen's Method in [[MATLAB]].

<source lang="matlab">

function Steffensen(f,p0,tol)

% This function takes as inputs: a fixed point iteration function, f,

% and initial guess to the fixed point, p0, and a tolerance, tol.

% The fixed point iteration function is assumed to be input as an

% inline function.

% This function will calculate and return the fixed point, p,

% that makes the expression f(x) = p true to within the desired

% tolerance, tol.

format compact % This shortens the output.

format long % This prints more decimal places.

for i=1:1000 % get ready to do a large, but finite, number of iterations.

% This is so that if the method fails to converge, we won't

% be stuck in an infinite loop.

p1=f(p0); % calculate the next two guesses for the fixed point.

p2=f(p1);

p=p0-(p1-p0)^2/(p2-2*p1+p0) % use Aitken's delta squared method to

% find a better approximation to p0.

if abs(p-p0)<tol % test to see if we are within tolerance.

break % if we are, stop the iterations, we have our answer.

end

p0=p; % update p0 for the next iteration.

end

if abs(p-p0)>tol % If we fail to meet the tolerance, we output a

% message of failure.

'failed to converge in 1000 iterations.'

end

</source>

==Generalisation==