Mathematical Tools for Physics

11—Numerical Analysis 319

Such iterative procedures are ideal for use on a computer, but use them with caution, as a simple example
shows:
f(x) =x^1 /^3.

Instead of the rootx= 0, the iterations in this first graph carry the supposed solution infinitely far away. This
happens here because the higher derivatives neglected in the straight line approximation are large near the root.

A milder form of non-convergence can occur if at the root the curvature changes sign and is large, as in
the second graph. This can lead to a limit cycle where the iteration simply oscillates from one side of the root to
the other without going anywhere.
A non-graphical derivation of this method starts from a Taylor series: Ifz 0 is an approximate root and
z 0 +is a presumed exact root, then

f(z 0 +) = 0 =f(z 0 ) +f′(z 0 ) +···.

Neglecting higher terms then,

=−f(z 0 )/f′(z 0 ), and z 1 =z 0 +=z 0 −f(z 0 )/f′(z 0 ), (5)

as before. I usezinstead ofxthis time to remind you that this method is just as valid for complex functions as
for real ones (and has as many pitfalls).
There is a simple variation on Newton’s method that can be used to speed convergence where it is poor or
to bring about convergence where the technique would otherwise break down.

x 1 =x 0 −wf(x 0 )/f′(x 0 ). (6)