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Unformatted text preview: 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 Newtons method. c. f ( x ) = x 3- x- 1 solution: To apply Newtons method, we need to do a fixed-point iteration on the func- tion g ( x ) = x- f ( x ) f ( x ) = x- x 3- x- 1 3 x 2- 1 . It remains to choose the starting points. Lets start with x = 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 ( p n ) using Horners 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 Newtons method by starting with a complex number. The latter strategy is easiest if we have already written a function in MATLAB for Newtons method. In any case the additional complex roots are- . 6624 + 0 . 5623 i and- . 6624 + 0 . 5623 i . Starting from i or- i Newtons method converges to within 10- 4 of the appropriate one of these two roots after 5 iterations. 4. Repeat exercise 2 using Mullers method....
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