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convert/ratpoly

convert series to a rational polynomial

 

Calling Sequence

Parameters

Description

Examples

References

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 Tk,x 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=orderseries 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

seriesⅇx,x

1+x+12x2+16x3+124x4+1120x5+Ox6

(1)

convert,ratpoly

1+35x+320x2+160x3125x+120x2

(2)

Digits5:

numapprox[chebyshev]cosx,x

0.76520T0,x0.22981T2,x+0.0049533T4,x0.000041877T6,x

(3)

convert,ratpoly,2,2

0.76025T0,x0.19673T2,xT0,x+0.043088T2,x

(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.

See Also

convert

convert/confrac

series

type/laurent

 


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