Hi everybody,
I have a non linear system of 9 equations:
F=A(F)F+B(F)g(F)
F is a 9x1 vector of the unknowns, A(F) and B(F) are 9x9 matrixes which depend on F ang g(F) is a 9x1 vector which depends on F.
My problem is that i can't give an explicit expression to A and B in terms of F because they require the inversion of symbolic matrices which depend on F, and it is computationally unfeasible.
So I was looking for an iterative solution, basically evaluating A and B in a known status F0, solving F=A(F0)F+B(F0)g(F) and the updating iteratively until two consecutive solutions converge.
Is this approach well-founded? will it converge to the solution?
You are talking about the fixed point iteration method.
See
https://en.wikipedia.org/wiki/Fixed-point_iteration
Another solution is to write your system as
A(F)F+B(F)g(F)-F = 0
and to consider root finding methods, such as Newton's method.
You have to numerically estimate the derivative.
best
I fully agree with Simone Orcioni . As an additional method you can use a graphical solution of the equation A(F)F+B(F)g(F)-F = 0
Fixed point iterative schemes will do for this.
Let me recommend the following scheme;
Given
F=A(F)F + B(F)g(F) ------------------------------- (1)
with the initial guess F0
Start by setting
Fn+1 = (1 - a) Fn + aFn, 0
Understand the non linear system. Use linear approximations to get estimates of final answers. Know boundaries of possible answers. Grid search multiple dimensions to narrow answer. Use standard methods to adjust results.
If the matrices A, B, and g are differential with respect to F then I would try Newton's Method!
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