Difference Equations to Differential Equations (16).pdf

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 £© Section

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 £© Newton’s Method
Many problems in mathematics involve, at some point or another, solving an equation
for an unknown quantity. An equation of the form f(x) = 0 may be solved for x by
simple algebra if f is an affine function and by the quadratic formula if f is a quadratic
polynomial. There are formulas similar to the quadratic formula for both cubic and quartic
polynomials, but they are, in general, very cumbersome. One of the most interesting
results of mathematics, due to Niels Henrik Abel (1802-1829), is that there does not exist
an analogue of the quadratic formula for quintic polynomials. For this and other reasons,
it turns out that in many situations solving an equation f(x) = 0 for x requires using a
method which can approximate the solutions to a predetermined level of accuracy.
In Section we discussed one such method, the bisection algorithm, for approximat-
ing the solutions of an equation. The strong point of the bisection algorithm is that, once
an appropriate starting interval has been found, the method will always find a solution to
any desired level of accuracy; its weakness lies in the slowness with which the essive
approximations approach the solution. In this section we will discuss another method,
known as Newton’s method, for approximating solution