scholarly journals Orthogonal polynomials and generalized Gauss-Rys quadrature formulae

2021 ◽  
Vol 49 (1) ◽  
Author(s):  
Gradimir V. Milovanovic ◽  
◽  
Nevena Vasovic ◽  
Keyword(s):  
Orthogonal Polynomials ◽  
Quadrature Rules ◽  
Quadrature Formulas ◽  
Numerical Construction ◽  
Quadrature Formulae ◽  
Gaussian Type ◽  
Modified Moments

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.

scholarly journals Quadrature with multiple nodes, power orthogonality, and moment-preserving spline approximation, part II

2019 ◽  
Vol 13 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Gradimir Milovanovic ◽  
Miroslav Pranic ◽  
Miodrag Spalevic
Keyword(s):  
Numerical Analysis ◽  
Error Analysis ◽  
Orthogonal Polynomials ◽  
Spline Approximation ◽  
Gauss Quadrature ◽  
Quadrature Formulas ◽  
Numerical Construction ◽  
Appl Math

The paper deals with new contributions to the theory of the Gauss quadrature formulas with multiple nodes that are published after 2001, including numerical construction, error analysis and applications. The first part was published in Numerical analysis 2000, Vol. V, Quadrature and orthogonal polynomials (W. Gautschi, F. Marcellan, and L. Reichel, eds.) [J. Comput. Appl. Math. 127 (2001), no. 1-2, 267-286].

scholarly journals Trigonometric multiple orthogonal polynomials of semi-integer degree and the corresponding quadrature formulas

2014 ◽  
Vol 96 (110) ◽  
pp. 211-226 ◽  
Author(s):  
Gradimir Milovanovic ◽  
Marija Stanic ◽  
Tatjana Tomovic
Keyword(s):  
Orthogonal Polynomials ◽  
Trigonometric Polynomials ◽  
Quadrature Rules ◽  
Quadrature Formulas ◽  
Numerical Examples ◽  
Multiple Orthogonal Polynomials ◽  
Optimal Set ◽  
Numer Math ◽  
Theoretical Results

An optimal set of quadrature formulas with an odd number of nodes for trigonometric polynomials in Borges? sense [Numer. Math. 67 (1994), 271-288], as well as trigonometric multiple orthogonal polynomials of semi-integer degree are defined and studied. The main properties of such a kind of orthogonality are proved. Also, an optimal set of quadrature rules is characterized by trigonometric multiple orthogonal polynomials of semiinteger degree. Finally, theoretical results are illustrated by some numerical examples.

scholarly journals Closed expressions for coefficients in weighted Newton-Cotes quadratures

Filomat ◽  
2013 ◽  
Vol 27 (4) ◽  
pp. 649-658 ◽  
Author(s):  
Mohammad Masjed-Jamei ◽  
Gradimir Milovanovic ◽  
M.A. Jafari
Keyword(s):  
Short Note ◽  
Stirling Numbers ◽  
Quadrature Formulas ◽  
Numerical Examples ◽  
Quadrature Formulae ◽  
Equidistant Nodes

In this short note, we derive closed expressions for Cotes numbers in the weighted Newton-Cotes quadrature formulae with equidistant nodes in terms of moments and Stirling numbers of the first kind. Three types of equidistant nodes are considered. The corresponding program codes in Mathematica Package are presented. Finally, in order to illustrate the application of the obtained quadrature formulas a few numerical examples are included.

Padé approximation and orthogonal polynomials

1974 ◽  
Vol 10 (2) ◽  
pp. 263-270 ◽  
Author(s):  
G.D. Allen ◽  
C.K. Chui ◽  
W.R. Madych ◽  
F.J. Narcowich ◽  
P.W. Smith
Keyword(s):  
Power Series ◽  
Variational Method ◽  
Orthogonal Polynomials ◽  
Gaussian Quadrature ◽  
Quadrature Formulas ◽  
The Past ◽  
Gaussian Quadrature Formulas ◽  
Poles And Zeros ◽  
Formal Power ◽  
Determinant Theory

By using a variational method, we study the structure of the Padé table for a formal power series. For series of Stieltjes, this method is employed to study the relations of the Padé approximants with orthogonal polynomials and gaussian quadrature formulas. Hence, we can study convergence, precise locations of poles and zeros, monotonicity, and so on, of these approximants. Our methods have nothing to do with determinant theory and the theory of continued fractions which were used extensively in the past.

Sign in / Sign up

Export Citation Format

Share Document