In a book for solving a nonlinear differential equation with $N+1$ points, $u_{xx} = e^{u}, u(-1)=u(1) = 0$, in $[-1,1]$ with homogeneous Dirichlet boundary conditions, the fixed point iteration is used. The MATLAB code is as follows:

u = zeros(N-1,1);
change = 1; it = 0;
while change > 1e-15
       unew = D2\exp(u);
       change1 = change;
       change = norm(unew-u,inf);
       u = unew; it = it+1;

where $D^2$ is a differential matrix. In each step the error is reduced by a factor of (change/change1) 0.2943. In the book it is asked how this factor is obtained? and what is its relationship with the eigenvalues of the $D^2$ matrix?

I evaluated the eigenvalues of the matrix $D^2$, but could not find a relation. How is the reduction factor in fixed point method obtained?

Thanks in advance.


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