I have the equation of the form:
$$ \textbf{X}_{n+1} = \textbf{X}_{n} + \textbf{X}_{0} $$
This equation come from physical optics when trying to solve the MFIE (below) iteratively.
$$ \textbf{J}(\textbf{r}) = 2\vec{n} \times \textbf{H}^{inc}(\textbf{r}) + 2\vec{n} \times \textbf{H}(\textbf{r}) $$
Where $\vec{X}_{0}$ is my incident plane wave (This remains a constant vector) and the $\vec{X}_{n}$ is the current which has been calcuated. The sum of these gives the next iteration of the current, $\vec{X}_{n+1}$. This was just some background to help get a better picture.
My questions are:
- What type of equation is this?
- What methods are there to solve an equation like this iteratively? Some of the methods I have looked at are only valid for linear equations of this type, $Ax = b$.
This ''vectorial affine recurrence relationship'' (such is ''a'' name that can be given):
$$\tag{1}V_{n+1}=AV_{n}+b$$
has a standard method of resolution.
First of all, look at the ''candidate vector'' say $V_{\infty}$ that will be the limit if any. $V_{\infty}$ should verify the fixed point equation:
$$\tag{2}V_{\infty} \ = \ A V_{\infty}+b$$
(and $V_{\infty}$ will be found as a solution to the equivalent linear system $(I_n-A)V_{\infty}=b$).
Now, make the subtraction (1)-(2):
$$\tag{3}V_{n+1}-V_{\infty} \ = \ A(V_{n+1}-V_{\infty})$$
where $b$ has disappeared; (3) can be written so, with an auxiliary sequence:
$$\tag{4}W_{n+1} \ = \ A W_{n+1} \ \ \text{with} \ \ W_{n}:=V_{n}-V_{\infty}$$
We are now faced with a classical problem, with a discussion on the eigenvalues of $A$:
Let us call ''spectral radius'' (and denote by $r$) the maximum of the absolute values of the eigenvalues of $A$:
if $r<1$, $W_n \to 0$, then $V_n \to V_{\infty}$.
if $r>1$ $W_n$ is divergent, then $V_n$ is divergent.
if $r=1$, different cases may arise but in general there is no convergence either
(for example if there is an eigenvalue equal to -1, one can observe permanent oscillations).