[petsc-dev] How do you get RIchardson?

[Matthew Knepley]

On Fri, Sep 16, 2011 at 6:21 PM, Sean Farley <sean at mcs.anl.gov> wrote:

> If one defines a Picard method as any fixed-point iteration then x^{n+1} =
>>> x^{n} - J(x^{n})^{-1} F(x^{n}) is a Picard iteration for the equation x = x
>>> - J(x)^{-1} F(x) in other words Newtons' method is a Picard method; is this
>>> true? Is Picard algorithm a synonym for fixed point iteration?
>>>
>>
>> http://en.wikipedia.org/wiki/Picard_iteration (redirects to "Fixed point
>> iteration")
>>
>> Also, Tim Kelley's book describes "fixed point iteration" as "also called
>> nonlinear Richardson iteration, Picard iteration, or the method of
>> successive substitution".
>>
>
> Actually, the first (in recorded history, of course) version of fixed point
> iteration "originated in antiquity, appearing, for example, in the writings
> of Heron of Alexandria [1]." Perhaps, we should, in the interest of not
> being historically blind, call it the Heron method.
>

Heron defined it only for a particular function.

   Matt


> [1] E. T. Bell. The Development of Mathematics. Second ed. McGraw-Hill, New
> York 1945.
>



-- 
What most experimenters take for granted before they begin their experiments
is infinitely more interesting than any results to which their experiments
lead.
-- Norbert Wiener
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.mcs.anl.gov/pipermail/petsc-dev/attachments/20110916/2c645981/attachment.html>