共查询到20条相似文献,搜索用时 15 毫秒
1.
We characterize the supports of the measures having quadrature formulae with similar exactness as Gauss’ theorem. Indeed we obtain the supports of the measures from which an m-point quadrature formula can be obtained such that it exactly integrates functions in the space ? m?k,m?k [ $ \bar z $ , z]. We also give a method for obtaining the nodes and the quadrature coefficients in all the cases and, as a consequence, we solve the same problem in the space of trigonometric polynomials. 相似文献
2.
Rüdiger Brombeer 《Numerische Mathematik》1978,30(2):149-164
Summary For some special Hilbert-spaces of periodic analytic functions it is known that quadrature formulae of minimal norm with preassigned equidistant nodes are even so-called Wilf-formulae, i.e. they satisfy necessary conditions for minimal norm with respect to their nodes. By simple examples, however, it can be shown that equidistant Wilf-formulae are not necessarily optimal. In this paper the question of optimality of equidistant nodes in quadrature formulae for rather general Hilbert-spaces of periodic analytic functions is answered by giving sufficient conditions which can be interpreted as conditions on the size of the regularity-regions of the functions belonging to the Hilbert-spaces under consideration. Examples prove these conditions to be quite sharp.In addition the trapezoidal-rule is shown to be only optimal formula (with respect to the nodes and coefficients) of orderk.Finally the trapezoidal-rule is shown to be asymptotically optimal for wide classes of Hilbert-spaces of periodic functions. 相似文献
3.
A. van der Sluis 《Numerische Mathematik》1972,19(1):49-55
Summary It is shown that the remainder term of any quadrature formula has an asymptotic expansion in terms of the step size; the occurrence of remainder terms of formc f
(n+1) () is discussed. 相似文献
4.
Dr. Franz Locher 《Numerische Mathematik》1972,20(4):317-326
Summary Usually the errorR
n
(j) of a quadrature formula is estimated with the aid of theL
1-norm of the Peano kernel. It is shown that this term may be estimated rather sharp using the norm Q
n
of the quadrature rule. Then it follows that formulas with non-negative weights are favourable also in the sense of minimizing theL
1-norm of the kernel. A remainder term of the typeR
n
(f)=cf(n+1) () is possible iff the kernel is definite. In the case of an interpolatory formula this definiteness is usually shown by an application of the so-called V-method. We determine the optimal formulas in the sense of this method. Then we analyse the influence of the structure of the mesh on the norm of a formula. We find that on an equidistant mesh withm nodes there exists a rule with a small norm if the order is not greater than
. 相似文献
5.
Hans Joachim Schmid 《Numerische Mathematik》1978,31(3):281-297
Summary In this paper an approach is outlined to the two-dimensional analogon of the Gaussian quadrature problem. The main results are necessary and sufficient conditions for the existence of cubature formulae which are exact for all polynomials of degree m and which have a minimal number of 1/2k(k+1) knots,k=[m/2]+1. Ifm is odd, similar results are due to I.P. Mysovskikh ([5, 6]) which will be derived in a new way as a special case of the general characterization given here. Furthermore, it will be shown how this characterization can be used to construct minimal formulae of even degree. 相似文献
6.
Peter Köhler 《Aequationes Mathematicae》1990,39(1):6-18
LetC
m
be a compound quadrature formula, i.e.C
m
is obtained by dividing the interval of integration [a, b] intom subintervals of equal length, and applying the same quadrature formulaQ
n
to every subinterval. LetR
m
be the corresponding error functional. Iff
(r)
> 0 impliesR
m
[f] > 0 (orR
m
[f] < 0),=" then=" we=" say=">C
m
is positive definite (or negative definite, respectively) of orderr. This is the case for most of the well-known quadrature formulas. The assumption thatf
(r)
> 0 may be weakened to the requirement that all divided differences of orderr off are non-negative. Thenf is calledr-convex. Now letC
m
be positive definite or negative definite of orderr, and letf be continuous andr-convex. We prove the following direct and inverse theorems for the errorR
m
[f], where , denotes the modulus of continuity of orderr:
相似文献
7.
