rational polynomials - Maple Help

convert/ratpoly

convert series to a rational polynomial

 Calling Sequence convert(series, ratpoly, numdeg, dendeg)

Parameters

 series - series; type 'laurent' or a Chebyshev series numdeg - integer; specify numerator degree dendeg - integer; specify denominator degree

Description

 • 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.

Examples

 > $\mathrm{series}\left(\mathrm{exp}\left(x\right),x\right)$
 ${1}{+}{x}{+}\frac{{1}}{{2}}{}{{x}}^{{2}}{+}\frac{{1}}{{6}}{}{{x}}^{{3}}{+}\frac{{1}}{{24}}{}{{x}}^{{4}}{+}\frac{{1}}{{120}}{}{{x}}^{{5}}{+}{O}{}\left({{x}}^{{6}}\right)$ (1)
 > $\mathrm{convert}\left(,\mathrm{ratpoly}\right)$
 $\frac{{1}{+}\frac{{3}}{{5}}{}{x}{+}\frac{{3}}{{20}}{}{{x}}^{{2}}{+}\frac{{1}}{{60}}{}{{x}}^{{3}}}{{1}{-}\frac{{2}}{{5}}{}{x}{+}\frac{{1}}{{20}}{}{{x}}^{{2}}}$ (2)
 > $\mathrm{Digits}≔5:$
 > $\mathrm{numapprox}\left[\mathrm{chebyshev}\right]\left(\mathrm{cos}\left(x\right),x\right)$
 ${0.76520}{}{T}{}\left({0}{,}{x}\right){-}{0.22981}{}{T}{}\left({2}{,}{x}\right){+}{0.0049533}{}{T}{}\left({4}{,}{x}\right){-}{0.000041877}{}{T}{}\left({6}{,}{x}\right)$ (3)
 > $\mathrm{convert}\left(,\mathrm{ratpoly},2,2\right)$
 $\frac{{0.76025}{}{T}{}\left({0}{,}{x}\right){-}{0.19673}{}{T}{}\left({2}{,}{x}\right)}{{T}{}\left({0}{,}{x}\right){+}{0.043088}{}{T}{}\left({2}{,}{x}\right)}$ (4)

References

 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.