Page 74 - Numerical Methods for Chemical Engineering

Newton’s method for a single equation 63

where the truncation error is
n
1 d f n
R n (x) = (x − x 0 ) for some ζ ∈ [x 0 , x] (2.13)
n! dx n ζ
When |x − x 0 | is very small,
2 3
|x − x 0 | |x − x 0 | |x − x 0 | · · · (2.14)
and we expect that, unless the higher-order derivatives become very large at x 0 , the truncated
expansion should be a reasonable description of the local behavior of f (x) near x 0 . The
smaller the truncation order, in general, the nearer that we will have to be to x 0 for the
approximation to be accurate.

Newton’s method for a single equation

We now use this truncated Taylor series to develop an iterative technique for solving a non-
linear algebraic equation f (x) = 0 known as Newton’s method. To start Newton’s method,
we make an initial guess x [0] of the solution that we hope is close to the true value x s where
[0]
f (x s ) = 0. We use a Taylor series to approximate f (x) in the vicinity of x ,
2

[0] df [0] 1 d f [0] 2

f (x) = f x + x − x + x − x + ··· (2.15)
dx 2! dx 2
x [0] x [0]
At the solution, f (x s ) = 0, and the Taylor series yields
2

[0] df [0] 1 d f [0] 2

0 = f x + x s − x + x s − x + ··· (2.16)
dx 2! dx 2
x [0] x [0]
Now, if x [0] is sufficiently close to x s , then
2 3

x s − x [0] x s − x [0] x s − x [0] ··· (2.17)
[0]
In this case, as long as the first derivative is nonzero at x , we obtain a reasonable approx-
[1]
imation of the solution, x , from the rule

[0] df [1] [0]
0 = f x + x − x (2.18)
dx
x [0]
Successive application of this rule yields Newton’s method for solving a single nonlinear
algebraic equation,
[k]
f x
x [k+1] = x [k] − (2.19)
f (1) x [k]
where we have used the notation f (m) (x) for the mth derivative of f (x). The iterations are
stopped when the function value satisfies

f x [k] and/or f x [k] ≤ δ rel f x [0] (2.20)
≤ δ abs