+1-617-874-1011 (US)
+61-7-5641-0117 (AU)
+44-117-230-1145 (UK)
Live Chat

CS 261/MATH 261 Methods of Numerical Analysis

Assignment 3

(8 marks) Let the functions g1 and g2 be defined by

CS 261/MATH 261 Methods of Numerical Analysis Image 1
  • Show that 1 is a fixed point of g1 and g2, and that any fixed point of g1 or g2 is a root of the equation x3 + 4x2 − 5 = 0.
  • Suppose that we take x0 sufficiently close to 1 (but not equal to 1) and would like to generate a sequence {xn} by the fixed-point iteration xn = g1(xn−1) (n ≥ 1) or by the fixed-point iteration xn = g2(xn−1) (n ≥ 1). If you wish to have faster convergence (to the point 1), which of the two fixed-point iterations would you choose? Justify your answer. (You need to make your choice by a theoretical result, not by comparing the first few iterates.)
  1. (6 marks) Let g(x) = (2x2 + 3)1/4.
    • Let p0 = 1. Find p10 by the fixed-point iteration

pn+1 = g(pn) (n ≥ 0).

(You should use MATLAB to do this. But just report p10 with at least 10 digits. )

  • Let = 1. Use Steffensen’s method to find and. (You should use MATLAB to do this. But just report and, with at least 10 digits. )
  • Show that √3 is a fixed point of g. Which of p10 and gives a better approximation to this fixed point?
  1. (6 marks) Let f(x) = x4 + 3x3 − 4x2 + 3x − 2.

Find f(1.6) and f(1.6) by Horner’s method using a table form (with the help of a calculator).

  1. (5 marks) Let f(x) = x3 x−1 and x0 = 1.1,x1 = 1.2,x2 = 1.3 be approximations to a solution of f(x) = 0. Use Mu¨ller’s method to find the next approximation x3.
  2. (5 marks) Suppose that Newton’s method is applied to find the solution p = 0 of the equation = 0. It is known that, starting with any p0 > 0, the sequence {pn} produced by the Newton’s method is monotonically decreasing (i.e., p0 > p1 > p2 > ··) and converges to 0.

Prove that {pn} converges to 0 linearly with rate 2/3. (hint: You need to have the patience to use L’Hospital rule repeatedly. )

Improve Your Grades with Custom Writing Help
Homework Help
Writing Help
Editing Services
Plagiarism check
Proofreading services
Research Project help
Custom writing services
E learning blogs

Disclaimer : The study tools and academic assistance/guidance through online tutoring sessions provided by AssignmentHelp.Net is to help and enable students to compete academically. The website does not provide ghostwriting services and has ZERO TOLERANCE towards misuse of the services. In case any user is found misusing our services, the user's account will be immediately terminated.