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1.
Summary. Using a method based on quadratic nodal spline interpolation, we define a quadrature rule with respect to arbitrary nodes,
and which in the case of uniformly spaced nodes corresponds to the Gregory rule of order two, i.e. the Lacroix rule, which
is an important example of a trapezoidal rule with endpoint corrections. The resulting weights are explicitly calculated,
and Peano kernel techniques are then employed to establish error bounds in which the associated error constants are shown
to grow at most linearly with respect to the mesh ratio parameter. Specializing these error estimates to the case of uniform
nodes, we deduce non-optimal order error constants for the Lacroix rule, which are significantly smaller than those calculated
by cruder methods in previous work, and which are shown here to compare favourably with the corresponding error constants
for the Simpson rule.
Received July 27, 1998/ Revised version received February 22, 1999 / Published online January 27, 2000 相似文献
2.
J. M. Carnicer 《Numerical Algorithms》1992,3(1):125-132
In this paper the necessary and sufficient conditions for given data to admit a rational interpolant in
k,1 with no poles in the convex hull of the interpolation points is studied. A method for computing the interpolant is also provided.Partially supported by DGICYT-0121. 相似文献
3.
Solveig Bruvoll 《Journal of Computational and Applied Mathematics》2010,233(7):1631-1639
Mean value interpolation is a simple, fast, linearly precise method of smoothly interpolating a function given on the boundary of a domain. For planar domains, several properties of the interpolant were established in a recent paper by Dyken and the second author, including: sufficient conditions on the boundary to guarantee interpolation for continuous data; a formula for the normal derivative at the boundary; and the construction of a Hermite interpolant when normal derivative data is also available. In this paper we generalize these results to domains in arbitrary dimension. 相似文献
4.
Annie Cuyt 《BIT Numerical Mathematics》1988,28(1):98-112
The problem of constructing a univariate rational interpolant or Padé approximant for given data can be solved in various equivalent ways: one can compute the explicit solution of the system of interpolation or approximation conditions, or one can start a recursive algorithm, or one can obtain the rational function as the convergent of an interpolating or corresponding continued fraction.In case of multivariate functions general order systems of interpolation conditions for a multivariate rational interpolant and general order systems of approximation conditions for a multivariate Padé approximant were respectively solved in [6] and [9]. Equivalent recursive computation schemes were given in [3] for the rational interpolation case and in [5] for the Padé approximation case. At that moment we stated that the next step was to write the general order rational interpolants and Padé approximants as the convergent of a multivariate continued fraction so that the univariate equivalence of the three main defining techniques was also established for the multivariate case: algebraic relations, recurrence relations, continued fractions. In this paper a multivariate qd-like algorithm is developed that serves this purpose. 相似文献
5.
Ghislain Franssens 《Advances in Computational Mathematics》1999,10(3-4):367-388
A new C
∞ interpolant is presented for the univariate Hermite interpolation problem. It differs from the classical solution in that
the interpolant is of non‐polynomial nature. Its basis functions are a set of simple, compact support, transcendental functions.
The interpolant can be regarded as a truncated Multipoint Taylor series. It has essential singularities at the sample points,
but is well behaved over the real axis and satisfies the given functional data. The interpolant converges to the underlying
real‐analytic function when (i) the number of derivatives at each point tends to infinity and the number of sample points
remains finite, and when (ii) the spacing between sample points tends to zero and the number of specified derivatives at each
sample point remains finite.
A comparison is made between the numerical results achieved with the new method and those obtained with polynomial Hermite
interpolation. In contrast with the classical polynomial solution, the new interpolant does not suffer from any ill conditioning,
so it is always numerically stable. In addition, it is a much more computationally efficient method than the polynomial approach.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
6.
We discuss multivariate interpolation with some radial basis function, called radial basis function under tension (RBFT). The RBFT depends on a positive parameter which provides a convenient way of controlling the behavior of the interpolating surface. We show that our RBFT is conditionally positive definite of order at least one and give a construction of the native space, namely a semi-Hilbert space with a semi-norm, minimized by such an interpolant. Error estimates are given in terms of this semi-norm and numerical examples illustrate the behavior of interpolating surfaces. 相似文献
7.
