On 06/02/2013 17:42, Phil Hobbs wrote:

You can often use it to derive a perturbation series for those times
when linearizing isn't quite enough.

I highly recommend Bender & Orszag's "Advanced Mathematical Methods for
Scientists and Engineers", which has all that sort of stuff--boundary
layer theory, steepest descents, and a whole lot of other asymptotic
methods. Arfken's applied math book is good as well--it used to be the
standard textbook for physics undergraduates.

Our local choice was Matthews & Walker Mathematical Methods of Physics.

It's often possible to derive a series solution, which may or may not
converge, and then convert to a continued fraction or apply convergence
tricks such as Shanks's algorithm or Richardson extrapolation. When
those work, which isn't always, they can be practically supernatural.

Always used to be popular with the turbulent flow brigade - often gave
divergent power series that could only be tamed with Shank's. The pure
mathematicians used to cringe at the abuse of method but they could not
deny that the results it predicted matched experimental data well!

Works best on alternating series with poor convergence or divergence!

I was once very interested in extended convergence tricks and have used
the rational approximation for Log(1+x) in anger several times.

Log(1+x) = x(6+x)/(6+4x) where -1/2 x 1

The other common one sometimes useful is

Sqrt(x) = (1+3x)/(3+x) where -1/2 x 2

Only really any good if you have fast hardware divide.

These tricks sometimes allow a rough polynomial approximation soluble
analytically to provide a much better initial input guess for more rapid
convergence of an iterative method like N-R.

--
Regards,
Martin Brown