共查询到20条相似文献,搜索用时 15 毫秒
1.
2.
G. Mühlbach 《Numerische Mathematik》1979,32(4):393-408
Summary In this note an ultimate generalization of Newton's classical interpolation formula is given. More precisely, we will establish the most general linear form of a Newton-like interpolation formula and a general recurrence relation for divided differences which are applicable whenever a function is to be interpolated by means of linear combinations of functions forming a ebyev-system such that at least one of its subsystems is again a ebyev-system. The theory is applied to trigonometric interpolation yielding a new algorithm which computes the interpolating trigonometric polynomial of smallest degree for any distribution of the knots by recurrence. A numerical example is given. 相似文献
3.
Previously D. P. Laurie has introduced a new and sharper error estimate for adaptive quadrature routines with the attractive property that the error is guaranteed to be in a small interval if some constraints are satisfied. In this paper we discuss how to test whether or not the constraints are satisfied, and we report a selection of results from our tests with one dimensional integrals to see how the error estimate works in practice. It turns out that we get a more economic routine using this error estimate, but the loss in reliability, even with the new tests, can be catastrophic.This work was supported by the Norwegian Research Council for Sciences and Huminaties. 相似文献
4.
Klaus-Jürgen Förster 《BIT Numerical Mathematics》1988,28(2):360-363
The purpose of this note is to give an example which demonstrates that one can achieve much higher algebraic precision with a quadrature rule with small but not minimal variance than with a Chebyshev rule with minimal variance. 相似文献
5.
N.I. Ioakimidis 《Journal of Computational and Applied Mathematics》1984,11(3):267-276
A Cauchy type singular integral equation of the first or the second kind can be numerically solved either directly or after its reduction (by the usual regularization procedure) to an equivalent Fredholm integral equation of the second kind. The equivalence of these two methods (that is, the equivalence both of the systems of linear algebraic equations to which the singular integral equation is reduced and of the natural interpolation formulae) is proved in this paper for a class of Cauchy type singular integral equations of the first kind and of the second kind (but with constant coefficients) for general interpolatory quadrature rules under sufficiently mild assumptions. The present results constitute an extension of a series of previous results concerning only Gaussian quadrature rules, based on the corresponding orthogonal polynomials and their properties. 相似文献
6.
Gradimir V. Milovanović Aleksandar S. Cvetković Marija P. Stanić 《Numerische Mathematik》2009,112(3):425-448
In this paper we consider interpolatory quadrature formulae with multiple nodes, which have the maximal trigonometric degree
of exactness. Our approach is based on a procedure given by Ghizzeti and Ossicini (Quadrature formulae, Academie-Verlag, Berlin,
1970). We introduce and consider the so-called σ-orthogonal trigonometric polynomials of semi-integer degree and give a numerical method for their construction. Also, some numerical examples are included.
The authors were supported in part by the Serbian Ministry of Science and Technological Development (Project: Orthogonal Systems
and Applications, grant number #144004) and the Swiss National Science Foundation (SCOPES Joint Research Project No. IB7320-111079
“New Methods for Quadrature”). 相似文献
7.
Let ℂ denote the complex numbers and
denote the ring of complex-valued Laurent polynomial functions on ℂ\{0}. Furthermore, we denote by
the subsets of Laurent polynomials whose restriction to the unit circle is real, nonnegative, respectively. We prove that
for any two Laurent polynomials
, which have no common zeros in ℂ\{0} there exists a pair of Laurent polynomials
satisfying the equation Q
1
P
1 + Q
2
P
2 = 1. We provide some information about the minimal length Laurent polynomials Q
1 and Q
2 with these properties and describe an algorithm to compute them. We apply this result to design a conjugate quadrature filter
whose zeros contain an arbitrary finite subset Λ⊂ℂ\{0} with the property that for every
implies
and
.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
8.
We present a geometric exposition of S. Lie's and E. Cartan's theory of explicit integration of finite-type (in particular, ordinary) differential equations. Numerous examples of how this theory works are given. In one of these, we propose a method of hunting for particular solutions of partial differential equations via symmetry preserving overdetermination. 相似文献
9.
Pierre Verlinden 《Numerical Algorithms》1999,22(2):183-192
Consider a weight
and a rational modification of it
is a polynomial. An algorithm is stabilized for the following problem: “Given the coefficients of the three-term recurrence
equation satisfied by the orthogonal polynomials relative to
, compute the coefficients of the recurrence equation satisfied by the orthogonal polynomials relative to
.”
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
10.
Serge Dubuc Daniel Lemire Jean-Louis Merrien 《Journal of Fourier Analysis and Applications》2001,7(5):537-552
Two subdivision schemes with Hermite data on ℤ are studied. These schemes use 2 or 7 parameters respectively depending on
whether Hermite data involve only first derivatives or include second derivatives. For a large region in the parameter space,
the schemes are convergent in the space of Schwartz distributions. The Fourier transform of any interpolating function can
be computed through products of matrices of order 2 or 3. The Fourier transform is related to a specific system of functional
equations whose analytic solution is unique except for a multiplicative constant. The main arguments for these results come
from Paley-Wiener-Schwartz theorem on the characterization of the Fourier transforms of distributions with compact support
and a theorem of Artzrouni about convergent products of matrices. 相似文献
11.
