26 research outputs found

    Non-iterative computation of Gauss-Jacobi quadrature

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    Asymptotic approximations to the zeros of Jacobi polynomials are given, with methods to obtain the coefficients in the expansions. These approximations can be used as standalone methods for the noniterative computation of the nodes of Gauss--Jacobi quadratures of high degree (n100n\ge 100). We also provide asymptotic approximations for functions related to the first-order derivative of Jacobi polynomials which are used for computing the weights of the Gauss--Jacobi quadrature. The performance of the asymptotic approximations is illustrated with numerical examples, and it is shown that nearly double precision relative accuracy is obtained for both the nodes and the weights when n100n\ge 100 and 1<α,β5-1< \alpha, \beta\le 5. For smaller degrees the approximations are also useful as they provide 101210^{-12} relative accuracy for the nodes when n20n\ge 20, and just one Newton step would be sufficient to guarantee double precision accuracy in that cases

    Error estimates of gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results

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    This paper presents a survey of recent results on error estimates of Gaussian-type quadrature formulas for analytic functions on confocal ellipses

    Error estimates of gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results

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    This paper presents a survey of recent results on error estimates of Gaussian-type quadrature formulas for analytic functions on confocal ellipses

    Fast, reliable and unrestricted iterative computation of Gauss-Hermite and Gauss-Laguerre quadratures

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    Methods for the computation of classical Gaussian quadrature rules are described which are effective both for small and large degree. These methods are reliable because the iterative computation of the nodes has guaranteed convergence, and they are fast due to their fourth-order convergence and its asymptotic exactness for an appropriate selection of the variables. For Gauss?Hermite and Gauss?Laguerre quadratures, local Taylor series can be used for computing efficiently the orthogonal polynomials involved, with exact initial values for the Hermite case and first values computed with a continued fraction for the Laguerre case. The resulting algorithms have almost unrestricted validity with respect to the parameters. Full relative precision is reached for the Hermite nodes, without any accuracy loss and for any degree, and a mild accuracy loss occurs for the Hermite and Laguerre weights as well as for the Laguerre nodes. These fast methods are exclusively based on convergent processes, which, together with the high order of convergence of the underlying iterative method, makes them particularly useful for high accuracy computations. We show examples of very high accuracy computations (of up to 1000 digits of accuracy)

    The remainder term of certain types of Gaussian quadrature formulae with specific classes of weight functions.

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    Integracija ima xiroku primenu prilikom matematiqkog mode- lovanja mnogih pojava koje se javljaju u prirodnim, tehniqkim naukama, ekonomiji i drugim oblastima. Kada se vrednost integrala ne moe analitiqki izraqunati, potrebno je kon- struisati formulu koja aproksimira njegovu vrednost sa prih- vatljivom taqnoxu. Pored tradicionalnih formula koje se ko- riste, tendencije u razvoju ove oblasti odnose se na poveanje taqnosti formule i ocenu grexke nastale kada se integral za- meni konaqnom sumom...Mathematical modeling of many phenomena which occur in the natural, technical sciences, economy requires signicant knowledge of the theory of numerical integration. In the situations where the integral cannot be determined analytically, it is necessary to construct the for- mula which approximates its value with acceptable error. Besides the traditional formulae, the tendencies in the development of this area refer to increment of algebraic degree of precision of the quadrature formula and its error estimation..

    Gradimir Milovanovic - a master in approximation and computation part ii

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