Vrftjkry

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 $x_0$.