J. M. de Villiers 《Numerische Mathematik》1993,66(1):123-137
Summary The Gregory rule is a well-known example in numerical quadrature of a trapezoidal rule with endpoint corrections of a given order. In the literature, the methods of constructing the Gregory rule have, in contrast to Newton-Cotes quadrature,not been based on the integration of an interpolant. In this paper, after first characterizing an even-order Gregory interpolant by means of a generalized Lagrange interpolation operator, we proceed to explicitly construct such an interpolant by employing results from nodal spline interpolation, as established in recent work by the author and C.H. Rohwer. Nonoptimal order error estimates for the Gregory rule of even order are then easily obtained. 相似文献
12.
Anry Nersessian 《Numerical Functional Analysis & Optimization》2013,34(1-2):227-240
The general scheme, suggested in [1] using a basis of an infinite-dimensional space and allowing to construct finite-dimensional orthogonal systems and interpolation formulas, is improved in the paper. This results particularly in a generalization of the well-known scheme by which periodic interpolatory wavelets are constructed. A number of systems which do not satisfy all the conditions for multiresolution analysis but have some useful properties are introduced and investigated. Starting with general constructions in Hilbert spaces, we give a more careful consideration to the case connected with the classic Fourier basis. Convergence of expansions which are similar to partial sums of the summation method of Fourier series, as well as convergence of interpolation formulas are considered. Some applications to fast calculation of Fourier coefficients and to solution of integrodifferential equations are given. The corresponding numerical results have been obtained by means of MATHEMATICA 3.0 system. 相似文献
13.
Ruymán Cruz-Barroso Leyla Daruis Pablo González-Vera Olav NjÅstad 《Journal of Computational and Applied Mathematics》2007
In this paper, the construction of orthogonal bases in the space of Laurent polynomials on the unit circle is considered. As an application, a connection with the so-called bi-orthogonal systems of trigonometric polynomials is established and quadrature formulas on the unit circle based on Laurent polynomials are studied. 相似文献
14.
Ayse Alaylioglu 《Journal of Computational and Applied Mathematics》1983,9(4):305-313
A computationally efficient algorithm for evaluating Fourier integrals ∫1?1?(x)eiωxdx using interpolatory quadrature formulas on any set of collocation points is presented. Examples are given to illustrate the performances of interpolatory formulas which are based on the applications of the Fejér, Clenshaw—Curtis, Basu and the Newton—Cotes points. Initially, the formulas for nonoscillatory integrals are generated and then generalizations to finite Fourier integrals are made. Extensions of this algorithm to some other weighted integrals are also considered. 相似文献
15.
Summary A method of integrating a function over a simplex is described in which (i) the simplex is first transformed into a right-angled isosceles simplex; (ii) this simplex is dissected into small cubes and truncated cubes; (iii) the integration over the truncated cubes is performed by the centroid method or by Stroud's method, and this requires the use of formulae for the moments of a truncated cube. These formulae are developed and are expressed in terms of Eulerian numbers. In the special case when the truncated cube is itself a right-angled isoceles simplex a new algorithm is given, depending on the discrete Fourier transform, for calculating the moments as polynomials inn wheren is the dimensionality. 相似文献
16.
C. Lubich 《Numerische Mathematik》1988,52(2):129-145
Numerical methods are derived for problems in integral equations (Volterra, Wiener-Hopf equations) and numerical integration (singular integrands, multiple time-scale convolution). The basic tool of this theory is the numerical approximation of convolution integrals
相似文献
17.
Yuan Xu 《Numerische Mathematik》1994,69(2):233-241
Summary.
The existence of Gaussian cubature for a given measure
depends on whether the corresponding multivariate orthogonal polynomials have
enough common zeros. We examine a class of orthogonal
polynomials of two variables generated from that of one variable.
Received February 9, 1993 / Revised version received
January 18, 1994 相似文献
18.
Summary. We generalize earlier results concerning
an asymptotic error expansion of wavelet
approximations. The properties of the monowavelets,
which are the building
blocks for the error expansion, are studied in more
detail, and connections
between spline wavelets and Euler and
Bernoulli polynomials are pointed out.
The expansion is used to compare the
error for different wavelet families.
We prove that the leading terms of the
expansion only depend on the multiresolution
subspaces and not
on how the complementary subspaces
are chosen.
Consequently, for a fixed set of
subspaces , the leading
terms do not depend on the fact whether
the wavelets are orthogonal or not.
We also show that Daubechies' orthogonal wavelets need,
in general, one level more than spline wavelets to obtain an
approximation with a prescribed accuracy.
These results are illustrated with numerical examples.
Received May 3, 1993 / Revised version received January 31, 1994 相似文献
19.
We study cyclicity of operators on a separable Banach space which admit a bicyclic vector such that the norms of its images
under the iterates of the operator satisfy certain growth conditions. A simple consequence of our main result is that a bicyclic
unitary operator on a Banach space with separable dual is cyclic. Our results also imply that if is the shift operator acting on the weighted space of sequences , if the weight ω satisfies some regularity conditions and ω(n) = 1 for nonnegative n, then S is cyclic if . On the other hand one can see that S is not cyclic if the series diverges. We show that the question of Herrero whether either S or S* is cyclic on admits a positive answer when the series is convergent. We also prove completeness results for translates in certain Banach spaces of functions on . 相似文献
20.
We present an extrapolation type algorithm for the numerical solution of fractional order differential equations. It is based
on the new result that the sequence of approximate solutions of these equations, computed by means of a recently published
algorithm by Diethelm [6], possesses an asymptotic expansion with respect to the stepsize. From this we conclude that the
application of extrapolation is justified, and we obtain a very efficient differential equation solver with practically no
additional numerical costs. This is also illustrated by a number of numerical examples.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
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