f = @(x) x.^2 - 4*x + 3.5;
r = roots([1 -4 3.5]);
Here is the fixed point iteration. This time we keep track of the whole sequence of approximations.
g = @(x) x - f(x);
x = 2.1;
for k = 1:12
x(k+1) = g(x(k));
end
It's easiest to construct and plot the sequence of errors.
err = abs(x-r(1));
semilogy(err,'.-'), axis tight
xlabel('iteration'), ylabel('error')
title('Convergence of fixed point iteration')
It's quite clear that the convergence quickly settles into a linear rate. We could estimate this rate by doing a least-squares fit to a straight line using MATLAB's built-in polyfit. Keep in mind that the values for small k should be left out of the computation, as they don't represent the linear trend.
p = polyfit(5:12,log(err(5:12)),1)
The first value is the slope, which must be negative. We can exponentiate it to get the convergence constant σ.
sigma = exp(p(1))
The numerical values of the error should decrease by a factor of σ at each iteration. We can check this easily with an elementwise division.
err(9:12) ./ err(8:11)
The methods for finding σ agree well.