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 lowest degree $v$ appearing in the series is negative, then the denominator of every rational approximation has degree at least $-v$, and an error is raised if $\mathrm{dendeg}+v<0$. If $v\&gt;\mathrm{numdeg}\ge 0$, the return value is $0$.

If the third and fourth arguments are not specified, then if $v=0$ 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 $m=n+1$ (otherwise, if $v\&gt;0$, then always $m\ge v$, and if $v<0$, then $n\ge -v$ and $m+n+1+v=\mathrm{order}\left(\mathrm{series}\right)$). 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.

$s≔\mathrm{series}\left(\exp \left(x\right)\&comma;x\right)$

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

$\mathrm{convert}\left(s\&comma;\mathrm{ratpoly}\right)$

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

$\mathrm{convert}\left(s\&comma;\mathrm{ratpoly}\&comma;2\&comma;3\right)$

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

$\mathrm{convert}\left(s\&comma;\mathrm{ratpoly}\&comma;3\&comma;3\right)$

Error, (in `convert/ratpoly`) series order too small for specified degrees

$t≔\mathrm{series}\left(\exp \left(x\right)x^4\&comma;x\&comma;7\right)$

$t≔x^4+x^5+\frac{1}{2}{x}^6+O\left(x^7\right)$

$\mathrm{convert}\left(t\&comma;\mathrm{ratpoly}\&comma;3\&comma;3\right)$

$0$

$\mathrm{convert}\left(t\&comma;\mathrm{ratpoly}\&comma;4\&comma;2\right)$

$\frac{x^4}{1-x+\frac{1}{2}{x}^2}$

$\mathrm{convert}\left(t\&comma;\mathrm{ratpoly}\right)$

$\frac{x^4}{1-x+\frac{1}{2}{x}^2}$

$u≔\mathrm{series}\left(\frac{\exp \left(x\right)}{x^3}\&comma;x\right)$

$u≔x^{\mathrm{-3}}+x^{\mathrm{-2}}+\frac{1}{2}{x}^{\mathrm{-1}}+\frac{1}{6}+\frac{1}{24}x+\frac{1}{120}{x}^2+O\left(x^3\right)$

$\mathrm{convert}\left(u\&comma;\mathrm{ratpoly}\&comma;2\&comma;3\right)$

$\frac{1+x+\frac{1}{2}{x}^2}{x^3}$

$\mathrm{convert}\left(u\&comma;\mathrm{ratpoly}\&comma;3\&comma;2\right)$

Error, (in `convert/ratpoly`) no rational approximation with denominator degree <= 2

$\mathrm{convert}\left(u\&comma;\mathrm{ratpoly}\right)$

$\frac{1+\frac{4}{5}x+\frac{3}{10}{x}^2+\frac{1}{15}{x}^3+\frac{1}{120}{x}^4}{-\frac{1}{5}{x}^4+x^3}$

$\mathrm{Digits}≔5\&colon;$

$\mathrm{numapprox}\left[\mathrm{chebyshev}\right]\left(\cos \left(x\right)\&comma;x\right)$

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

$\mathrm{convert}\left(\&comma;\mathrm{ratpoly}\&comma;2\&comma;2\right)$

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

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.