Unknown

5.1. Approximation of Roots 163

unknown zero T of p(t). This method will enable us to (often) find a

better approximation to r.

(a) Use Taylor’s Theorem to argue that 0 = p(r) is approximately

equal top(u)+p’(u)(r-u), and therefore r will be approximately

equal to u - p(u)/p’(u).

Thus, we can start with an approximation ui = ‘u, and for i =

2,3,4,... , move to successive approximations ui, where

Uj = Uj-1 - P(w-l)/P’(w-1).

(b) There is a zero of the polynomial 2t6 - 7t5 + t4 + t3 - 12t2 - 5t + 1

between t = 3 and t = 4. Explain and verify the following table

created in approximating this zero; give the zero to two decimal

places.

t p(t) p’(t) p(t)lp’(t)

3 -257 139 -1.849

4.849 7594.21 13224.74 0.574

4.275 2385.60 5703.92 0.418

3.857 691.35 2673.82 0.259

3.598 160.02 1504.00 0.106

3.492 20.18 1144.69 0.018

3.474 0.073 1089.66 0.000

(c) There is also a zero of the same polynomial between 0.125 and

0.25. Using each of these values as first approximations, deter-

mine this zero to three decimal places.

(d) Use Newton’s Method to approximate the zeros of the other

polynomials in Exercise 2(a) and 2(b).

(e) We can get a geometric picture of Newton’s Method. Show that

the tangent to the graph of y = p(x) through the point (u,p(u))

intercepts the z-axis at the point (r~ - p(u)/p’(u),O). For each

of the following graphs, the initial approximation is indicated.

Indicate where the next few approximations will lie.