Experimental mathematics involving orthogonal polynomials. (English) Zbl 1215.65041
Gautschi, Walter (ed.) et al., Approximation and computation. In honor of Gradimir V. Milovanović. Most papers based on the presentations at the international conference, Niš, Serbia, August 25–29, 2008. Dordrecht: Springer (ISBN 978-1-4419-6593-6/hbk; 978-1-4419-6594-3/ebook). Springer Optimization and Its Applications 42, 117-134 (2011).
Summary: In Wikipedia, the term “experimental mathematics” is defined as follows: “Experimental mathematics is an approach to mathematics in which numerical computation is used to investigate mathematical objects and identify properties and patterns.” The ultimate goal of experimental mathematics is to encourage, and provide direction for, purely mathematical research, in the hope of thereby extending the boundaries of mathematical knowledge. It is in this spirit that we are going to deal here with a few special topics that involve orthogonal polynomials.
A key to experimental mathematics is numerical computation, and that presupposes the existence of a body of reliable computational tools that allows us to generate numerically all entities of interest. In the realm of orthogonal polynomials, we are in the fortunate position of having at disposal a number of well-tested computational techniques for this purpose, supported by a package of software in bMatlab, OPQ (Orthogonal Polynomials and Quadrature), in the public domain (http://WWW.GS-purdue.edu/archives/2002/wxg/codes). This not only enables but also encourages experimentation in this area of mathematics.
The mathematical objects we want to investigate are, on the one hand, Jacobi polynomials and, on the other hand, quadrature formulae. The properties and patterns to be identified are, in the former case, inequalities and respective domains of validity-inequalities for zeros of Jacobi polynomials and Bernstein’s inequality for Jacobi polynomials – and positivity in the latter case – positivity of Newton-Cotes formulae on zeros of Jacobi polynomials and positivity of generalized Gauss-Radau and Gauss-Lobatto formulae. We also report on experiments with Gaussian quadrature formulae corresponding to exotic weight functions, for example weight functions exhibiting super-exponential decay at infinity or densely oscillatory behavior at zero. Both are of interest in computing integral transforms that involve modified Bessel functions \(K_\nu\) of complex order \(\nu=\alpha+i\beta\).
For the entire collection see [Zbl 1203.65006].
A key to experimental mathematics is numerical computation, and that presupposes the existence of a body of reliable computational tools that allows us to generate numerically all entities of interest. In the realm of orthogonal polynomials, we are in the fortunate position of having at disposal a number of well-tested computational techniques for this purpose, supported by a package of software in bMatlab, OPQ (Orthogonal Polynomials and Quadrature), in the public domain (http://WWW.GS-purdue.edu/archives/2002/wxg/codes). This not only enables but also encourages experimentation in this area of mathematics.
The mathematical objects we want to investigate are, on the one hand, Jacobi polynomials and, on the other hand, quadrature formulae. The properties and patterns to be identified are, in the former case, inequalities and respective domains of validity-inequalities for zeros of Jacobi polynomials and Bernstein’s inequality for Jacobi polynomials – and positivity in the latter case – positivity of Newton-Cotes formulae on zeros of Jacobi polynomials and positivity of generalized Gauss-Radau and Gauss-Lobatto formulae. We also report on experiments with Gaussian quadrature formulae corresponding to exotic weight functions, for example weight functions exhibiting super-exponential decay at infinity or densely oscillatory behavior at zero. Both are of interest in computing integral transforms that involve modified Bessel functions \(K_\nu\) of complex order \(\nu=\alpha+i\beta\).
For the entire collection see [Zbl 1203.65006].