P. Köhler 《Numerische Mathematik》1995,72(1):93-116
Summary.
We show that, for integrals with arbitrary integrable weight functions,
asymptotically best quadrature formulas with equidistant nodes can be
obtained by applying a certain scheme of piecewise polynomial interpolation
to the function
to be integrated, and then integrating this interpolant.
Received August 7, 1991 相似文献
8.
J. M. Carnicer 《Advances in Computational Mathematics》1995,3(1):395-404
A set of multivariate data is called strictly convex if there exists a strictly convex interpolant to these data. In this
paper we characterize strict convexity of Lagrange and Hermite multivariate data by a simple property and show that for strict
convex data and given smoothness requirements there exists a smooth strictly convex interpolant. We also show how to construct
a multivariate convex smooth interpolant to scattered data.
Partially supported by DGICYT PS93-0310 and by the EC project CHRX-CT94-0522. 相似文献
9.
Rational interpolation through the optimal attachment of poles to the interpolating polynomial 总被引:1,自引:0,他引:1
After recalling some pitfalls of polynomial interpolation (in particular, slopes limited by Markov's inequality) and rational
interpolation (e.g., unattainable points, poles in the interpolation interval, erratic behavior of the error for small numbers
of nodes), we suggest an alternative for the case when the function to be interpolated is known everywhere, not just at the
nodes. The method consists in replacing the interpolating polynomial with a rational interpolant whose poles are all prescribed,
written in its barycentric form as in [4], and optimizing the placement of the poles in such a way as to minimize a chosen
norm of the error.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
10.
Gašper Jakli? Jernej Kozak Marjeta Krajnc Emil ?agar 《Journal of Computational and Applied Mathematics》2010,233(7):1704-164
In this paper, (d+1)-pencil lattices on simplicial partitions in Rd are studied. The barycentric approach naturally extends the lattice from a simplex to a simplicial partition, providing a continuous piecewise polynomial interpolant over the extended lattice. The number of degrees of freedom is equal to the number of vertices of the simplicial partition. The constructive proof of this fact leads to an efficient computer algorithm for the design of a lattice. 相似文献
11.
R. CairaF. Dell’Accio F. Di Tommaso 《Journal of Computational and Applied Mathematics》2012,236(7):1691-1707
We propose a new combination of the bivariate Shepard operators (Coman and Trîmbi?a?, 2001 [2]) by the three point Lidstone polynomials introduced in Costabile and Dell’Accio (2005) [7]. The new combination inherits both degree of exactness and Lidstone interpolation conditions at each node, which characterize the interpolation polynomial. These new operators find application to the scattered data interpolation problem when supplementary second order derivative data are given (Kraaijpoel and van Leeuwen, 2010 [13]). Numerical comparison with other well known combinations is presented. 相似文献
12.
In this paper, we study the global behavior of a function that is known to be small at a given discrete data set. Such a function
might be interpreted as the error function between an unknown function and a given approximant. We will show that a small
error on the discrete data set leads under mild assumptions automatically to a small error on a larger region. We will apply
these results to spline smoothing and show that a specific, a priori choice of the smoothing parameter is possible and leads
to the same approximation order as the classical interpolant. This has also a surprising application in stabilizing the interpolation
process by splines and positive definite kernels. 相似文献
13.
J. M. Carnicer 《Numerical Algorithms》1991,1(2):155-176
For givenk-convex data, ak-convex interpolant is sought, so that a certain convex functional related with thek-th derivative is minimized.Partially supported by C.I.C.Y.T. PS87/0060. 相似文献
14.
In this note interpolation by real polynomials of several real variables is treated. Existence and unicity of the interpolant for knot systems being the perspective images of certain regular knot systems is discussed. Moreover, for such systems a Newton interpolation formula is derived allowing a recursive computation of the interpolant via multivariate divided differences. A numerical example is given.Partially supported by CICYT Res. Grant PS 87/0060 and by a Europe Travel Grant CAI-CONAI, Spain, 1988. 相似文献
15.
