Solution of a system of linear equations when matrix is a function of a vector variable

Question

My problem is actually a nonlinear equation:

$A(x)x=0$, where x is a vector and matrix A is a function of vector variable x.

Are there any works related to this kind of problem??

Answer 1

You would probably need to use some variant of the Newton's method. Depending on how non-linear is $x$ in the function $A(x)x$.

If it happen that $A(x)x$ is just quadratic on $x$ then Gauss-Newton method would suffice:

https://en.m.wikipedia.org/wiki/Gauss-Newton_algorithm

The general approach for non-linear systems is to find a first "guess" for $x$, called $x_0$. You also need to linearize the function $A(x)x$, that means picking a first or second order approximation of $A(x)x$ using the Taylor's expansion of it. In practice you often need to find just the Jacobian matrix. Once you have the $x_0$ and the linearization of $A(x)x$ then the method consist of solving a series of linear systems iteratively to find better approximations of $x_i$ such that $A(x_i) x_i = 0$