[petsc-dev] How do you get RIchardson?

[Jed Brown]

On Fri, Sep 16, 2011 at 23:00, Matthew Knepley <knepley at gmail.com> wrote:

> It is the name for this generic concept:
>
>   Solve F(u) = 0, by successive approximations u^{n+1} = F(u^n)
>

You have defined F(u^n) = x_n - f(x_n). Why is this the One True Way to
solve f(x) = 0?

When you were a child and learned fixed-point iteration for scalar problems,
you learned to reformulate f(x) = 0 as x = F(x) so that the fixed point
iteration was convergent. This step was crucial because for any given
problem, there are lots of was to formulate the fixed point iteration and
most of them don't work. A worked example for f(x) = x^3 - sin x is in these
notes:

http://pages.cs.wisc.edu/~amos/412/lecture-notes/lecture03.pdf


I don't see why systems are any different. Why do you assume that when given
f(x), the only mathematically pure formulation is as x = x - f(x)?
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