共查询到20条相似文献,搜索用时 633 毫秒
1.
Among all integration rules with n points, it is well-known that n-point Gauss–Legendre quadrature rule∫−11f(x) dx∑i=1nwif(xi)has the highest possible precision degree and is analytically exact for polynomials of degree at most 2n−1, where nodes xi are zeros of Legendre polynomial Pn(x), and wi's are corresponding weights.In this paper we are going to estimate numerical values of nodes xi and weights wi so that the absolute error of introduced quadrature rule is less than a preassigned tolerance ε0, say ε0=10−8, for monomial functionsf(x)=xj, j=0,1,…,2n+1.(Two monomials more than precision degree of Gauss–Legendre quadrature rules.) We also consider some conditions under which the new rules act, numerically, more accurate than the corresponding Gauss–Legendre rules. Some examples are given to show the numerical superiority of presented rules. 相似文献
2.
Let f be an arithmetical function and S={x 1,x 2,…,xn } a set of distinct positive integers. Denote by [f(xi ,xj }] the n×n matrix having f evaluated at the greatest common divisor (xi ,xj ) of xi , and xj as its i j-entry. We will determine conditions on f that will guarantee the matrix [f(xi ,xj )] is positive definite and, in fact, has properties similar to the greatest common divisor (GCD) matrix [(xi ,xj )] where f is the identity function. The set S is gcd-closed if (xi ,xj )∈S for 1≤ i j ≤ n. If S is gcd-closed, we calculate the determinant and (if it is invertible) the inverse of the matrix [f(xi ,xj )]. Among the examples of determinants of this kind are H. J. S. Smith's determinant det[(i,j)]. 相似文献
3.
Birkholl quadrature formulae (q.f.), which have algebraic degree of precision (ADP) greater than the number of values used, are studied. In particular, we construct a class of quadrature rules of ADP = 2n + 2r + 1 which are based on the information {ƒ(j)(−1), ƒ(j)(−1), j = 0, ..., r − 1 ; ƒ(xi), ƒ(2m)(xi), i = 1, ..., n}, where m is a positive integer and r = m, or r = m − 1. It is shown that the corresponding Birkhoff interpolation problems of the same type are not regular at the quadrature nodes. This means that the constructed quadrature formulae are not of interpolatory type. Finally, for each In, we prove the existence of a quadrature formula based on the information {ƒ(xi), ƒ(2m)(xi), i = 1, ..., 2m}, which has algebraic degree of precision 4m + 1. 相似文献
4.
J. Musielak 《Journal of Approximation Theory》1987,50(4)
Let l be a generalized Orlicz sequence space generated by a modular (x) = ∑i − 0∞ i(¦ti¦), X = (ti), with s-convex functions i, 0 < s 1, and let Kw,j: R+ → R+ for j=0,1,2,…, w ε Wwhere
is an abstract set of indices. Assuming certain singularity assumptions on the nonlinear kernel Kw,j and setting Twx = ((Twx)i)i = 0∞, with (Twx)i = ∑j = 0i Kw,i − j(¦tj¦) for x = (tj), convergence results: Twx → x in l are obtained (both modular convergence and norm convergence), with respect to a filter
of subsets of the set
. 相似文献
5.
Jean B. Lasserre 《Optimization Letters》2011,5(4):549-556
We consider the convex optimization problem P:minx {f(x) : x ? K}{{\rm {\bf P}}:{\rm min}_{\rm {\bf x}} \{f({\rm {\bf x}})\,:\,{\rm {\bf x}}\in{\rm {\bf K}}\}} where f is convex continuously differentiable, and
K ì \mathbb Rn{{\rm {\bf K}}\subset{\mathbb R}^n} is a compact convex set with representation
{x ? \mathbb Rn : gj(x) 3 0, j = 1,?,m}{\{{\rm {\bf x}}\in{\mathbb R}^n\,:\,g_j({\rm {\bf x}})\geq0, j = 1,\ldots,m\}} for some continuously differentiable functions (g
j
). We discuss the case where the g
j
’s are not all concave (in contrast with convex programming where they all are). In particular, even if the g
j
are not concave, we consider the log-barrier function fm{\phi_\mu} with parameter μ, associated with P, usually defined for concave functions (g
j
). We then show that any limit point of any sequence (xm) ì K{({\rm {\bf x}}_\mu)\subset{\rm {\bf K}}} of stationary points of fm, m? 0{\phi_\mu, \mu \to 0} , is a Karush–Kuhn–Tucker point of problem P and a global minimizer of f on K. 相似文献
6.
