The convert/ratpoly function converts a series to a rational polynomial (rational function). If the first argument is a Taylor or Laurent series then the result is a Pade approximation, and if it is a Chebyshev series then the result is a Chebyshev-Pade approximation.

The first argument must be either of type 'laurent' (hence a Laurent series) or else a Chebyshev series (represented as a sum of products in terms of the basis functions $T\left(k\,x\right)$ for integers k).

If the third and fourth arguments appear, they must be integers specifying the desired degrees of numerator and denominator, respectively. (Note:  The actual degrees appearing in the approximant may be less than specified if there exists no approximant of the specified degrees.)

If the third and fourth arguments are not specified then the degrees of numerator and denominator are chosen to be m and n, respectively, such that $m+n+1=\mathrm{order}\left(\mathrm{series}\right)$ and either $m=n$ or $m=n+1.$ (The order of a Chebyshev series is defined to be $d+1$ where d is the highest-degree term which appears.)

For the Pade case, two different algorithms are implemented. For the pure univariate case where the coefficients contain no indeterminates and no floating-point numbers, a ``fast'' algorithm due to Cabay and Choi is used. Otherwise, an algorithm due to Geddes based on fraction-free symmetric Gaussian elimination is used.

For the Chebyshev-Pade case, the method used is based on transforming the Chebyshev series to a power series with the same coefficients, computing a Pade approximation for the power series, and then converting back to the appropriate Chebyshev-Pade approximation.

$\mathrm{series}\left({\ⅇ}^x\,x\right)$

$1+x+\frac{1}{2}{x}^2+\frac{1}{6}{x}^3+\frac{1}{24}{x}^4+\frac{1}{120}{x}^5+O\left(x^6\right)$

$\mathrm{convert}\left(\,\mathrm{ratpoly}\right)$

$\frac{1+\frac{3}{5}x+\frac{3}{20}{x}^2+\frac{1}{60}{x}^3}{1-\frac{2}{5}x+\frac{1}{20}{x}^2}$

$\mathrm{Digits}≔5\:$

${\mathrm{numapprox}}_{\mathrm{chebyshev}}\left(\cos \left(x\right)\,x\right)$

$0.76520T\left(0\,x\right)-0.22981T\left(2\,x\right)+0.0049533T\left(4\,x\right)-0.000041877T\left(6\,x\right)$

$\mathrm{convert}\left(\,\mathrm{ratpoly}\,2\,2\right)$

$\frac{0.76025T\left(0\,x\right)-0.19673T\left(2\,x\right)}{T\left(0\,x\right)+0.043088T\left(2\,x\right)}$

Cabay, S., and Choi, D. K. "Algebraic Computations of Scaled Pade Fractions." SIAM J. Comput. Vol. 15(1), (Feb. 1986): 243-270.

Geddes, K. O. "Block Structure in the Chebyshev-Pade Table." SIAM J. Numer. Anal. Vol. 18(5), (Oct. 1981): 844-861.

Geddes, K. O. "Symbolic Computation of Pade Approximants." ACM Trans. Math. Software, Vol. 5(2), (June 1979): 218-233.