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1.
Summary We present a LagrangeC
2-interpolant to scattered convex data which preserves convexity. We also present a LagrangeC
2-interpolant to uniformly spaced monotone data sites which preserves monotonicity. In both cases no further conditions are required on the data values. These interpolants are explicitely described and local. Error isO(h
3) when the function to be interpolated isC
3. 相似文献
2.
We present a method to interpolate scattered monotone data in R
s
using a variational approach. We present both theoretical and practical properties and give a dual algorithm allowing us to compute the resulting function whens=2. The method is specially suited for scattered data but comparison with existing methods for data on grids shows that it is a valid approach even in that case.Communicated by Wolfgang Dahmen. 相似文献
3.
We propose a parametric tensioned version of the FVS macro-element to control the shape of the composite surface and remove artificial oscillations, bumps and other undesired behaviour. In particular, this approach is applied to C1 cubic spline surfaces over a four-directional mesh produced by two-stage scattered data fitting methods. 相似文献
4.
We study numerical integration for functions f with singularities. Nonadaptive methods are inefficient in this case, and we show that the problem can be efficiently solved by adaptive quadratures at cost similar to that for functions with no singularities.
Consider first a class of functions whose derivatives of order up to r are continuous and uniformly bounded for any but one singular point. We propose adaptive quadratures Q*n, each using at most n function values, whose worst case errors are proportional to n−r. On the other hand, the worst case error of nonadaptive methods does not converge faster than n−1.
These worst case results do not extend to the case of functions with two or more singularities; however, adaption shows its
power even for such functions in the asymptotic setting. That is, let F∞r be the class of r-smooth functions with arbitrary (but finite) number of singularities. Then a generalization of Q*n yields adaptive quadratures Q**n such that |I(f)−Q**n(f)|=O(n−r) for any f ∈ F∞r. In addition, we show that for any sequence of nonadaptive methods there are `many' functions in F∞r for which the errors converge no faster than n−1.
Results of numerical experiments are also presented.
The authors were partially supported, respectively, by the State Committee for Scientific Research of Poland under Project
1 P03A 03928 and by the National Science Foundation under Grant CCR-0095709. 相似文献
5.
Let {q}
j
=0n–1
be a family of polynomials that satisfy a three-term recurrence relation and let {t
k
}
k
=1n
be a set of distinct nodes. Define the Vandermonde-like matrixW
n
=[w
jk
]
k,j
=1n
,w
jk
=q
j–1(t
k
). We describe a fast algorithm for computing the elements of the inverse ofW
n
inO(n
2) arithmetic operations. Our algorithm generalizes a scheme presented by Traub [22] for fast inversion of Vandermonde matrices. Numerical examples show that our scheme often yields higher accuracy than the LINPACK subroutine SGEDI for inverting a general matrix. SGEDI uses Gaussian elimination with partial pivoting and requiresO(n
3) arithmetic operations.Dedicated to Gene H. Golub on his 60th birthdayResearch supported by NSF grant DMS-9002884. 相似文献
6.
For evaluation schemes based on the Lagrangian form of a polynomial with degreen, a rigorous error analysis is performed, taking into account that data, computation and even the nodes of interpolation might be perturbed by round-off. The error norm of the scheme is betweenn
2 andn
2+(3n+7)
n
, where
n
denotes the Lebesgue constant belonging to the nodes. Hence, the error norm is of least possible orderO(n
2) if, for instance, the nodes are chosen to be the Chebyshev points or the Fekete points. 相似文献
7.
Franz Peherstorfer 《BIT Numerical Mathematics》1990,30(1):145-151
In this note it is shown that for weight functions of the formw(t)=(1 –t
2)1/2/s
m
(t), wheres
m
is a polynomial of degreem which is positive on [–1, +1], successive Kronrod extension of a certain class ofN-point interpolation quadrature formulas, including theN-point Gauss-formula, is always possible and that each Kronrod extension has the positivity and interlacing property. 相似文献
8.
