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128ahw4sum10

# 128ahw4sum10 - MATH 128A SUMMER 2010 HOMEWORK 4 SOLUTION...

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MATH 128A, SUMMER 2010, HOMEWORK 4 SOLUTION BENJAMIN JOHNSON Homework 4: Due Wednesday, July 7 2.6; 1c, 4h 3.1; 1c, 3c, 12 section 2.6 1. Find the approximations to within 10 - 4 of all the real zeros of the following polynomials using Newton’s method. c. f ( x ) = x 3 - x - 1 solution: To apply Newton’s method, we need to do a fixed-point iteration on the func- tion g ( x ) = x - f ( x ) f 0 ( x ) = x - x 3 - x - 1 3 x 2 - 1 . It remains to choose the starting points. Let’s start with x 0 = 1. If we do this, we generate the sequence h p n i = h 1 , 1 . 5000 , 1 . 3478 , 1 . 3252 , 1 . 3247 i ; and 1.3247 is correct to within 10 - 4 . [Ideally we should have made an effort to compute all the values f ( p n ) and f 0 ( p n ) using Horner’s method, but since this is such a simple function, and my computer is fast, I did not bother to do that.] The remaining roots are complex, and to get these we can either divide f ( x ) by ( x - 1 . 3247) and use the quadratic formula, or we can continue using Newton’s method by starting with a complex number. The latter strategy is easiest if we have already written a function in MATLAB for Newton’s method. In any case the additional complex roots are - 0 . 6624 + 0 . 5623 i and - 0 . 6624 + 0 . 5623 i .

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128ahw4sum10 - MATH 128A SUMMER 2010 HOMEWORK 4 SOLUTION...

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