I understand the steps in solving a system of equations
$$\mathbf{A}(\mathbf{x})\mathbf{x} = \mathbf{b}$$
Where $\mathbf{A}(\mathbf{x})$ is a matrix that depends on the solution vector $\mathbf{x}$
i) Start with an initial guess for $\mathbf{x}$, $\mathbf{x_0}$
ii) Compute the residual $\mathbf{r} = \mathbf{A}(\mathbf{x})\mathbf{x} - \mathbf{b}$
iii) Repeat the following three steps until convergence is obtained
- Solve $\mathbf{A}(\mathbf{x})\mathbf{w} = \mathbf{b}$
- Set $\mathbf{x}$ to $\mathbf{w}$
- Compute the residual $\mathbf{r} = \mathbf{A}(\mathbf{x})\mathbf{x} - \mathbf{b}$
My question: Can I adapt these steps for a system of semilinear equations
$$\mathbf{A}\mathbf{x} = \mathbf{b}(\mathbf{x})$$
I.e.
i) Start with an initial guess for $\mathbf{x}$, $\mathbf{x_0}$
ii) Compute the residual $\mathbf{r} = \mathbf{A}\mathbf{x} - \mathbf{b}(\mathbf{x})$
iii) Repeat the following three steps until convergence is obtained
- Solve $\mathbf{A}\mathbf{w} = \mathbf{b}(\mathbf{x})$
- Set $\mathbf{x}$ to $\mathbf{w}$
- Compute the residual $\mathbf{r} = \mathbf{A}\mathbf{x} - \mathbf{b}(\mathbf{x})$
It may work. I assume $A$ is nonsingular. Suppose the true solution is $x_0$ and you are at $x = x_0 + y$ with $y$ small. Let $J$ be the Jacobian of $b$ at $x = x_0$.
Then you get $w = A^{-1} b(x_0 + y) \approx A^{-1}(b(x_0) + J y) = x_0 + A^{-1} J y$. If all eigenvalues of $A^{-1} J$ have absolute value less than $1$, you will converge to the solution $x_0$ provided you start close enough. If there is an eigenvalue with absolute value $> 1$, you will almost certainly not converge to $x_0$.