Leyla Daruis Pablo Gonzlez-Vera Francisco Marcelln 《Journal of Computational and Applied Mathematics》2002,140(1-2)
Let μ be a probability measure on [0,2π]. In this paper we shall be concerned with the estimation of integrals of the form
Full-size image 8.
This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the \(L_{2}^{(m)}(0,1)\) 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+1≥m, 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 \(L_{2}^{(m)}(0,1)\) space are exact for P m?1(x), where P m?1(x) is a polynomial of degree m?1. Furthermore, we present some numerical results, which confirm the obtained theoretical results. 相似文献
9.
Manuel Gräf 《Numerical Functional Analysis & Optimization》2013,34(7-8):665-688
In this paper, we construct sampling sets over the rotation group SO(3). The proposed construction is based on a parameterization, which reflects the product nature 2 × 1 of SO(3) very well, and leads to a spherical Pythagorean-like formula in the parameter domain. We prove that by using uniformly distributed points on 2 and 1, we obtain uniformly sampling nodes on the rotation group SO(3). Furthermore, quadrature formulae on 2 and 1 lead to quadratures on SO(3), as well. For scattered data on SO(3), we give a necessary condition on the mesh norm such that the sampling nodes possess nonnegative quadrature weights. We propose an algorithm for computing the quadrature weights for scattered data on SO(3) based on fast algorithms. We confirm our theoretical results with examples and numerical tests. 相似文献
10.
Birkholl quadrature formulae (q.f.), which have algebraic degree of precision (ADP) greater than the number of values used, are studied. In particular, we construct a class of quadrature rules of ADP = 2n + 2r + 1 which are based on the information {ƒ(j)(−1), ƒ(j)(−1), j = 0, ..., r − 1 ; ƒ(xi), ƒ(2m)(xi), i = 1, ..., n}, where m is a positive integer and r = m, or r = m − 1. It is shown that the corresponding Birkhoff interpolation problems of the same type are not regular at the quadrature nodes. This means that the constructed quadrature formulae are not of interpolatory type. Finally, for each In, we prove the existence of a quadrature formula based on the information {ƒ(xi), ƒ(2m)(xi), i = 1, ..., 2m}, which has algebraic degree of precision 4m + 1. 相似文献
11.
Dr. Gerhard Schmeisser 《Numerische Mathematik》1976,27(3):355-358
Summary We show that the remainder of the Maclaurin quadrature formula belonging to oddn (n+1 is the number of nodes) can be represented asR
n
(f)=c
n
f
(n+1) (), wheneverf
(n+1) exists and is continuous The corresponding problem for evenn has already been settled by A. Walther in 1925. 相似文献
12.
《Journal of Computational and Applied Mathematics》1998,94(2):199-214
We study interpolatory quadrature formulae, relative to the Legendre weight function on [−1,1], having as nodes the zeros of any one of the four Chebyshev polynomials of degree n plus one of the points 1 or −1. In particular, we derive explicit formulae for the weights and examine their positivity, we determine the precise degree of exactness, we obtain asymptotically optimal error bounds, and we examine the definiteness f these quadrature formulae. In addition, we establish their convergence for Riemann integrable functions on [−1, 1] as well as for functions having a monotonic singularity at −1 or 1. 相似文献
13.
H. Fiedler 《Numerische Mathematik》1987,51(5):571-581
Summary Interpolatory quadrature formulae consist in replacing
by
wherep
f
denotes the interpolating polynomial off with respect to a certain knot setX. The remainder
may in many cases be written as
wherem=n resp. (n+1) forn even and odd, respectively. We determine the asymptotic behaviour of the Peano kernelP
X
(t) forn for the quadrature formulae of Filippi, Polya and Clenshaw-Curtis. 相似文献
14.
The extremal problem of minimizing the error of approximate evaluation of a line integral of the first kind is considered for certain classes of functions and spatial curves determined by moduli of continuity.It is proved that if the endpoints of the interval [0, L] (where L is the length of the curve along which the integration is performed) are not included in the set of nodes of a quadrature formula for evaluating the line integral of the first kind, then the best quadrature formula for the classes m(p) ρ of functions and \({H^{{\omega _1}, \ldots ,{\omega _m}}}\) of curves is the midpoint rectangle formula. If the extreme points x = 0 and x = L of the interval are included in the set of nodes of a quadrature formula for approximately evaluating the line integral (such formulas are said to be Markov-type), then, for these classes, the best formula is the trapezoidal rule. Sharp error estimates for all considered classed of functions and curves are calculated and a generalization to more general classes is given. 相似文献
15.