MSC:
| 65D20 | Computation of special functions and constants, construction of tables |
| 65R10 | Numerical methods for integral transforms |
| 65D32 | Numerical quadrature and cubature formulas |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 33F05 | Numerical approximation and evaluation of special functions |
| 41A55 | Approximate quadratures |
| 44A15 | Special integral transforms (Legendre, Hilbert, etc.) |
Keywords:
numerical examples; experimental mathematics; orthogonal polynomials; Jacobi polynomials; quadrature formulae; Newton-Cotes formulae; zeros of Jacobi polynomials; Gauss-Radau and Gauss-Lobatto formulae; Gaussian quadrature formulae; integral transforms; Bessel functionsSoftware:
OPQReferences:
| [1] | Askey, Richard. Email of May 13, 2008. |
| [2] | Bernstein, Serge, Sur les polynomes orthogonaux relatifs a un segment fini, J. Math., 10, 219-286 (1931) · Zbl 0003.00702 |
| [3] | Berrut, Jean-Paul and Lloyd N. Trefethen. Barycentric Lagrange interpolation, SIAM Rev. 46 (2004)(3), 501-517. · Zbl 1061.65006 |
| [4] | Chow, Yunshyong, L. Gatteschi, and R. Wong. A Bernstein-type inequality for the Jacobi polynomial, Proc. Am. Math. Soc. 121 (1994)(3), 703-709. · Zbl 0802.33010 |
| [5] | Fejér, L. Mechanische Quadraturen mit positiven Cotesschen Zahlen, Math. Z. 37 (1933), 287-309. · JFM 59.0261.03 |
| [6] | Gatteschi, Luigi. On the zeros of Jacobi polynomials and Bessel functions. In: International conference on special functions: theory and computation (Turin, 1984). Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 149-177 (1985) · Zbl 0597.33014 |
| [7] | Gautschi, Walter. Moments in quadrature problems. Approximation theory and applications, Comput. Math. Appl. 33 (1997)(1-2), 105-118. · Zbl 0930.65011 |
| [8] | Gautschi, Walter. Orthogonal polynomials: computation and approximation, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2004. · Zbl 1130.42300 |
| [9] | Gautschi, Walter. Generalized Gauss-Radau and Gauss-Lobatto formulae, BIT Numer. Math. 44 (2004)(4), 711-720. · Zbl 1076.65023 |
| [10] | Gautschi, Walter. Computing polynomials orthogonal with respect to densely oscillating and exponentially decaying weight fuinctions and related integrals, J. Comput. Appl. Math. 184 (2005)(2), 493-504. · Zbl 1076.65019 |
| [11] | Gautschi, Walter. Numerical quadrature computation of the Macdonald function for complex orders, BIT 45 (2005)(3), 593-603. · Zbl 1082.65026 |
| [12] | Gautschi, Walter. Computing the Kontorovich-Lebedev integral transforms and their inverses, BIT 46 (2006)(1), 21-40. · Zbl 1091.65128 |
| [13] | Gautschi, Walter. On a conjectured inequality for the largest zero of Jacobi polynomials, Numer. Algorithm 49 (2008)(1-4), 195-198. · Zbl 1167.33003 |
| [14] | Gautschi, Walter. On conjectured inequalities for zeros of Jacobi polynomials, Numer. Algorithm 50 (2009)(1), 93-96. · Zbl 1165.33009 |
| [15] | Gautschi, Walter. New conjectured inequalities for zeros of Jacobi polynomials, Numer. Algorithm 50 (2009)(3), 293-296. · Zbl 1168.33001 |
| [16] | Gautschi, Walter. How sharp is Bernstein’s inequality for Jacobi polynomials?, Electr. Trans. Numer. Anal. 36 (2009), 1-8. · Zbl 1189.33016 |
| [17] | Gautschi, Walter. High-order generalized Gauss-Radau and Gauss-Lobatto formulae for Jacobi and Laguerre weight functions, Numer. Algorithm 51 (2009)(2), 143-149. · Zbl 1167.65347 |
| [18] | Gautschi, Walter and Carla Giordano. Luigi Gatteschi’s work on aymptotics of special functions and their zeros, Numer. Algorithm 49 (2008)(1-4), 11-31. · Zbl 1170.26007 |
| [19] | Gautschi, Walter and Paul Leopardi. Conjectured inequalities for Jacobi polynomials and their largest zeros, Numer. Algorithm 45(1-4)(2007), 217-230. · Zbl 1124.33009 |
| [20] | http://en.Wikipedia.org/wiki/Experimental_mathematics |
| [21] | Joulak, Hédi and Bernhard Beckermann, On Gautschi’s conjecture for generalized Gauss-Radau and Gauss-Lobatto formulae, J. Comput. Appl. Math., 233, 3, 768-774 (2009) · Zbl 1194.41040 |
| [22] | Mastroianni, G. and G.V. Milovanović. Interpolation processes: basic theory and applications, Springer Monographs in Mathematics, Springer, Berlin, 2009. · Zbl 1154.41001 |
| [23] | Milovanović, Gradimir V. Personal communication, December 1993. |
| [24] | Szegö, Gabor. Orthogonal polynomials, 4th ed., American Mathematical Society, Colloquium Publications, Vol. 23, Amer. Math. Soc., Providence, RI, 1975. · Zbl 0305.42011 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
