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 $f:(0,\frac{1}{2}\pi )\to ℝ$ is given by $f\left(x\right)=\frac{1}{2}\tan x$. Estimate numerically the solution to $x=f\left(x\right)$ with $x\in (0,\frac{1}{2}\pi )$ using Picard iteration.
	Let $f:(0,\frac{1}{2}\pi )\to ℝ$ is given by $f\left(x\right)=\frac{1}{2}\tan x$. Estimate numerically the solution to $x=f\left(x\right)$ with $x\in (0,\frac{1}{2}\pi )$ using Newton iteration (let $F\left(x\right)=x-f\left(x\right)$).
	Show that the equation $g\left(x\right)=x^3+x-1=0$ has a solution between 0 and 1. Transform the equation to the form $x=f\left(x\right)$ for a suitable function $f:[0,1]\to [0,1]$. Use Picard iteration to find the solution to 3 decimal places. (Try $f\left(x\right)=1/(x^2+1)$).
	Show that the equation $g\left(x\right)=x^4-4x^2-x+4=0$ has a solution between $\sqrt{3}$ and 2. Transform the equation to the form $x=f\left(x\right)$ for a suitable function $f:[\sqrt{3},2]\to [\sqrt{3},2]$. Use Picard iteration to find the solution to 3 decimal places. (Try $f\left(x\right)=\sqrt{2+\sqrt{x}}$).

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)