Miodrag M. Spalević

Error bounds for Gauss-Turán quadrature formulae of analytic functions

We study the kernels of the remainder term Rn,s(f) of Gauss-Thran quadrature formulas ∫-11f(t)w(t)dt = Ai,vf(i)(τv)+Rn,s(f) (n ∈ N; s ∈ N0) for classes of analytic functions on elliptical contours with foci at ± 1, when the weight w is one of the special Jacobi weights w(α,β)(t) = (1 - t)α(1 + t)β (α = β = -1/2; α = β = 1/2 + s; α = -1/2, β = 1/2 + s; α = 1/2 + s, β = -1/2). We investigate the location on the contour where the modulus of the kernel attains its maximum value. Some numerical examples are included.

ON GENERALIZED AVERAGED GAUSSIAN FORMULAS

We present a simple numerical method for constructing the optimal (generalized) averaged Gaussian quadrature formulas which are the optimal stratified extensions of Gauss quadrature formulas. These extensions exist in many cases in which real positive Kronrod formulas do not exist. For the Jacobi weight functions w(x)= w (α,β) (x) =(1- x)α(1 + x) β (α,β > -1) we give a necessary and sufficient condition on the parameters a and β such that the optimal averaged Gaussian quadrature formulas are internal.

Quadrature formulae connected to σ-orthogonal polynomials

Let dλ(t) be a given nonnegative measure on the real line R, with compact or infinite support, for which all moments µk = ∫R tk dλ(t) k = 0, 1, ..... exist and are finite, and µ0 > 0. Quadrature formulas of Chakalov-Popoviciu type with multiple nodes ∫R f(t)dλ(t) = Σv=1n, Σi=02sv Ai,vf(i) (τv) + R(f),where σ = σn = (s1,s2,...,sn) is a given sequence of nonnegative integers, are considered. A such quadrature formula has maximum degree of exactness dmax = 2 Σv=1n sv 2n-1 if and only if ∫R Πv=1n (t-τv)2sv+1 tk dλ(t) = 0, k = 0, 1,..., n-1.The proof of the uniqueness of the extremal nodes τ1, τ2,...,τn, was given first by Ghizzetti and Ossicini (Rend. Mat. 6(8) (1975) 1-15). Here, an alternative simple proof of the existence and the uniqueness of such quadrature formulas is presented. In a study of the error term R(f), an influence function is introduced, its relevant properties are investigated, and in certain classes of functions the error estimate is given. A numerically stable iterative procedure, with quadratic convergence, for determining the nodes τv, v= 1,2 ..... n, which are the zeros of the corresponding σ-orthogonal polynomial, is presented. Finally, in order to show a numerical efficiency of the proposed procedure, a few numerical examples are included.

A note on generalized averaged Gaussian formulas

We have recently proposed a very simple numerical method for constructing the averaged Gaussian quadrature formulas. These formulas exist in many more cases than the real positive Gauss–Kronrod formulas. In this note we try to answer whether the averaged Gaussian formulas are an adequate alternative to the corresponding Gauss–Kronrod quadrature formulas, to estimate the remainder term of a Gaussian rule.

Error bounds of certain Gaussian quadrature formulae

We study the kernel of the remainder term of Gauss quadrature rules for analytic functions with respect to one class of Bernstein-Szego weight functions. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective error bounds of the corresponding Gauss quadratures.

Error analysis in some Gauss-Turán-Radau and Gauss-Turán-Lobatto quadratures for analytic functions

We consider the generalized Gauss-Turan quadrature formulae of Radau and Lobatto type for approximating ∫-11 f(t)w(t)dt. The aim of this paper is to analyze the remainder term in the case when f is an analytic function in some region of the complex plane containing the interval [- 1, 1] in its interior. The remainder term is presented in the form of a contour integral over confocal ellipses (cf. SIAM J. Numer. Anal. 80 (1983) 1170). Sufficient conditions on the convergence for some of such quadratures, associated with the generalized Chebyshev weight functions, are found. Using some ideas from Hunter (BIT 35 (1995) 64) we obtain new estimates of the remainder term, which are very exact. Some numerical results and illustrations are shown.

Maximum of the modulus of kernels in Gauss-Turán quadratures

We study the kernels K n,s (z) in the remainder terms R n,s (f) of the Gauss-Turan quadrature formulae tor analytic functions on elliptical contours with foci at ±1, when the weight ω is a generalized Chebyshev weight function. For the generalized Chebyshev weight of the first (third) kind, it is shown that the modulus of the kernel |K n,s (z)| attains its maximum on the real axis (positive real semi-axis) for each n > no, nO = n 0 (ρ, s). It was stated as a conjecture in [Math. Comp. 72 (2003), 1855-1872]. For the generalized Chebyshev weight of the second kind, in the case when the number of the nodes n in the corresponding Gauss-Turan quadrature formula is even, it is shown that the modulus of the kernel attains its maximum on the imaginary axis for each n ≥ no, no = n 0 (p, s). Numerical examples are included.

Error estimates for Gaussian quadratures of analytic functions

For analytic functions the remainder term of Gaussian quadrature formula and its Kronrod extension can be represented as a contour integral with a complex kernel. We study these kernels on elliptic contours with foci at the points +/-1 and the sum of semi-axes @r>1 for the Chebyshev weight functions of the first, second and third kind, and derive representation of their difference. Using this representation and following Kronrods method of obtaining a practical error estimate in numerical integration, we derive new error estimates for Gaussian quadratures.

Kronrod extensions with multiple nodes of quadrature formulas for Fourier coefficients

Abstract. We continue with analyzing quadrature formulas of high degree of precision for computing the Fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials, started recently by Bojanov and Petrova [Quadrature formulae for Fourier coefficients, J. Comput. Appl. Math. 231 (2009), 378–391] and we extend their results. Construction of new Gaussian quadrature formulas for the Fourier coefficients of a function, based on the values of the function and its derivatives, is considered. We prove the existence and uniqueness of Kronrod extensions with multiple nodes of standard Gaussian quadrature formulas with multiple nodes for several weight functions, in order to construct some new generalizations of quadrature formulas for the Fourier coefficients. For the quadrature formulas for the Fourier coefficients based on the zeros of the corresponding orthogonal polynomials we construct Kronrod extensions with multiple nodes and highest algebraic degree of precision. For this very desirable kind of extension there do not exist any results in the theory of standard quadrature formulas.

Error bounds of Gaussian quadrature formulae for one class of Bernstein-Szegő weights

The kernels Kn(z) in the remainder terms Rn(f) of the Gaussian quadrature formulae for analytic functions f inside elliptical contours with foci at ∓1 and a sum of semi-axes ρ > 1, when the weight function w is of Bernstein-Szegő type w(t) ≡ w(−1/2,1/2) γ (t) = √ 1 + t 1− t · 1 1− 4γ (1 + γ)2 t , t ∈ (−1, 1), γ ∈ (−1, 0), are studied. Sufficient conditions are found ensuring that the kernel attains its maximal absolute value at the intersection point of the contour with the positive real semi-axis. This leads to effective error bounds of the corresponding Gauss quadratures. The quality of the derived bounds is analyzed by a comparison with other error bounds intended for the same class of integrands. In part our analysis is based on the well-known Cardano formulas, which are not very popular among mathematicians.