Kronrod Extensions of Gaussian Quadratures with Multiple Nodes

2006 ◽  
Vol 6 (3) ◽  
pp. 291-305 ◽  
Author(s):  
G.V. Milovanović ◽  
M.M. Spalević ◽  
L.J. Galjak
Keyword(s):  
Error Bounds ◽  
Existence And Uniqueness ◽  
Gaussian Quadrature ◽  
Weight Functions ◽  
Quadrature Formulas ◽  
Gaussian Quadrature Formulas ◽  
Gaussian Quadratures ◽  
Explicit Expressions

Abstract In this paper, general real Kronrod extensions of Gaussian quadrature formulas with multiple nodes are introduced. A proof of their existence and uniqueness is given. In some cases, the explicit expressions of polynomials, whose zeros are the nodes of the considered quadratures, are determined. Very effective error bounds of the Gauss — Turán — Kronrod quadrature formulas, with Gori — Micchelli weight functions, for functions analytic on confocal ellipses, are derived.

scholarly journals Generalized Gaussian quadratures for integrals with logarithmic singularity

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 1111-1126 ◽  
Author(s):  
Gradimir Milovanovic
Keyword(s):  
Logarithmic Singularity ◽  
Gaussian Quadrature ◽  
Quadrature Rules ◽  
Quadrature Formulas ◽  
Short Account ◽  
Numerical Examples ◽  
Gaussian Quadrature Formulas ◽  
Gaussian Quadratures ◽  
Logarithmic Singularities

A short account on Gaussian quadrature rules for integrals with logarithmic singularity, as well as some new results for weighted Gaussian quadrature formulas with respect to generalized Gegenbauer weight x |? |x|(1-x2)?, ? > -1, on (-1,1), which are appropriated for functions with and without logarithmic singularities, are considered. Methods for constructing such kind of quadrature formulas and some numerical examples are included.

scholarly journals Construction of Gaussian quadrature formulas for even weight functions

2017 ◽  
Vol 11 (1) ◽  
pp. 177-198 ◽  
Author(s):  
Mohammad Masjed-Jamei ◽  
Gradimir Milovanovic
Keyword(s):  
Weight Function ◽  
Jacobi Matrix ◽  
Gaussian Quadrature ◽  
Quadrature Rule ◽  
Weight Functions ◽  
Quadrature Formulas ◽  
Numerical Examples ◽  
Numerical Instabilities ◽  
Gaussian Quadrature Formulas ◽  
Free Parameters

Instead of a quadrature rule of Gaussian type with respect to an even weight function on (?a, a) with n nodes, we construct the corresponding Gaussian formula on (0, a2) with only [(n+1)/2] nodes. Especially, such a procedure is important in the cases of nonclassical weight functions, when the elements of the corresponding three-diagonal Jacobi matrix must be constructed numerically. In this manner, the influence of numerical instabilities in the process of construction can be significantly reduced, because the dimension of the Jacobi matrix is halved. We apply this approach to Pollaczek?s type weight functions on (?1, 1), to the weight functions on R which appear in the Abel-Plana summation processes, as well as to a class of weight functions with four free parameters, which covers the generalized ultraspherical and Hermite weights. Some numerical examples are also 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.

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