Systems of nonlinear equations

The problem

Many engineering models (coupled systems of differential equations, discretized partial differential equations, boundary problems) end in a system of $n$ nonlinear equations with $n$ unknowns:

where the $f_i$ are the coordinate functions of $F$ and $D$ is open and convex. Just as in the scalar case, the solution $\alpha\in\mathbb{R}^n$ is approximated with fixed-point iterative methods, now described by a vector function $G:\mathbb{R}^n\to\mathbb{R}^n$: $x^{(k+1)}=G(x^{(k)})$, $k=0,1,2,\dots$

Convergence order with norms

The role of absolute values in the scalar case is taken over by vector norms. The fixed-point order theorem also generalizes: the scheme $x^{(k+1)}=G(x^{(k)})$ has order $p$ if $G(\alpha)=\alpha$ and all partial derivatives of the components $g_i$ up to order $p-1$ vanish at $\alpha$, with some order-$p$ one nonzero. The error equation reads $e^{(k+1)}=L\,(e^{(k)})^p+\mathcal{O}\bigl((e^{(k)})^{p+1}\bigr)$, where $e^{(k)}=x^{(k)}-\alpha$ and $L$ is a $p$-linear function derived from the multivariate Taylor expansion of $F$.

Without knowing $\alpha$, the order is estimated with the multidimensional ACOC, identical to the scalar one but with norms:

Stopping criteria

• The last two iterates are very close: $\|x^{(k+1)}-x^{(k)}\|<\varepsilon$.
• The residual is very small: $\|F(x^{(k+1)})\|<\varepsilon$.
• The maximum number of iterations has been reached without converging.