We discuss the evaluation of the Hilbert transformf
–1
1
(t-)–1
w(, )(t)dt,–1<<1, of the Jacobi weight functionw(, )(t)=(1–t))(1+t) by analytic and numerical means and also comment on the recursive computation of the quantitiesf
–1
1
)(t–)–1
n
(t;w
(, ))
w
(, )(t)dt,n=0, 1, 2, ..., where
n
(·;w
(, )) is the Jacobi polynomial of degreen.The work of the first author was supported in part by the National Science Foundation under grant DCR-8320561. The work of the second author was supported by the National Science Foundation under grant DMS-8419086. 相似文献
9.
M. N. El Tarazi 《BIT Numerical Mathematics》1990,30(3):484-489
The interpolation problem at uniform mesh points of a quadratic splines(x
i)=f
i,i=0, 1,...,N ands(x
0)=f0 is considered. It is known that s–f=O(h
3) and s–f=O(h
2), whereh is the step size, and that these orders cannot be improved. Contrary to recently published results we prove that superconvergence cannot occur for any particular point independent off other than mesh points wheres=f by assumption. Best error bounds for some compound formulae approximatingf
i
andf
i
(3)
are also derived. 相似文献
10.
Summary We continue the work of Part I, treating in detail the theory of numerical quadrature over a square [0, 1]2 using anm
2 copy,Q
(m), of a one-point quadrature rule. As before, we determine the nature of an asymptotic expansion for the quadrature error functionalQ
(m) F—IF in inverse powers ofm and related functions, valid for specified classes of the integrand functionF. The extreme case treated here is one in which the integrand function has a full-corner algebraic singularity. This has the formx
y r, (x, y). Here , , and need not be integer, andr
is (x
2+y
2)/2 or some other similar homogeneous function. The error expansion forms the theoretic basis for the use of extrapolation, for this kind of integrand.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38 相似文献
11.
Geno Nikolov 《Numerische Mathematik》1992,62(1):557-565
Summary LetI(f)L(f)=
k=0
r
=0
vk–1
a
k
f
()(X
k
) be a quadrature formula, and let {S
n
(f)}
n=1
be successive approximations of the definite integralI(f)=
0
1
f(x)dx obtained by the composition ofL, i.e.,S
n(f)=L(
n
), where
.We prove sufficient conditions for monotonicity of the sequence {S
n
(f)}
n=1
. As particular cases the monotonicity of well-known Newton-Cotes and Gauss quadratures is shown. Finally, a recovery theorem based on the monotonicity results is presented 相似文献
12.
This paper addresses the problem of constructing some free-form curves and surfaces from given to different types of data: exact and noisy data. We extend the theory of Dm-splines over a bounded domain for noisy data to the smoothing variational vector splines. Both results of convergence for respectively the exact and noisy data are established, as soon as some estimations of errors are given. 相似文献
13.
Summary A common strategy in the numerical integration over ann-dimensional hypercube or simplex, is to consider a regular subdivision of the integration domain intom
n
subdomains and to approximate the integral over each subdomain by means of a cubature formula. An asymptotic error expansion whenm is derived in case of an integrand with homogeneous boundary singularities. The error expansion also copes with the use of different cubature formulas for the boundary subdomains and for the interior subdomains. 相似文献
14.
Rémi Arcangéli María Cruz López de Silanes Juan José Torrens 《Numerische Mathematik》2005,101(4):573-599
We derive error estimates in W2,∞-semi-norms for multivariate discrete D2-splines that interpolate an unknown function at the vertices of given triangulations. These results are widely based on the
construction of approximation operators and linear projectors onto piecewise polynomial spaces having weakly stable local
bases. 相似文献
15.
