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hw4 - Homework 4 Foundations of Computational Math 2 Spring...

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Homework 4 Foundations of Computational Math 2 Spring 2011 Solutions will be posted Monday, 2/7/11 Problem 4.1 Suppose we want to approximate a function f ( x ) on the interval [ a,b ] with a piecewise quadratic interpolating polynomial with a constant spacing, h , of the interpolation points a = x 0 < x 1 ... < x n = b . That is, for any a x b , the value of f ( x ) is approximated by evaluating the quadratic polynomial that interpolates f at x i 1 , x i , and x i +1 for some i with x = x i + sh , x i 1 = x i - h , x i +1 = x i + h and - 1 s 1. (How i is chosen given a particular value of x is not important for this problem. All that is needed is the condition x i 1 x x i +1 .) Suppose we want to guarantee that the relative error of the approximation is less than 10 d , i.e., d digits of accuracy. Specifically, | f ( x ) - p ( x ) | | f ( x ) | 10 d . (It is assumed that | f ( x ) | is sufficiently far from 0 on the interval [ a,b ] for relative accuracy to be a useful value.) Derive a bound on h that guarantees the desired accuracy and apply
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