11—Numerical Analysis 320W is a factor that can be chosen greater than one to increase the correction or less than one to decrease it.
Which one to do is more an art than a science (1.5 and 0.5 are common choices). You can easily verify that any
choice ofwbetween 0 and 2/3 will cause convergence for the solution ofx^1 /^3 = 0. You can also try this method
on the solution off(x) =x^2 = 0. A straight-forward Newton method will certainly converge, but with painful
slowness. The choice ofw > 1 improves this considerably.
When Newton’s method works well, it will typically double the number of significant figures at each iteration.
A drawback to Newton’s method is that it requires knowledge off′(x), and that may not be simple. An
alternate approach that avoids this starts from the picture in which a secant through the curve is used in place
of a tangent at a point.
Givenf(x 1 )andf(x 2 ), construct a straight liney−f(x 2 ) =[
f(x 2 )−f(x 1 )
x 2 −x 1]
(x−x 2 ).
x 1 x 2This has its root aty= 0, or
x=x 2 −f(x 2 )x 2 −x 1
f(x 2 )−f(x 1 ). (7)
This root is taken asx 3 and the method is iterated, substitutingx 2 andx 3 forx 1 andx 2. As with Newton’s
method, when it works, it works very well, but you must look out for the same type of non-convergence problems.
This is called the secant method.
11.3 Differentiation
Given tabular or experimental data, how can you compute its derivative?
Approximating the tangent by a secant, a good estimate for the derivative off at the midpoint of the
(x 1 ,x 2 )interval is
[
f(x 2 )−f(x 1 )]
/(x 2 −x 1 )