Marie-Laurence Mazure 《Numerische Mathematik》2008,109(3):459-475
Via blossoms we analyse the dimension elevation process from to , where is spanned over [0, 1] by 1, x,..., x
n-2, x
p
, (1 − x)
q
, p, q being any convenient real numbers. Such spaces are not Extended Chebyshev spaces but Quasi Extended Chebyshev spaces. They
were recently introduced in CAGD for shape preservation purposes (Costantini in Math Comp 46:203–214; 1986, Costantini in
Advanced Course on FAIRSHAPE, pp. 87–114 in 1996; Costantini in Curves and Surfaces with Applications in CAGD, pp. 85–94,
1997). Our results give a new insight into the special case p = q for which dimension elevation had already been considered, first when p = q was supposed to be an integer (Goodman and Mazure in J Approx Theory 109:48–81, 2001), then without the latter requirement
(Costantini et al. in Numer Math 101:333–354, 2005). The question of dimension elevation in more general Quasi Extended Chebyshev
spaces is also addressed. 相似文献
16.
In convex interpolation the curvature of the interpolants should be as small as possible. We attack this problem by treating
interpolation subject to bounds on the curvature. In view of the concexity the lower bound is equal to zero while the upper
bound is assumed to be piecewise constant. The upper bounds are called fair with respect to a function class if the interpolation
problem becomes solvable for all data sets in strictly convex position. We derive fair a priori bounds for classes of quadraticC
1, cubicC
2, and quarticC
3 splines on refined grids. 相似文献
17.
Summary A functionf C (),
is called monotone on if for anyx, y the relation x – y
+
s
impliesf(x)f(y). Given a domain
with a continuous boundary and given any monotone functionf on we are concerned with the existence and regularity ofmonotone extensions i.e., of functionsF which are monotone on all of and agree withf on . In particular, we show that there is no linear mapping that is capable of producing a monotone extension to arbitrarily given monotone boundary data. Three nonlinear methods for constructing monotone extensions are then presented. Two of these constructions, however, have the common drawback that regardless of how smooth the boundary data may be, the resulting extensions will, in general, only be Lipschitz continuous. This leads us to consider a third and more involved monotonicity preserving extension scheme to prove that, when is the unit square [0, 1]2 in 2, strictly monotone analytic boundary data admit a monotone analytic extension.Research supported by NSF Grant 8922154Research supported by DARPA: AFOSR #90-0323 相似文献
18.
A method is described for the evaluation of a Cauchy principal value integral of the formf
0
p
f(t)dt, wheref is analytic in the interval [0,p] except at a simple pole at an unknown point in (0,p), with an unknown residue. The method is based on the trapezoidal rule. 相似文献
19.
John C. Clements 《Numerische Mathematik》1992,63(1):165-171
Summary AC
2 parametric rational cubic interpolantr(t)=x(t)
i+y(t)
j,t[t
1,t
n] to data S={(xj, yj)|j=1,...,n} is defined in terms of non-negative tension parameters
j
,j=1,...,n–1. LetP be the polygonal line defined by the directed line segments joining the points (x
j
,y
j
),t=1,...,n. Sufficient conditions are derived which ensure thatr(t) is a strictly convex function on strictly left/right winding polygonal line segmentsP. It is then proved that there always exist
j
,j=1,...,n–1 for whichr(t) preserves the local left/righ winding properties of any polygonal lineP. An example application is discussed.This research was supported in part by the natural Sciences and Engineering Research Council of Canada. 相似文献
20.
In this paper the interpolation byG
2 continuous planar cubic Bézier spline curves is studied. The interpolation is based upon the underlying curve points and
the end tangent directions only, and could be viewed as an extension of the cubic spline interpolation to the curve case.
Two boundary, and two interior points are interpolated per each spline section. It is shown that under certain conditions
the interpolation problem is asymptotically solvable, and for a smooth curvef the optimal approximation order is achieved. The practical experiments demonstrate the interpolation to be very satisfactory.
Supported in prat by the Ministry of Science and Technology of Slovenjia, and in part by the NSF and SF of National Educational
Committee of China. 相似文献