A procedure to construct quadrature formulae which are exact for solutions of linear differential equations and are optimal in the sense ofSard is discussed. We give necessary and sufficient conditions under which such formulae do exist. Several formulae obtained by applying this method are considered and compared with well known formulae.

/pdf/on-optimal-quadrature-formulae-11kh823odo.pdf

A. Ghizzetti/A. Ossicini, Quadrature Formulae. 192 S. Berlin 1970. Akademie-Verlag. Preis geb. 28,50 M

On optimal quadrature formulae

In this paper we construct an optimal quadrature formula in the sense of Sard in the Hilbert space K 2(P 2). Using S.L. Sobolev's method we obtain new optimal quadrature formula of such type and give explicit expressions for the corresponding optimal coefficients. Furthermore, we investigate order of the convergence of the optimal formula and prove an asymptotic optimality of such a formula in the Sobolev space $L_2^{(2)}(0,1)$ . The obtained optimal quadrature formula is exact for the trigonometric functions sinx and cosx. Also, we include a few numerical examples in order to illustrate the application of the obtained optimal quadrature formula.

/pdf/on-an-optimal-quadrature-formula-in-the-sense-of-sard-523q3offrp.pdf

On an optimal quadrature formula in the sense of Sard

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.

http://www.mi.sanu.ac.rs/~gvm/radovi/gvm-mms5f.pdf

Error bounds for Gauss-Turán quadrature formulae of analytic functions

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the L2(m)(0,1)$L_{2}^{(m)}(0,1)$ space for numerical calculation of Fourier coefficients. Using the S.L.Sobolev's method, we obtain new optimal quadrature formulas of such type for N+1źm, where N+1 is the number of nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We study the order of convergence of the optimal formula for the case m=1. The obtained optimal quadrature formulas in the L2(m)(0,1)$L_{2}^{(m)}(0,1)$ space are exact for Pmź1(x), where Pmź1(x) is a polynomial of degree mź1. Furthermore, we present some numerical results, which confirm the obtained theoretical results.

Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space L2(m)(0,1)$L_{2}^{(m)}(0,1)$

Our aim in this paper is to obtain error expansions in the Gauss–Turan quadrature formula ∫−11f(t)w(t) dt=∑ν=1n∑i=02sAi,νf(i)(τν)+Rn,s(f), in the case when f is an analytic function in some region of the complex plane containing the interval [−1,1] in its interior. Using a representation of the remainder term Rn,s(f) in the form of contour integral over confocal ellipses, we obtain Rn,1(f) for the four Chebyshev weights and Rn,2(f) for the Chebyshev weight of the first kind. Also, we get a few new L1-estimates of the remainder term, which are stronger than the previous ones. Some numerical results, illustrations and comparisons are also given.

A. Ghizzetti/A. Ossicini, Quadrature Formulae. 192 S. Berlin 1970. Akademie-Verlag. Preis geb. 28,50 M

Citations

On optimal quadrature formulae

On an optimal quadrature formula in the sense of Sard

Error bounds for Gauss-Turán quadrature formulae of analytic functions

Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space L2(m)(0,1)$L_{2}^{(m)}(0,1)$

An Error Expansion for some Gauss–Turán Quadratures and L1-Estimates of the Remainder Term

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