Question

2. (a) Explain Newton's Method, which lets you improve approximations to roots of a function f(x) by following the tangent line down to the x-axis.

(b) What if, instead of following a best fit straight line, you were to follow a best fit parabola? What's the equation of this parabola, and of its intersection with the x-axis? Compared with Newton's Method, how quickly do the approximate roots computed using this method typically converge to the exact root?

(c) The method in part (b) streaked across the sky shortly after Newton's Method did. What if, instead of using the best fit parabola, we use the best fit cubic or the best fit quartic or the best fit polynomial of degree 2019? Is this a good idea, or not?

Answer 1

In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function. It is one example of a root-finding algorithm.

The Newton–Raphson method in one variable is implemented as follows:

The method starts with a function f defined over the real numbers x, the function's derivative f ′, and an initial guess x₀ for a root of the function f. If the function satisfies the assumptions made in the derivation of the formula and the initial guess is close, then a better approximation x₁ is

{\displaystyle x_{1}=x_{0}-{\frac {f(x_{0})}{f'(x_{0})}}\,.} $x_{1}=x_{0}-{\frac {f(x_{0})}{f'(x_{0})}}\,.$

Geometrically, (x₁, 0) is the intersection of the x-axis and the tangent of the graph of f at (x₀, f (x₀)).

The process is repeated as

{\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}\,} $x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}\,$

until a sufficiently accurate value is reached.

This algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functions and to systems of equations.