The Computable Differential Equation Lecture ... - Bruce E. Shapiro

CHAPTER 1. CLASSIFYING THE PROBLEM 23
Here is a more LISP-like implementation that returns the same, identical results:
EulersMethod2[f , t0 , y0 , tmax , h ] :=
Module[{Euler},
Euler[{x , u }] := {x + h, u + h * f[x, u]}; ⇐=
Return[NestList[Euler, {t0, y0}, Round[tmax/h]]]
]
⇐=
This makes use of the function NestList . NestList implements fixed point
iteration in Mathematica. For example,
NestList[f, a, 3]
returns three fixed point iterations of the function f starting at the point a:
{a, f[a], f[f[a]], f[f[f[a]]]}
In EulersMethod2 the fixed point iteration is being performed on the locally defined
function Euler. Eulertakes a pair of numbers {t, y} and returns a new pair of
numbers
{t + h, y + hf(t, y)}
These are same two numbers that are calculated at each step during the previous
Euler’s method implementation. They are then used recursively by NestList
as the new values of the parameters x and u. The number of iterations of
NestList is determined by rounding the ratio t max /h, which might not otherwise
be an integer.
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge