Using of Padé approximations in mathematical and physical geodesy

Besides the widely used Taylor expansion for analytical functions there is more complex form to represent some different expansions using the rational function [P^L/Q^M] in the form of relation of the two polynomials P and Q of degrees L for numerator and M for denominator.
The convergence rate of such an approximation is faster than of the ordinary Taylor expansion, while the coefficients of Padé approximation are calculated based on the Taylor expansion.

For example, let show the possibilities of the Padé approximations for expansion of the elliptical integrals, let's consider the well-known length X of an meridian arc from equator to geodetic latitude B on the reference ellipsoid with semi-major axis a and eccentricity e.

After standard Taylor expansion and integration we obtain expansion up to 11th degree:

 

 

The corresponding Padé approximation:

 

 

Regardless of the coefficients in front of the powers of the cosine of the latitude, 
we can see that the maximum order of the cosine of the latitude reaches only 4.

Taking for example the ellipsoid WGS-84 we get the length of meridian arc form equator to latitude 89 degrees:

precise (numerical integration): 9890270.31374637 m,

by formula (*):          9890270.31374637 m,

by formula (**):         9890270.31374636 m.

We see that using of the two polynomials of lower degree (max 4) provide the same accuracy than the usual expansion up to 11 degree!

There are possibility to develop the method using special type of Padé approximations for the

- orthogonal functions, and in particular

- orthogonal polynomials.

In physical geodesy the Padé approximations could be used in the 

- representing of the normal field characteristics, expanded into Taylor series, e.g. length of the coordinate line of the spheroidal system used in normal height calculation,

- gravity field modelling using Padé approximations with orthogonal functions,

- solving of the integral Fredholm equation of the second type by successive approximations.

The deficiencies of this method are related to the poles - points where the denominator turns into zero.

Literature:

G.A. Baker, P. Graves-Morris. Padé approximations. Part 1: Basic theory. Encyclopedia of Mathematics and its applications, Addison-Wesley, Reading, 1981.