Sotirios E. Notaris 《BIT Numerical Mathematics》2002,42(2):440-446
We consider interpolatory quadrature formulae, relative to the Legendre weight function w(t) = 1 on [–1, 1], having as nodes the zeros of the nth degree Jacobi polynomial P
n
(, ) plus the points 1 and –1. We show that in specific domains of and gb the weights of these formulae are almost all positive, exceptions occurring only with the weights corresponding to 1 and –1. 相似文献
16.
Summary In order to compute an integralI[f], one needs at least two cubature formulaeQ
j
,j{1, 2}. |Q
1[f]–Q
2[f]| can be used as an error estimate for the less precise cubature formula. In order to reduce the amount of work, one can try to reuse some of the function evaluations needed forQ
1, inQ
2. The easiest way to construct embedded cubature formulae is: start with a high degree formulaQ
1, drop (at least) one knot and calculate the weights such that the new formulaQ
2 is exact for as much monomials as possible. We describe how such embedded formulae with positive weights can be found. The disadvantage of such embedded cubature formulae is that there is in general a large difference in the degree of exactness of the two formulae. In this paper we will explain how the high degree formula can be chosen to obtain an embedded pair of cubature formulae of degrees 2m+1/2m–1. The method works for all regions
n
,n2. We will also show the influence of structure on the cubature formulae. 相似文献
17.
H. Strauss 《Numerische Mathematik》1984,44(3):337-347
Summary This paper deals with quadrature formulae of Gauss type corresponding to subspaces of spline functionsS
m–1, k
of degreem–1 withk fixed knots. We shall show monotonicity of the quadrature formulae for functions which are contained in the so-called convexity cone ofS
m–1,k
Moreover, we apply these results to monosplines and establish comparison theorems for these splines. 相似文献
18.
Franz Peherstorfer 《Numerische Mathematik》1987,52(3):317-327
Summary It is well known that the Chebyshev weight function (1–x
2)–1/2 is the only weight function (up to a linear transformation) for which then point Gauss quadrature formula has equal weights for alln. In this paper we describe all weight functions for which thenm point Gauss quadrature formula has equal weights for alln, wherem is fixed. 相似文献
19.
Dušan Lj. Djukić Aleksandar V. Pejčev Miodrag M. Spalević 《Numerical Algorithms》2018,77(4):1003-1028
We consider the Gauss-Kronrod quadrature formulae for the Bernstein-Szeg? weight functions consisting of any one of the four Chebyshev weights divided by the polynomial \(\rho (t)=1-\frac {4\gamma }{(1+\gamma )^{2}}\,t^{2},\quad t\in (-1,1),\ -1<\gamma \le 0\). For analytic functions, the remainder term of this quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points ? 1 and sum of semi-axes ρ > 1, for the given quadrature formula. Starting from the explicit expression of the kernel, we determine the locations on the ellipses where maximum modulus of the kernel is attained. So we derive effective error bounds for this quadrature formula. An alternative approach, which has initiated this research, has been proposed by S. Notaris (Numer. Math. 103, 99–127, 2006). 相似文献
20.
Summary We prove that the error inn-point Gaussian quadrature, with respect to the standard weight functionw1, is of best possible orderO(n
–2) for every bounded convex function. This result solves an open problem proposed by H. Braß and published in the problem section of the proceedings of the 2. Conference on Numerical Integration held in 1981 at the Mathematisches Forschungsinstitut Oberwolfach (Hämmerlin 1982; Problem 2). Furthermore, we investigate this problem for positive quadrature rules and for general product quadrature. In particular, for the special class of Jacobian weight functionsw
, (x)=(1–x)(1+x), we show that the above result for Gaussian quadrature is not valid precisely ifw
, is unbounded.Dedicated to Prof. H. Braß on the occasion of his 55th birthday 相似文献
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