Newton Method

[Namir]

Hello Dieter,

Last year I was reading "Practical Numerical Methods for Chemical Engineers", third edition, by Richard Davis. When this author discusses Newton's method he uses the following very interesting variant on page 195:

x(i) = x(i-1) - f(x(i-1)) * f'(x(i-1)( / ([f'(x(i-1)]^2 + delta)

Davis uses delta = 1e-5 to protected against the slope f'(x) (approximated using any one of our techniques) being zero. The expression ([f'(x(i-1)]^2 + delta) is always positive.I am guessing that the delta term also stabilizes convergence as f;(x) gets very small too.

The above equation requires a few more calculations and probably one more storage register/variable, but adds more stability.