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Stieltjes Polynomials
In mathematics, the Stieltjes polynomials ''E''''n'' are polynomials associated to a family of orthogonal polynomials ''P''''n''. They are unrelated to the Stieltjes polynomial solutions of differential equations. Stieltjes originally considered the case where the orthogonal polynomials ''P''''n'' are the Legendre polynomials. The Gauss–Kronrod quadrature formula uses the zeros of Stieltjes polynomials. Definition If ''P''0, ''P''1, form a sequence of orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the class ... for some inner product, then the Stieltjes polynomial ''E''''n'' is a degree ''n'' polynomial orthogonal to ''P''''n''–1(''x'')''x''''k'' for ''k'' = 0, 1, ..., ''n'' – 1. References *{{eom, id=s/s120250, title=Sti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Kronrod Quadrature Formula
The Gauss–Kronrod quadrature formula is an adaptive method for numerical integration. It is a variant of Gaussian quadrature, in which the evaluation points are chosen so that an accurate approximation can be computed by re-using the information produced by the computation of a less accurate approximation. It is an example of what is called a nested quadrature rule: for the same set of function evaluation points, it has two quadrature rules, one higher order and one lower order (the latter called an ''embedded'' rule). The difference between these two approximations is used to estimate the calculational error of the integration. These formulas are named after Alexander Kronrod, who invented them in the 1960s, and Carl Friedrich Gauss. Description The problem in numerical integration is to approximate definite integrals of the form :\int_a^b f(x)\,dx. Such integrals can be approximated, for example, by ''n''-point Gaussian quadrature :\int_a^b f(x)\,dx \approx \sum_^n w_i f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by P. L. Chebyshev and was pursued by A. A. Markov and T. J. Stieltjes. They appear in a wide variety of fields: numerical analysis ( quadrature rules), probability theory, representation theory (of Lie groups, quantum groups, and related objects), enumerative combinatorics, algebraic combinatorics, mathematical physics (the theory of random matr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |