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Fixed Point Iteration in Numerical Analysis - Lab 3 | MATH 4340, Lab Reports of Mathematical Methods for Numerical Analysis and Optimization

Material Type: Lab; Professor: Ginting; Class: Numerical Analysis; Subject: Mathematics; University: University of Wyoming; Term: Spring 2009;

Typology: Lab Reports

Pre 2010

Uploaded on 08/19/2009

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CHE/COSC/MATH 4340-01 Numerical Analysis
Computer Laboratory Assignment 03
Due Date: Thursday, 02/05/09
Fixed Point Iteration
This problem concerns with finding zeros of f(x) = x23x+2 using fixed point iteration. We have
learned in class that using iteration xn+1=g(xn)with g(x) = x2+2
3would give us r1=1, while with
g(x) = 3x2 would give us r2=2.
Using the original f(x), find two more g(x). Then write a MATLAB function to compute the fixed
point iteration with this g(x)and give an observation about the convergence of fixed point iteration
with these g(x). For example, you might want to comment about to which exact zero it converges,
how flexible is the initial iterate.
Let en=rxnbe the error at iteration level n. For each g(x)that you have discovered (there are
four of them), plot the ratio en/en1against index nwith n=1,2,3,···40. Thus you should report
four different plots. What do you observe? Relate your assessment with the error analysis that we
established in class.
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CHE/COSC/MATH 4340-01 Numerical Analysis

Computer Laboratory Assignment 03

Due Date: Thursday, 02/05/

Fixed Point Iteration

This problem concerns with finding zeros of f ( x ) = x^2 − 3 x + 2 using fixed point iteration. We have

learned in class that using iteration xn + 1 = g ( xn ) with g ( x ) =

x^2 + 2 3

would give us r 1 = 1, while with

g ( x ) =

3 x − 2 would give us r 2 = 2.

  • Using the original f ( x ), find two more g ( x ). Then write a MATLAB function to compute the fixed point iteration with this g ( x ) and give an observation about the convergence of fixed point iteration with these g ( x ). For example, you might want to comment about to which exact zero it converges, how flexible is the initial iterate.
  • Let en = rxn be the error at iteration level n. For each g ( x ) that you have discovered (there are four of them), plot the ratio en / en − 1 against index n with n = 1 , 2 , 3 , · · ·40. Thus you should report four different plots. What do you observe? Relate your assessment with the error analysis that we established in class.