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Numerical Analysis

David Gleich

Purdue University

Spring 2016

Course number CS-51400, MATH-51400

Tuesday and Thursday, noon-1:15pm

Location Lawson B155

Homework 7

Please answer the following questions in complete sentences in a typed manuscript and submit the solution on blackboard by on April 18th at noon.

Problem 0: Homework checklist

Problem 1: Gautschi Exercise 4.19

Consider the equation

  1. Show graphically, as simply as possible, that in the interval there is exactly one root .

  2. Does Newton's method converge to a root if the initial approximation is taken to be ? Justify your answer.

Problem 2: Gautschi Exercise 4.25

Consider Newton's method for computing . Let .

  1. Show that

  2. Use (1) to show that Discuss the significance of this result with regard to the overall behavior of Newton's iteration.

Problem 3: Nonlinear systems

Express the Newton iteration for solving these systems of equations and show (analytically or experimentally) what happens if you start from .

  1. ,

  2. ,

Problem 4: Fixed point methods

Can either of the two systems above be solved via the fixed-point method? Two variations on the fixed point method to consider are:

Do either of those modifications help? (This problem requires you to implement and investigate.)