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1.
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
2.
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
3.
Bernard Bialecki 《BIT Numerical Mathematics》1989,29(3):464-476
Sinc function approach is used to obtain a quadrature rule for estimating integrals of functions with poles near the are of integration. Special treatment is given to integration over the intervals (–, ), (0, ), and (–1, 1). It is shown that the error of the quadrature rule converges to zero at the rateO(exp(–cN)) asN , whereN is the number of nodes used, and wherec is a positive constant which is independent ofN. 相似文献
4.
Summary This paper is concerned with the practical implementation of a product-integration rule for approximating
, wherek is integrable andf is continuous. The approximation is
, where the weightsw
ni
are such as to make the rule exact iff is any polynomial of degree n. A variety of numerical examples, fork(x) identically equal to 1 or of the form |–x| with >–1 and ||1, or of the form cosx or sinx, show that satisfactory rates of convergence are obtained for smooth functionsf, even ifk is very singular or highly oscillatory. Two error estimates are developed, and found to be generally safe yet quite accurate. In the special casek(x)1, for which the rule reduces to the Clenshaw-Curtis rule, the error estimates are found to compare very favourably with previous error estimates for the Clenshaw-Curtis rule. 相似文献
5.
Summary This paper is concerned with the theoretical properties of a productintegration method for the integral
, wherek is absolutely integrable andf is continuous. The integral is approximated by
, where the points are given byx
ni
=cos(i/n, 0in, and where the weightsw
ni
are chosen to make the rule exact iff is any polynomial of degree n. The principal result is that ifkL
p
[–1, 1] for somep>1, then the rule converges to the exact result asn for all continuous (or indeed R-integrable) functionsf, and moreover that the sum of the absolute values of the weights converges to the least possible value, namely
. A limiting expression for the individual weights is also obtained, under certain assumptions. The results are exteded to other point sets of a similar kind, including the classical Chebyshev points. 相似文献
6.
H. W. J. Lenferink 《Numerische Mathematik》1989,55(2):213-223
Summary We investigate contractivity properties of explicit linear multistep methods in the numerical solution of ordinary differential equations. The emphasis is on the general test-equation
, whereA is a square matrix of arbitrary orders1. The contractivity is analysed with respect to arbitrary norms in thes-dimensional space (which are not necessarily generated by an inner product). For given order and stepnumber we construct optimal multistep methods allowing the use of a maximal stepsize.This research has been supported by the Netherlands organisation for scientific research (NWO) 相似文献
7.
LetJ
n
(z) be the Bessel function of the first kind and ordern, and letf(z) be an analytic function in|z|r (r>0); then it is known that the Bessel expansion
相似文献
8.
Improved estimates of statistical regularization parameters in fourier differentiation and smoothing
Summary We investigate the statistical methods of cross-validation (CV) and maximum-likelihood (ML) for estimating optimal regularization parameters in the numerical differentiation and smoothing of non-exact data. Various criteria for optimality are examined, and the (asymptotic) notions of strong optimality, weak optimality and suboptimality are introduced relative to these criteria. By restricting attention to certainN-dimensional Hilbert spaces of smooth and stochastic functions, whereN is the number of data, we give regularity conditions on the data under which CV, the regularization parameter predicted by CV, is strongly optimal with respect to the predictive mean-square signal error. We show that ML is at best weakly optimal with respect to this criterion but is strongly optimal with respect to the innovation variance of the data. For numerical differentiation, CV and ML are both shown to be suboptimal with respect to the predictive mean-square derivative error. 相似文献
9.
We develop an approach to multivariable cubature based on positivity, extension, and completion properties of moment matrices. We obtain a matrix-based lower bound on the size of a cubature rule of degree 2n + 1; for a planar measure , the bound is based on estimating
where C:=C# [ ] is a positive matrix naturally associated with the moments of . We use this estimate to construct various minimal or near-minimal cubature rules for planar measures. In the case when C = diag(c1,...,cn) (including the case when is planar measure on the unit disk), (C) is at least as large as the number of gaps ck >ck+1. 相似文献
10.
Martin Kütz 《Numerische Mathematik》1982,39(3):421-428
Summary Let
, be holomorphic in an open disc with the centrez
0=0 and radiusr>1. LetQ
n
(n=1, 2, ...) be interpolatory quadrature formulas approximating the integral
. In this paper some classes of interpolatory quadratures are considered, which are based on the zeros of orthogonal polynomials corresponding to an even weight function. It is shown that the sequencesQ
n
9f] (n=1, 2, ...) are monotone. Especially we will prove monotony in Filippi's quadrature rule and with an additional assumption onf monotony in the Clenshaw-Curtis quadrature rule. 相似文献
11.
Shiquan WU 《应用数学学报(英文版)》1996,12(4):377-383
Letn, s
1,s
2, ... ands
n
be positive integers. Assume
is an integer for eachi}. For
,
, and
, denotes
p
(a)={j|1jn,a
j
p},
, and
.
is called anI
t
p
-intersecting family if, for any a,b
,a
i
b
i
=min(a
i
,b
i
)p for at leastt i's.
is called a greedyI
t
P
-intersecting family if
is anI
t
p
-intersecting family andW
p
(A)W
p
(B+A
c
) for anyAS
p
(
) and any
with |B|=t–1.In this paper, we obtain a sharp upper bound of |
| for greedyI
t
p
-intersecting families in
for the case 2ps
i
(1in) ands
1>s
2>...>s
n
.This project is partially supported by the National Natural Science Foundation of China (No.19401008) and by Postdoctoral Science Foundation of China. 相似文献
12.
The classical weighted spline introduced by Ph. Cinquin (1981), (see also K. Salkauskas (1984) and T.A. Foley (1986)) consists in minimizing
a
b
w(t)(x(t))2 dt under the conditionsx(t
i
)=y
i
,i=1,...,n, where the functionw is piecewise constant on the subdivisiona<t
1<t
2<...<t
n
<b. The solution is a cubic spline, but it is notC
2. We consider here the minimization of
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