[petsc-users] fixed point interations

[Dominik Szczerba]

On Sun, Nov 6, 2011 at 10:40 PM, Matthew Knepley <knepley at gmail.com> wrote:
> On Sun, Nov 6, 2011 at 9:34 PM, Dominik Szczerba <dominik at itis.ethz.ch>
> wrote:
>>
>> On Sun, Nov 6, 2011 at 6:59 PM, Matthew Knepley <knepley at gmail.com> wrote:
>> > On Sun, Nov 6, 2011 at 5:52 PM, Dominik Szczerba <dominik at itis.ethz.ch>
>> > wrote:
>> >>
>> >> >>> I want to start small by porting a very simple code using fixed
>> >> >>> point
>> >> >>> iterations as follows: A(x)x = b(x) is approximated as A(x0)x =
>> >> >>> b(x0),
>> >> >>> then solved by KSP for x, then x0 is updated to x, then repeat
>> >> >>> until
>> >> >>> convergence.
>> >> >
>> >> > Run the usual "Newton" methods with A(x) in place of the true
>> >> > Jacobian.
>> >>
>> >> When I substitute A(x) into eq. 5.2 I get:
>> >>
>> >> A(x) dx = -F(x) (1)
>> >> A(x) dx = -A(x) x + b(x) (2)
>> >> A(x) dx + A(x) x = b(x) (3)
>> >> A(x) (x+dx) = b(x) (4)
>> >>
>> >> My questions:
>> >>
>> >> * Will the procedure somehow optimally group the two A(x) terms into
>> >> one, as in 3-4? This requires knowledge, will this be efficiently
>> >> handled?
>> >
>> > There is no grouping. You solve for dx and do a vector addition.
>> >
>> >>
>> >> * I am solving for x+dx, while eq. 5.3 solves for dx. Is this, and
>> >> how, correctly handled? Should I somehow disable the update myself?
>> >
>> > Do not do any update yourself, just give the correct A at each iteration
>> > in
>> > your FormJacobian routine.
>> >    Matt
>>
>> OK, no manual update, this is clear now. What is still not clear is
>> that by substituting A for F' I arrive at an equation in x+dx (my eq.
>> 4), and not dx (Petsc eq. 5.3)...
>
> Newton's equation is for dx. Then you add that to x to get the next guess.
> This
> is described in any book on numerical analysis, e.g. Henrici.
>    Matt

I understand that, but I am asking something else... when taking F' =
A I arrive at an equation in x+dx, which is not Newton equation. What
is wrong in this picture?

Dominik