Principal lattices are classical simplicial configurations of nodes suitable for multivariate polynomial interpolation in
n dimensions. A principal lattice can be described as the set of intersection points of n + 1 pencils of parallel hyperplanes. Using a projective point of view, Lee and Phillips extended this situation to n + 1 linear pencils of hyperplanes. In two recent papers, two of us have introduced generalized principal lattices in the
plane using cubic pencils. In this paper we analyze the problem in n dimensions, considering polynomial, exponential and trigonometric pencils, which can be combined in different ways to obtain
generalized principal lattices.We also consider the case of coincident pencils. An error formula for generalized principal
lattices is discussed.
Partially supported by the Spanish Research Grant BFM2003-03510, by Gobierno de Aragón and Fondo Social Europeo. 相似文献
16.
Jean-Paul Berrut 《Numerical Algorithms》2000,24(1-2):17-29
Among the representations of rational interpolants, the barycentric form has several advantages, for example, with respect to stability of interpolation, location of unattainable points and poles, and differentiation. But it also has some drawbacks, in particular the more costly evaluation than the canonical representation. In the present work we address this difficulty by diminishing the number of interpolation nodes embedded in the barycentric form. This leads to a structured matrix, made of two (modified) Vandermonde and one Löwner, whose kernel is the set of weights of the interpolant (if the latter exists). We accordingly modify the algorithm presented in former work for computing the barycentric weights and discuss its efficiency with several examples. 相似文献
17.
This is the second part of a note on interpolation by real polynomials of several real variables. For certain regular knot systems (geometric or regular meshes, tensor product grids), Neville-Aitken algorithms are derived explicitly. By application of a projectivity they can be extended in a simple way to arbitrary (k+1)-pencil lattices as recently introduced by Lee and Phillips. A numerical example is given.Partially supported by CICYT Res. Grant PS87-0060.Travel Grant Programa Europa 1991, CAI Zaragoza. 相似文献
18.
We discuss polynomial interpolation in several variables from a polynomial ideal point of view. One of the results states
that if I is a real polynomial ideal with real variety and if its codimension is equal to the cardinality of its variety, then for
each monomial order there is a unique polynomial that interpolates on the points in the variety. The result is motivated by
the problem of constructing cubature formulae, and it leads to a theorem on cubature formulae which can be considered an extension
of Gaussian quadrature formulae to several variables.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
19.
Daan Huybrechs 《Journal of Computational and Applied Mathematics》2009,231(2):933-947
Newton-Cotes quadrature rules are based on polynomial interpolation in a set of equidistant points. They are very useful in applications where sampled function values are only available on a regular grid. Yet, these rules rapidly become unstable for high orders. In this paper we review two techniques to construct stable high-order quadrature rules using equidistant quadrature points. The stability follows from the fact that all coefficients are positive. This result can be achieved by allowing the number of quadrature points to be larger than the polynomial order of accuracy. The computed approximations then implicitly correspond to the integral of a least squares approximation of the integrand. We show how the underlying discrete least squares approximation can be optimised for the purpose of numerical integration. 相似文献
20.
Shayne Waldron 《Numerische Mathematik》1998,80(3):461-494
Summary. The main result of this paper is an abstract version of the Kowalewski–Ciarlet–Wagschal
multipoint Taylor formula for representing the pointwise error in multivariate Lagrange interpolation. Several applications of this result are given
in the paper. The most important of these is the construction of a multipoint Taylor error formula for a general finite element, together with the corresponding –error bounds. Another application is the construction of a family of error formul? for linear interpolation (indexed by real
measures of unit mass) which includes some recently obtained formul?. It is also shown how the problem of constructing an
error formula for Lagrange interpolation from a D–invariant space of polynomials with the property that it involves only derivatives which annihilate the interpolating space
can be reduced to the problem of finding such a formula for a ‘simpler’ one–point interpolation map.
Received March 29, 1996 / Revised version received November 22, 1996 相似文献