Homework 7 Solution

Homework 7 Solution - Lecture 7 MEEN 357 Partial Homework...

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Lecture 7 MEEN 357 Partial Homework Solution 1 Lecture-07-Homework-Solution-2009.doc RMB 1. Problem 6.12 of the textbook. You are to use the Newton Raphson method to find the real root of the polynomial () 432 0.0074 0.284 3.355 12.183 5 fx x x x x =− + + (7.1) in the interval [ ] 15,20 . You are given the initial guess of 0 16.15 x = . In addition, you are asked to use MATLAB to calculate all of he roots. Solution : The first part of the problem is essentially a repeat, as far as method is concerned, of the second problem of the homework for Lecture 6. The following table represents a hand solution. i x i f(x i ) f '(x i ) ε a 0 16.15 -9.57445 -1.3536821 1 9.077102 8.678763 0.662596499 77.92% 2 -4.02101 128.6318 -54.86395947 325.74% 3 -1.67645 36.24995 -25.96598744 139.85% 4 -0.2804 8.686147 -14.13209509 497.89% 5 0.334244 1.292213 -10.03430353 183.89% 6 0.463023 0.050416 -9.25583635 27.81% 7 0.46847 8.81E-05 -9.223505336 1.16% Newton Raphson Method: Problem 6.16 on Page 163 f=0.0074x^4-0.284x^3+3.355x^2-12.183x+5 Clearly, the iteration did not work with the suggested starting point. We shall see that the method jumped to one of the other roots, namely 0.46847 r x = . If, contrary to the book’s advice, we start at 18 r x = , the result is i x i f(x i ) f '(x i ) ε a 0 18 -6.7396 5.1762 1 19.30204 4.646969 12.76866067 6.75% 2 18.9381 0.443455 10.36900748 1.92% 3 18.89533 0.005729 10.1016126 0.23% 4 18.89477 1E-06 10.09808721 0.00% 5 18.89477 -2.6E-13 10.09808659 0.00% 6 18.89477 1.14E-13 10.09808659 0.00% 7 18.89477 -8.5E-14 10.09808659 0.00% Newton Raphson Method: Problem 6.16 on Page 163 f=0.0074x^4-0.284x^3+3.355x^2-12.183x+5 which is a root in the specified range. The fact that the iteration scheme might converge to a different root from that sought, was discussed on page 147 of the textbook.
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This note was uploaded on 07/28/2010 for the course MEEN 357 taught by Professor Anamalai during the Fall '07 term at Texas A&M.

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Homework 7 Solution - Lecture 7 MEEN 357 Partial Homework...

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