On numerical improvement of Gauss–Radau quadrature rules

For the class of polynomial quadrature rules we show that conveniently chosen bases allow to compute both the weights and the theoretical error expression of a $n$-point rule via the undetermined coefficients method. As an illustration, the framework is applied to some classical rules such as Newton-Cotes, Adams-Bashforth, Adams-Moulton and Gaussian rules.

Applied Mathematics and Computation, 2006

In this paper we introduce an integration method with equal coefficient in the following form: Z b a wðxÞf ðxÞdx ' C n X n i¼1 f ðx i Þ. Then by using the formulas of Newton's equations and degree of precision we introduce a method which express the nodes and coefficients in this integration formulas. Finally some examples are presented to illustrate the procedure.

Calcolo, 2013

In this paper, quadrature formulas on the real line with the highest degree of accuracy, with positive weights, and with one or two prescribed nodes anywhere on the interval of integration are characterized. As an application, the same kind of rules but with one or both (finite) endpoints being fixed nodes and one or two more prescribed nodes inside the interval of integration are derived. An efficient computation of such quadrature formulas is analyzed by considering certain modified Jacobi matrices. Some numerical experiments are finally presented.

Kuwait Journal of Science, 2021

Orthogonal polynomials and the corresponding quadrature formulas of Gaussian type concerni λ ng > the 1 e / v 2 en wei x gh > t f 0 unction ω(t; x) = exp λ (−= xt 1 2) / ( 2 1 − t2)−1/2 on (−1, 1), with parameters − and , are considered. For these quadrature rules reduce to the socalled Gauss-Rys quadrature formulas, which were investigated earlier by several authors, e.g., Dupuis at al 1976 and 1983; Sagar 1992; Schwenke 2014; Shizgal 2015; King 2016; Milovanovic ´ 2018, etc. In this generalized case, the method of modified moments is used, as well as a transformation of quadratures on (−1, 1) with N nodes to ones on (0, 1) with only (N + 1)/2 nodes. Such an approach provides a stable and very efficient numerical construction.

Applied Mathematics and Computation, 2005

This paper discusses on numerical improvement of the Newton-Cotes integration rules, which are in forms of: Z b¼aþnh a f ðxÞ dx ' X n k¼0 B ðnÞ k f ða þ khÞ: It is known that the precision degree of above formula is n + 1 for even n's and is n for odd n's. However, if its bounds are considered as two additional variables (i.e. a and h in fact) we reach a nonlinear system that numerically improves the precision degree of above integration formula up to degree n + 2. In this way, some numerical examples are given to show the numerical superiority of our approach with respect to usual Newton-Cotes integration formulas.

Applied Mathematics and Computation, 2005

One of the quadrature rules is the ''Equal coefficients quadrature rules'' represented by Z b a wðxÞf ðxÞ dx ' C n X n i¼1 f ðx i Þ;

SIAM Journal on Numerical Analysis, 1996

In [6], a far-reaching generalization of the classical Gaussian quadrature rules is introduced, replacing the polynomials with a wide class of functions. While the rules of [6] possess most of the desirable properties of the classical Gaussian integration formulae (positivity of the weights, etc.), it is not clear from [6] how such quadrature rules can be obtained numerically. In this paper, we present a numerical scheme for the generation of such generalized Gaussian quadratures. The algorithm is applicable to a variety of functions, including smooth functions as well as functions with endpoint singularities. The performance of the algorithm is demonstrated with several numerical examples. Accesion For NTIS CRA&W DTIC TAB Unannounced -Justification By , Distribution, D T IC VMC Availability Codes ELECTE t Avail and/or ) E C 0 91993

Consider a hermitian positive-definite linear functional F, and assume we have m distinct nodes fixed in advance anywhere on the real line. In this paper we then study the existence and construction of nth rational Gauss-Radau (m = 1) and Gauss-Lobatto (m = 2) quadrature formulas that approximate F{f }. These are quadrature formulas with n positive weights and with the n−m remaining nodes real and distinct, so that the quadrature is exact in a (2n − m)dimensional space of rational functions.

Computers & Mathematics with Applications, 2009

In this paper we consider quadrature formulas which use the derivative of only an arbitrary fixed order (m) of function f at the nodes. One of the advantages of the new approach is that we can increase the precision degree of the n-point quadrature formulas from 2n − 1 to 2n + m − 1. Furthermore we give an asymptotic estimation for the rate of convergence of this formula. Some examples will be given to support the results.

Some Gauss-type quadrature rules over [0, 1], which involve values and/or the derivative of the integrand at 0 and/or 1, are investigated Comment: 13 pages, 3 figures, 5 tables

Mathematical and Computer Modelling, 2003

Mathematical Analysis, Approximation Theory and Their Applications, 2016

In this paper a brief historical survey of the development of quadrature rules with multiple nodes and the maximal algebraic degree of exactness is given. The natural generalization of such rules are quadrature rules with multiple nodes and the maximal degree of exactness in some functional spaces that are different from the space of algebraic polynomial. For that purpose we present a generalized quadrature rules considered by A. Ghizzeti and A. Ossicini [Quadrature Formulae, Academie-Verlag, Berlin, 1970] and apply their ideas in order to obtain quadrature rules with multiple nodes and the maximal trigonometric degree of exactness. Such quadrature rules are characterized by so called s− and σ −orthogonal trigonometric polynomials. Numerical method for the construction of such quadrature rules are given, as well as numerical the example which illustrate obtained theoretical results.

2009

— A form of Gauss-Quadrature rule over [0,1] has been investigated that involves the derivative of the integrand at the pre-assigned left or right end node. This situation arises when the underlying polynomials are orthogonal with respect to the weight function ω ( x): = 1 − x over [0,1]. Along the lines of Golub’s work, the nodes and weights of the quadrature rule are computed from a Jacobi-type matrix with entries related to simple rational sequences. The structure of these sequences is based on some characteristics of the identity-type polynomials recently developed by one of the authors. The devised rule has a slight advantage over that subject to the weight function ω ( x): = 1.

Applied Numerical Mathematics

Let w(x) = e −x β x α ,w(x) = xw(x) and let {p m (w)} m , {p m (w)} m be the corresponding sequences of orthonormal polynomials. Since the zeros of p m+1 (w) interlace those of p m (w), it makes sense to construct an interpolation process essentially based on the zeros of Q 2m+1 := p m+1 (w)p m (w), which is called "Extended Lagrange Interpolation". In this paper the convergence of this interpolation process is studied in suitable weighted L 1 spaces, in a general framework which completes the results given by the same authors in weighted L p u ((0, +∞)), 1 ≤ p ≤ ∞ (see [30], [27]). As an application of the theoretical results, an extended product integration rule, based on the aforesaid Lagrange process, is proposed in order to compute integrals of the type +∞ 0 f (x)k(x, y)u(x)dx, u(x) = e −x β x γ (1 + x) λ , γ > −1, λ ∈ IR + , where the kernel k(x, y) can be of different kinds. The rule, which is stable and fast convergent, is used in order to construct a computational scheme involving the single product integration rule studied in [22]. It is shown that the "compound quadrature sequence" represents an efficient proposal for saving 1/3 of the evaluations of the function f , under unchanged speed of convergence.

In this paper we study quadrature formulas with the higher degree of accuracy. We study the quasi-orthogonality of orthogonal poly-nomials and we give some results on the location of their zeros. Mathematics Subject Classification (2010): 41A55, 42C05.