N-th Order Linear Differential Equations 193If (3) x1 > fa, then fax1 >a and fa> a/x 1.
Case (2) states if x 1 is less than fa, then a/x 1 is greater than fa. Whilecase (3) states if x1 is greater than fa, then a/ x 1 is less than fa. In either
case, both x1 and a/x1 are approximations of fa. One is an under estimate
and the other is an over estimate. Notice that the product of x 1 and a/x 1 is
a. Let x2 denote the average of x1 and a/x 1. That is, let
Since x 1 and a/x 1 are estimates of fa, their average x 2 is also an estimate
of fa. As before, either x2 = fa, or x2 < fa, or x2 > fa. And if x 2 -1-fa,
thenwill be a new approximation of fa. Hence, we obtain the following iteration
procedure for calculating an approximate value for fa: Make an initial guess
x 1 > 0 which approximates fa. Then successively compute
Xn+l = (xn + a/xn)/2
for n = 1, 2, 3, ... until Xn+i is as good an approximation to fa as desired.
Neither we nor the Babylonians proved that for any initial guess x 1 > 0, as
n ---+ +oo, Xn ---+ fa; but it does.
We should also note that the ancient Babylonians knew how to compute
approximate roots of certain cubic polynomials.
In the third century A.D., Chinese mathematicians developed a method
known as fan fa for computing approximate roots of polynomials. In 1303,
Chu Shih-chieh computed roots of polynomials up to degree fourteen using
the method of fan fa. Unaware that the method of fan fa had been invented
in China and in use for nearly fifteen centuries, the English mathematician
W. G. Horner published the equivalent of this method in 1819. In the West
the method is called "Homer's method."
After inventing the calculus, Isaac Newton used the derivative of a function
in a procedure which he invented in 1669 to iteratively calculate approximate
roots of the function. As far as we know, Newton used his method, Newton's
method, to find only the positive root of just one polynomial, x^3 - 2x - 5.
Joseph Raphson simplified and improved Newton's method and published a
new version of the method in 1690. It is the Newton-Raphson method which
is currently often used to find roots of equations. Unfortunately, nineteenth
century textbook writers ignored Raphson's contribution and the Newton-
Raphson method is commonly called Newton's method today.
The Newton-Raphson method for finding a root of a differentiable function