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A fast numerical algorithm for the integration of rational functions

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Abstract

A new iterative method for high-precision numerical integration of rational functions on the real line is presented. The algorithm transforms the rational integrand into a new rational function preserving the integral on the line. The coefficients of the new function are explicit polynomials in the original ones. These transformations depend on the degree of the input and the desired order of the method. Both parameters are arbitrary. The formulas can be precomputed. Iteration yields an approximation of the desired integral with mth order convergence. Examples illustrating the automatic generation of these formulas and the numerical behaviour of this method are given.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Virginia Wesleyan College, Norfolk, VA, 23502, USA

    Dante V. Manna

  2. Department of Mathematics, Rutgers University, New Brunswick, NJ, 08854-8019, USA

    Luis A. Medina

  3. Department of Mathematics, Universidad de Puerto Rico, San Juan, PR, 00931, USA

    Luis A. Medina

  4. Department of Mathematics, Tulane University, New Orleans, LA, 70118, USA

    Victor H. Moll & Armin Straub

Authors
  1. Dante V. Manna
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  2. Luis A. Medina
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  3. Victor H. Moll
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  4. Armin Straub
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Correspondence to Dante V. Manna.

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Manna, D.V., Medina, L.A., Moll, V.H. et al. A fast numerical algorithm for the integration of rational functions. Numer. Math. 115, 289–307 (2010). https://doi.org/10.1007/s00211-009-0284-9

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  • DOI: https://doi.org/10.1007/s00211-009-0284-9

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