Historical Perspective
Not the least bit surprising, Isaac Newton
was the one to introduce the basic ideas of
Newton’s Method. However, his “method” at that time was not the Newton’s Method we use
today. Iteratively, Newton found better approximations for the solution of an equation.
He was able to find these approximations by using first-order approximations. In
Newton’s book, De Analysis per Aequationes Infinitas, he showed how to use this method
through the example of finding the solution to x3-2x-5=0 (Verbeke & Cools, 1995).
“Newton first noted that the integer part of the root is 2 setting x₀=2” (Deuflhard,
2012, p. 27). He then set x=2+p and substituted this into the original equation.
The resulting equation was: p3 + 6p2 +10p -1 = 0. He then proceeded to solve for
10p -1 = 0, using the first order term and constant. He got p= 0.1 + q. He then
substituted this value of p into p3 + 6p2 +10p -1 = 0. The resulting polynomial was
q3 + 6.3q2 +11.23q +0.061=0. He then continued this iterative process to get
q = -0.0054 + r and r = 0.00004853. He then summed all of these iterations,
x0 + p + q + r, to equal 2.09455147 as the solution to the cubic polynomial.
The algorithm Newton used did not incorporate the use of a derivative like the
final form of Newton’s Method that we use does (Deuflhard, 2012). That form
was introduced later on.
Newton’s Method is sometimes referred to as the Newton-Rahpson Method due to Joseph
Raphson’s contributions. Raphson eliminated the computing of multiple polynomials and,
instead, relied on the original polynomial for the computations (Deuflhard, 2012).
Thomas Simpson introduced the use of derivatives as a part of Newton’s Method. “He
wrote down the true iteration for one (nonpolynomial) equation and for a system of
two equations in two unknowns thus making the correct extension to systems for the
first time” (Deuflhard, 2012 p. 28). The form Simpson used is very similar to our
form of Newton’s Method. This is why some people suggest that the name Newton-Simpson
Method would be more appropriate than the name Newton-Raphson Method (Deuflhard, 2012).