> > 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. [1] E. T. Bell. The Development of Mathematics. Second ed. McGraw-Hill, New York 1945. -------------- next part -------------- An HTML attachment was scrubbed... URL: <http://lists.mcs.anl.gov/pipermail/petsc-dev/attachments/20110916/0537aa1a/attachment.html>