Matlab

function newton(fun, dfun, x0, TOL, DAMPED, nmax, numeric)

% Implementation of the well known Newton method.
%
% INPUT: fun the function to investigate, given as inline function
%dfun the derivative, given as inline function
% x0 starting guess
% TOL tolerance for declaration of convergence
% DAMPED = 0 for the standardundamped method
% = 1 for the damped method to achieve global
% convergence
% nmax maximum number of iteration steps
% numeric use numericaldifferentiation
%
% OUTPUT: Table giving the iterates and corresponding function values
% Plot visualizing the iteration steps
% Author: B. Flemisch

%clf;
small = 1e-8;
sigma =0.0001; % parameter for damped method
rho = 0.5; % parameter for damped method

xvec = x0;
x = x0;
err = abs(feval(fun, x0));
errvec = err;
errvec2 = [err^2];

nit = 0;
while((nit TOL))
nit = nit + 1;
if (numeric)
dx = max(sqrt(eps)*abs(x), 1e-8); % step width for central difference
dfx = (feval(fun, x + dx) - feval(fun, x - dx))/(2*dx);else
dfx = feval(dfun,x); % get derivative
end

if (abs(dfx) < small) % zero derivative -> divergence of the method
nit
x
dfx
disp('Zero derivative!');break;
else
d = - feval(fun,x)/dfx; % get search direction
alpha = 1.0; % damping parameter set to 1
xold = x;
x = x + alpha*d; % new iterate
if (DAMPED),
f2 =feval(fun,xold).^2;
f2n = feval(fun,x).^2;
while (f2n > f2 + sigma*alpha*feval(fun,xold)*d)
alpha = rho*alpha; % damping parameter is adjusted
x = xold + alpha*d; %adjust iterate
f2n = feval(fun,x).^2;
end
end
xvec = [xvec; x];
err = abs(feval(fun, x));
errvec = [errvec; err];
errvec2 = [errvec2; err^2];
end...

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