I am trying to numerically solve the following system of nonlinear equations
$x_1=f_1(x_2-x_1)$
$x_2=f_2(x_3-x_2)$
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.
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$x_{n-1}=f_{n-1}(x_n-x_{n-1})$
$x_n=f_n(x_1-x_n)$
the method that I am using is to assign an initial value to the array $[x_2-x_1,x_3-x_2,...,x_1-x_n]$, use the array to find the values of $x_1,...,x_n$ from the equation above, recalculate the array using the newly found values of $x_1,...,x_n$ , and repeat the process until it converges. However, during the iterations I come up with very large values (in the order of $10^{180}$) which disables the computer to continue. Is there a method to control the values at each iteration (such as adding normalization factor) to avoid this issue? I am sure that the solution to the system of equations exists.
Thanks.
If we rewrite the system using the new variables $$ z_1=x_2-x_1, z_2 = x_3-x_2, \cdots z_{n-1}=x_n-x_{n-1}, z_n = x_1-x_n $$
we get $$ \begin{cases} z_1 &= f_2(z_2)-f_1(z_1)\\ z_2 & = f_3(z_3)-f_2(z_2)\\ &\vdots\\ z_{n-1} & =f_n(z_n)-f_{n-1}(z_{n-1})\\ z_n &= f_1(z_1)-f_n(z_n) \end{cases} $$
The fixed point method amounts to start with an initial guess $z^{(0)}$ and iterate using $$ z_i^{(k+1)} = f_{i+1}(z_{i+1}^{(k)})-f_i(z_i^{(k)}), \quad i = 1, \cdots, n $$ (using the notation $f_{n+1} := f_1$ )
The convergence of this method depends heavily on the properties of $f_1, \cdots ,f_n$ and, without further information, it is not possible to help you. Normalization will not help you... If the method does not converge you need to rewrite the system and get a different iteration function.