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Chebfun and numerical quadrature. (English) Zbl 1257.41018

Summary: Chebfun is a Matlab-based software system that overloads Matlab’s discrete operations for vectors and matrices to analogous continuous operations for functions and operators. We begin by describing Chebfun’s fast capabilities for Clenshaw-Curtis and also Gauss-Legendre, -Jacobi, -Hermite, and -Laguerre quadrature, based on algorithms of Waldvogel and Glaser, Liu and Rokhlin. Then we consider how such methods can be applied to quadrature problems including 2D integrals over rectangles, fractional derivatives and integrals, functions defined on unbounded intervals, and the fast computation of weights for barycentric interpolation.

MSC:

41A55 Approximate quadratures
97N80 Mathematical software, computer programs (educational aspects)
26A33 Fractional derivatives and integrals

Software:

Chebfun; Matlab
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Full Text: DOI Link

References:

[1] Assheton P. Comparing Chebfun to Adaptive Quadrature Software. MS Thesis, Mathematical Modelling and Scientific Computing, Oxford University, 2008
[2] Berrut J P, Trefethen L N. Barycentric Lagrange interpolation. SIAM Rev, 2004, 46: 501–517 · Zbl 1061.65006 · doi:10.1137/S0036144502417715
[3] Clenshaw C W, Curtis A R. A method for numerical integration on an automatic computer. Numer Math, 1960, 2: 197–205 · Zbl 0093.14006 · doi:10.1007/BF01386223
[4] Espelid T O. Doubly adaptive quadrature routines based on Newton-Cotes rules. BIT Numer Math, 2003, 43: 319–337 · Zbl 1034.65017 · doi:10.1023/A:1026087703168
[5] Espelid T O. Extended doubly adaptive quadrature routines. Tech Rep 266. Department of Informatics, University of Bergen · Zbl 1034.65017
[6] Gentleman W M. Implementing Clenshaw-Curtis quadrature I and II. J ACM, 1972, 15: 337–346 · Zbl 0234.65024 · doi:10.1145/355602.361310
[7] Glaser A, Liu X, Rokhlin V. A fast algorithm for the calculation of the roots of special functions. SIAM J Sci Comp, 2007, 29: 1420–1438 · Zbl 1145.65015 · doi:10.1137/06067016X
[8] Golub G H, Welsch J H. Calculation of Gauss quadrature rules. Math Comp, 1969, 23: 221–230 · Zbl 0179.21901 · doi:10.1090/S0025-5718-69-99647-1
[9] Gonnet P. Increasing the reliability of adaptive quadrature using explicit interpolants. ACM Trans Math Softw, 2010, 37: 26:2–26:32 · Zbl 1364.65061
[10] Gonnet P. Battery test of Chebfun as an integrator. http://www.maths.ox.ac.uk/chebfun/examples/quad , 2010
[11] Hale N, Townsend A. Fast and accurate computation of Gauss-Jacobi nodes and weights. In preparation, 2012 · Zbl 1270.65017
[12] Higham N J. The numerical stability of barycentric Lagrange interpolation. IMA J Numer Anal, 2004, 2: 547–556 · Zbl 1067.65016 · doi:10.1093/imanum/24.4.547
[13] Octave software. http://www.octave.org/
[14] O’Hara H, Smith F J. Error estimation in the Clenshaw-Curtis quadrature formula. Comput J, 1968, 11: 213–219 · Zbl 0165.17901 · doi:10.1093/comjnl/11.2.213
[15] Oldham K B, Spanier J. The Fractional Calculus: Integrations and Differentiations of Arbitrary Order. New York-London: Academic Press, 1974 · Zbl 0292.26011
[16] Richardson M. Approximating Divergent Functions in the Chebfun System. MS Thesis, Mathematical Modelling and Scientific Computing, Oxford University, 2009
[17] Salzer H E. Lagrangian interpolation at the Chebyshev points x n,{\(\nu\)} = cos({\(\nu\)}{\(\pi\)}/n), {\(\nu\)} = 0(1)n; some unnoted advantages. Computer J, 1972, 15: 156–159 · Zbl 0242.65007 · doi:10.1093/comjnl/15.2.156
[18] Samko S G, Kilbas A A, Marichev O I. Fractional Integrals and Derivatives. New York: Gordon and Breach, 1993 · Zbl 0818.26003
[19] Szego G. Orthogonal Polynomials. Providence, RI: Amer Math Soc, 1939
[20] Trefethen L N. Is Gauss quadrature better than Clenshaw-Curtis. SIAM Rev, 2008, 50: 67–87 · Zbl 1141.65018 · doi:10.1137/060659831
[21] Trefethen L N. Six myths of polynomial interpolation and quadrature. Math Today, 2011, 47: 184–188
[22] Trefethen L N. Approximation Theory and Approximation Practice. Philadelphia: SIAM, in press
[23] Trefethen L N, et al. Chebfun Version 4.0, 2011, http://www.maths.ox.ac.uk/chebfun/
[24] Waldvogel J. Fast construction of the Fejér and Clenshaw-Curtis quadrature rules. BIT Numer Math, 2006, 46: 195–202 · Zbl 1091.65028 · doi:10.1007/s10543-006-0045-4
[25] Wang H, Xiang S. On the convergence rates of Legendre approximation. Math Comp, 2012, 81: 861–877 · Zbl 1242.41016 · doi:10.1090/S0025-5718-2011-02549-4
[26] Winston C. On mechanical quadratures formulae involving the classical orthogonal polynomials. Ann Math, 1934, 35: 658–677 · JFM 60.0294.01 · doi:10.2307/1968756
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