Page 78 - Numerical Methods for Chemical Engineering

Newton’s method for a single equation 67

are
eads initia
cn vered ess ar t
stins rit t ind

2
stin ve rit
1
ve rit

1 ve et
ve et
2
1 2
initia ess
Figure 2.4 Update function (line) of Newton’s method for various initial guesses for f (x) = (x − 3)
(x − 2) (x − 1). Converged solutions are plotted as dots vs. the initial guess.

return the root at x = 2. This demonstrates another feature of nonlinear equations: it is
usually very difficult to identify all solutions, and while guess after guess may yield one of
some particular set of solutions, this does not guarantee that there are no other solutions
waiting to be found. We describe here a simple technique to search for additional solutions.
Let us say that we have identified the roots at x = 1 and x = 3, but want to look for
additional solutions. We can factor these roots out formally,

f (x) = (x − 1)(x − 3)g(x) (2.25)
and then apply Newton’s method to solve
f (x)
g(x) = = 0 (2.26)
(x − 1)(x − 3)
Here, because f (x) is a polynomial, we can obtain g(x) analytically. In general, this is
not possible, and g(x) must be defined as above, in a form that behaves poorly (∼0/0) at
x = 1 and x = 3. Care must be taken to avoid these regions, or to provide an upper limit
on the allowable magnitude of g(x) to avoid numerical overflow. As Figure 2.5 shows, this
technique finds the third root at x = 2 from any initial guess except close to x = 1 and
x = 3 where the algorithm terminates without convergence.

Formal convergence properties of Newton’s method for a
single equation
We now consider the convergence properties of Newton’s method more formally. We have
seen that when the initial guess is not very close to a solution, Newton’s method behaves