Problem Set - Picard and Newton iteration

Problem Set - Picard and Newton iteration

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 7 December 2009

Picard and Newton iteration

Let f:(0,12π) is given by f(x)=12tanx. Estimate numerically the solution to x=f(x) with x(0,12π) using Picard iteration.

Let f:(0,12π) is given by f(x)=12tanx. Estimate numerically the solution to x=f(x) with x(0,12π) using Newton iteration (let F(x)=x-f(x)).

Show that the equation g(x)=x3+x-1=0 has a solution between 0 and 1. Transform the equation to the form x=f(x) for a suitable function f:[0,1][0,1]. Use Picard iteration to find the solution to 3 decimal places. (Try f(x)=1/(x2+1)).

Show that the equation g(x)=x4-4x2-x+4=0 has a solution between 3 and 2. Transform the equation to the form x=f(x) for a suitable function f:[3,2][3,2]. Use Picard iteration to find the solution to 3 decimal places. (Try f(x)=2+x).

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)