Quadrature Theory

Brass, Helmut; Petras, Knut

doi:10.1090/surv/178

Cited by 54 publications

(38 citation statements)

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“…Thus, the error estimate (1.5) by using the Lebesgue constant may be overestimated for some special points of sets for functions of limited regularities. Moreover, it has been observed, by Clenshaw-Curtis [10] and O'Hara and Smith [29], that n-point Gauss quadrature and n-point Clenshaw-Curtis quadrature have essentially the same accuracy, which has been showed recently by Trefethen [58,59], Brass and Petras [6] and Xiang and Bornemann [75]. Both of these two quadrature are derived from the interpolation polynomial…”

Section: (18)mentioning

confidence: 74%

On Interpolation Approximation: Convergence Rates for Polynomial Interpolation for Functions of Limited Regularity

Xiang¹

2016

SIAM J. Numer. Anal.

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Abstract. The convergence rates on polynomial interpolation in most cases are estimated by Lebesgue constants. These estimates may be overestimated for some special points of sets for functions of limited regularities. In this paper, by applying the Peano kernel theorem and Wainerman's lemma, new formulas on the convergence rates are considered. Based upon these new estimates, it shows that the interpolation at strongly normal pointsystems can achieve the optimal convergence rate, the same as the best polynomial approximation. Furthermore, by using the asymptotics on Jacobi polynomials, the convergence rates are established for Gauss-Jacobi, Jacobi-Gauss-Lobatto or Jacobi-Gauss-Radau pointsystems. From these results, we see that the interpolations at the Gauss-Legendre, Legendre-Gauss-Lobatto pointsystem, or at strongly normal pointsystems, has essentially the same approximation accuracy compared with those at the two Chebyshev piontsystems, which also illustrates the equally accuracy of the Gauss and Clenshaw-Curtis quadrature. In addition, numerical examples illustrate the perfect coincidence with the estimates, which means the convergence rates are optimal.

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Section: (18)mentioning

confidence: 74%

On Interpolation Approximation: Convergence Rates for Polynomial Interpolation for Functions of Limited Regularity

Xiang¹

2016

SIAM J. Numer. Anal.

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“…2 We also assume that for t ∈ [x, z], the function B n (t) is finite and has a finite inverse, that is,…”

Section: Iterative Methods For Root Finding and Quadrature Rulesmentioning

confidence: 99%

“…Numerical integration is discussed, for example, in [14] Ch. 6, [2] Ch. 5, and for Newton-Cotes quadrature rules, see for instance [7], Ch.…”

Section: Introductionmentioning

confidence: 99%

Root finding by high order iterative methods based on quadratures

Graça

Lima

2015

Applied Mathematics and Computation

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We discuss a recursive family of iterative methods for the numerical approximation of roots of nonlinear functions in one variable. These methods are based on Newton-Cotes closed quadrature rules. We prove that when a quadrature rule with n + 1 nodes is used the resulting iterative method has convergence order at least n + 2, starting with the case n = 0 (which corresponds to the Newton's method).

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“…The integrals in (21) can be approximated using the quadrature rule, see, e.g., (Golub and Meurant, 2010;Brass and Petras, 2011):…”

Section: Electricity Forward Contract Pricesmentioning

confidence: 99%