C&EE 103 Applied Numerical Computing and Modeling Spring 2017 Homework #2 Due at noon on Wednesday, April 19, 2017 Problem 1. Use the bisection method (" D 10 4 , a1 D 0, b1 D 2), Newton's method (" D 10 6 , x0 D 2 in Part a), x0 D 1 in Part b)), and the secant method (" D 10 6 , x0 D 0, x1 D 2) with a hand calculator or computer to find the indicated roots of the following equations for the given tolerances ". a) The real root of x 3 x2 x b) The smallest positive root of e 1 D 0. x Problem 2. Consider the function D sin x. 3 f .x/ D cos.2x/ C x 2 1; which has exactly one root D 0. a) For the given function f .x/, one possible start value x0 0 for Newton's method shall be determined so that the root D 0 can be found exactly with one iteration only. Turn this problem statement into a rootfinding problem of the form g.x/ D 0 with unknown root D x0 . b) Find one root D x0 of the function g.x/ as derived in Part a) using a modified Newton's method according to the modified general iteration formula xnC1 D xn g.xn / ; g0 .xN 0 / where xN 0 D \u00192 is the start value for the modified Newton's method (i.e. xN 0 x0 /. Perform two iterations to determine the value of x2 \u0019 . c) Give one advantage and one disadvantage of the modified Newton's method used in Part b) compared to (the original version of) Newton's method. Problem 3. Consider the function f .x/ D e x x 4 C xe x ; which has three real roots. a) Use Newton's method with a tolerance " D 10 12 and initial guesses x0 D 1, x0 D 2:5, and x0 D 5 to find all three real roots with the provided M ATLAB code. b) Modify Newton's method so that it is based on a quadratic Taylor polynomial p2 .x/ (rather than on .1/ a linear Taylor polynomial p1 .x/) about xn . Verify that the two real roots of p2 .x/, given by xnC1 .2/ and xnC1 , can be determined by .1/;.2/ xnC1 D xn f 0 .xn / p f 0 .xn /2 2f .xn /f 00 .xn / : f 00 .xn / .1/ .2/ Moreover, find a condition that needs to be satisfied to find the real roots xnC1 and xnC1 of p2 .x/, i.e. the roots should not be complex numbers. 1 C&EE 103 Applied Numerical Computing and Modeling Spring 2017 c) Write a M ATLAB code to implement the modified Newton's method based on the real roots of p2 .x/ as found in Part b). To this end, do the following: 1. Initialize n D 0. 2. Check the residual stopping criterion and stop the algorithm if the stopping criterion is satisfied. 3. Based on xn , determine xnC1 . If p2 .x/ has no real roots, determine xnC1 as the root of p1 .x/ (as obtained by the original version of Newton's method). If p2 .x/ has two real roots, choose the root that is closer to the root of p1 .x/ (as obtained by the original version of Newton's method). 4. Set n D n C 1 and return to 2. d) Use the modified Newton's method from Parts b) and c) with a tolerance " D 10 12 and initial guesses x0 D 1, x0 D 2:5, and x0 D 5 to find all three real roots with the M ATLAB code from Part c). e) Give one advantage and two disadvantages of the modified Newton's method used in Parts b) and c) compared to (the original version of) Newton's method. Turn in your M ATLAB code for this problem. Remember to copy all your files into a folder labeled LastnameFirstnameHW2, zip that folder, and upload it to CCLE. You are also required to turn in a hardcopy of all relevant materials including a print-out of your code. 2