# h1 - CS 510 Homework - due 9/28/10 I. (i) Find the Taylor...

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CS 510 Homework - due 9/28/10 I. (i) Find the Taylor polynomial which matches f ( x )= x and its ﬁrst two derivatives at x =4 . (ii) Suppose the polynomial in (i) is used to approximate 5. What is the resulting approximation? Bound its absolute error using the error formula for Taylor polynomials. II. Consider the nonlinear equation f ( x x 3 - x - 11 = 0. (i) Show that f ( x ) has a unique root x [2 , 3]. (ii) How many iterations of the bisection method are needed to compute x to within absolute error 10 - 6 starting with [ a, b ]=[2 , 3]? (iii) Compute the ﬁrst three iterates of regula falsi with [2 , 3] as the initial interval. III. Consider the following ﬁxed point reformulations of the equation f ( x )=0inII: (a) x = g 1 ( x x 3 - 11 (b) x = g 2 ( x )=( x + 11) 1 / 3 (c) x = g 3 ( x x 4 - x 2 11 (d) x = g 4 ( x x - x 3 - x - 11 3 x 2 - 1 (Newton’s method) (i) For which of the above g ’s will the corresponding ﬁxed point iteration be locally convergent to x ? Explain. (ii) Show that x k +1 = g 2 ( x k ) will converge to x for all x 0 [2 , 3].

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## This note was uploaded on 10/06/2010 for the course CS CS 510 taught by Professor Richter during the Fall '10 term at Rutgers.

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h1 - CS 510 Homework - due 9/28/10 I. (i) Find the Taylor...

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