164 5. Approximation and Location of Zeros
0)
(ii)
(iii) Y = P(X)
*X
- Let c # 1. Show that Newton’s Method applied to the polynomial
tc - 1 and any first approximation yields the zero c-l in one step.
With a diagram, explain why this occurs. - (a) Let c > 0. Sh ow that Newton’s Method applied to the poly-
nomial t2 - c yields from any positive approximation al, the
sequence {a,,} of successive approximations to 4, where
1
a,+1 = --(a, 2 + c/a,) (n 2 1).
(b) Find an expression which relates the difference ai+r - c to the
difference u: - c. Argue that, as n becomes larger, a, gets closer
and closer to JE. Show that {a,.,} is decreasing for n 2 2.
- Determine to three decimal places all of the real zeros of the polyno-
mial 3t4 - 2t3 - t2 - 3t + 1. Use all of the methods discussed in these
exercises. - Newton’s Method for successive approximations to the root of an
equation is a particular instance of a more general approach:
Given an equation p(t) = 0, derive from it an equation of
theformt = f(t) with which it shares a solution. Pick a first