共查询到20条相似文献,搜索用时 46 毫秒
1.
Erich Novak 《Numerische Mathematik》1986,50(2):245-252
Summary The definition of the average error of numerical methods (by example of a quadrature formula
to approximateS(f)= f d on a function classF) is difficult, because on many important setsF there is no natural probability measure in the sense of an equidistribution. We define the average a posteriori error of an approximation
by an averaging process over the set of possible information, which is used by
(in the example of a quadrature formula,N(F)={(f(a
1), ...,f/fF} is the set of posible information). This approach has the practical advantage that the averaging process is related only to finite dimensional sets and uses only the usual Lebesgue measure. As an application of the theory I consider the numerical integration of functions of the classF={f:[0,1]/f(x)–f(y)||x–y|}. For arbitrary (fixed) knotsa
i
we determine the optimal coefficientsc
i
for the approximation
and compute the resulting average error. The latter is minimal for the knots
. (It is well known that the maximal error is minimal for the knotsa
i
.) Then the adaptive methods for the same problem and methods for seeking the maximum of a Lipschitz function are considered. While adaptive methods are not better when considering the maximal error (this is valid for our examples as well as for many others) this is in general not the case with the average error. 相似文献
2.
In this paper, we analyze a class of methods for minimizing a proper lower semicontinuous extended-valued convex function
. Instead of the original objective function f, we employ a convex approximation f
k
+ 1 at the kth iteration. Some global convergence rate estimates are obtained. We illustrate our approach by proposing (i) a new family of proximal point algorithms which possesses the global convergence rate estimate
even it the iteration points are calculated approximately, where
are the proximal parameters, and (ii) a variant proximal bundle method. Applications to stochastic programs are discussed. 相似文献
3.
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
4.
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
5.
A. Hinkkanen 《Inventiones Mathematicae》1992,108(1):549-574
Letf be meromorphic in the plane. We find a sharp upper bound for the error term
相似文献
6.
Using the following notation: C is the space of continuous bounded functions f equipped with the norm
, V is the set of functions f such that
, the set E consists of fCV and possesses the following property:
7.
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
8.
M. S. Lyapina 《Journal of Mathematical Sciences》2004,120(2):1109-1116
A bi-Lipschitz continuous mapping of a space X is a bijection
such that
, where
. We write
if f is a Lipschitz (bi-Lipschitz) mapping of X into itself and denote by
the set of all bi-Lipschitz mappings of X that are not isometry. Thus,
if
and blip
. For X we consider a standard Cantor set K on the real line (with standard metric). The main result of this paper is formulated as follows:
where
Bibliography: 2 titles. 相似文献
9.
Ian H. Sloan 《Numerische Mathematik》1981,38(2):263-278
Summary This paper is concerned with a class of approximation methods for integral equations of the form
, wherea andb are finite,f andy are continuous and the kernelk may be weakly singular. The methods are characterized by approximate equations of the form
; such methods include the Nyström method and a variety of product-integration methods. A general convergence theory is developed for methods of this type. In suitable cases it has the feature that its application to a specific method depends only on a knowledge of convergence properties of the underlying quadrature rule. The theory is used to deduce convergence results, some of them new, for a number of specific methods.Work supported by the U.S. Department of Energy 相似文献
10.
Estimates for deviations are established for a large class of linear methods of approximation of periodic functions by linear combinations of moduli of continuity of different orders. These estimates are sharp in the sense of constants in the uniform and integral metrics. In particular, the following assertion concerning approximation by splines is proved: Suppose that
is odd,
. Then
11.
The paper deals with the system
12.
O. L. Vinogradov 《Journal of Mathematical Sciences》2001,107(4):3987-4001
In what follows, C is the space of
-periodic continuous real-valued functions with uniform norm,
is the first continuity modulus of a function
with step h, H
n is the set of trigonometric polynomials of order at most n,
is the set of linear positive operators
(i.e., of operators such that
for every
),
is the space of square-integrable functions on
,
13.
In what follows, $C$ is the space of
-periodic continuous functions; P is a seminorm defined on C, shift-invariant, and majorized by the uniform norm;
is the mth modulus of continuity of a function f with step h and calculated with respect to P;
,
(
),
,
14.
To an evolution family on the half-line
of bounded operators on a Banach space X we associate operators IX and IZ related
to the integral equation
and a closed
subspace Z of X. We characterize the exponential dichotomy of
by the
exponential dichotomy and the quasi-exponential dichotomy of the operators
X we associate operators IX and IZ, respectively. 相似文献
15.
For f L
n
(T
d
) and
, the modulus of smoothness
16.
Summary In this paper we consider the approximate evaluation of
, whereK(x) is a fixed Lebesgue integrable function, by product formulas of the form
based on cubic spline interpolation of the functionf.Generally, whenever it is possible, product quadratures incorporate the bad behaviour of the integrand in the kernelK. Here, however, we allowf to have a finite number of jump discontinuities in [a, b]. Convergence results are established and some numerical applications are given for a logarithmic singularity structure in the kernel.Work sponsored by the Ministero della Pubblica Istruzione of Italy 相似文献
17.
Václav Tryhuk 《Czechoslovak Mathematical Journal》2000,50(3):509-518
The paper describes the general form of an ordinary differential equation of the order n + 1 (n 1) which allows a nontrivial global transformation consisting of the change of the independent variable. A result given by J. Aczél is generalized. A functional equation of the form
where
are given functions,
is solved on
. 相似文献
18.
The one-step prediction problem is studied in the context ofP
n-weakly stationary stochastic processes
, where
is an orthogonal polynomial sequence defining a polynomial hypergroup on
. This kind of stochastic processes appears when estimating the mean of classical weakly stationary processes. In particular, it is investigated whether these processes are asymptoticP
n-deterministic, i.e. the prediction mean-squared error tends to zero. Sufficient conditions on the covariance function or the spectral measure are given for
being asymptoticP
n-deterministic. For Jacobi polynomialsP
n(x) the problem of
being asymptoticP
n-deterministic is completely solved. 相似文献
19.
D. F. Kuznetsov 《Journal of Mathematical Sciences》2002,109(6):2148-2165
An expansion of multiple Stratonovich stochastic integrals of multiplicity
, into multiple series of products of Gaussian random variables is obtained. The coefficients of this expansion are the coefficients of multiple Fourier-series expansion of a function of several variables relative to a complete orthonormal system in the space
. The convergence in mean of order
, is established. Some expansions of multiple Stratonovich stochastic integrals with the help of polynomial and trigonometric systems are considered. Bibliography: 8 titles. 相似文献
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