Summability of spherical h-harmonic expansions with respect to the weight function ∏j=1d |xj|2κj (κj0) on the unit sphere Sd−1 is studied. The main result characterizes the critical index of summability of the Cesàro (C,δ) means of the h-harmonic expansion; it is proved that the (C,δ) means of any continuous function converge uniformly in the norm of C(Sd−1) if and only if δ>(d−2)/2+∑j=1d κj−min1jd κj. Moreover, it is shown that for each point not on the great circles defined by the intersection of the coordinate planes and Sd−1, the (C,δ) means of the h-harmonic expansion of a continuous function f converges pointwisely to f if δ>(d−2)/2. Similar results are established for the orthogonal expansions with respect to the weight functions ∏j=1d |xj|2κj(1−|x|2)μ−1/2 on the unit ball Bd and ∏j=1d xjκj−1/2(1−|x|1)μ−1/2 on the simplex Td. As a related result, the Cesàro summability of the generalized Gegenbauer expansions associated to the weight function |t|2μ(1−t2)λ−1/2 on [−1,1] is studied, which is of interest in itself. 相似文献
7.
Michael Revers 《Journal of Approximation Theory》2000,103(2):385
In 1918 S. N. Bernstein published the surprising result that the sequence of Lagrange interpolation polynomials to |x| at equally spaced nodes in [−1, 1] diverges everywhere, except at zero and the end-points. In the present paper, we prove that the sequence of Lagrange interpolation polynomials corresponding to |x|α (0<α1) on equidistant nodes in [−1, 1] diverges everywhere in the interval except at zero and the end-points. 相似文献
8.
In a symmetrizable Kac–Moody algebra g(A), let α=∑i=1nkiαi be an imaginary root satisfying ki>0 and α,αi<0 for i=1,2,…,n. In this paper, it is proved that for any xαgα{0}, satisfying [xα,fn]≠0 and [xα,fi]=0 for i=1,2,…,n−1, there exists a vector y such that the subalgebra generated by xα and y contains g′(A), the derived subalgebra of g(A). 相似文献
9.
M. G. Pleshakov 《Journal of Approximation Theory》1999,99(2):519
Let 2s points yi=−πy2s<…<y1<π be given. Using these points, we define the points yi for all integer indices i by the equality yi=yi+2s+2π. We shall write fΔ(1)(Y) if f is a 2π-periodic continuous function and f does not decrease on [yi, yi−1], if i is odd; and f does not increase on [yi, yi−1], if i is even. In this article the following Theorem 1—the comonotone analogue of Jackson's inequality—is proved.
1. If fΔ(1)(Y), then for each nonnegative integer n there is a trigonometric polynomial τn(x) of order n such that τnΔ(1)(Y), and |f(x)−πn(x)|c(s) ω(f; 1/(n+1)), x
, where ω(f; t) is the modulus of continuity of f, c(s)=const. Depending only on s. 相似文献
10.
Oleg Burdakov 《BIT Numerical Mathematics》1997,37(3):591-599
The following problem is considered. Givenm+1 points {x
i
}0
m
inR
n
which generate anm-dimensional linear manifold, construct for this manifold a maximally linearly independent basis that consists of vectors
of the formx
i
−x
j
. This problem is present in, e.g., stable variants of the secant and interpolation methods, where it is required to approximate
the Jacobian matrixf′ of a nonlinear mappingf by using values off computed atm+1 points. In this case, it is also desirable to have a combination of finite differences with maximal linear independence.
As a natural measure of linear independence, we consider the hadamard condition number which is minimized to find an optimal
combination ofm pairs {x
i
,x
j
}. We show that the problem is not NP-hard, but can be reduced to the minimum spanning tree problem, which is solved by the
greedy algorithm inO(m
2) time. The complexity of this reduction is equivalent to onem×n matrix-matrix multiplication, and according to the Coppersmith-Winograd estimate, is belowO(n
2.376) form=n. Applications of the algorithm to interpolation methods are discussed.
Part of the work was done while the author was visiting DIMACS, an NSF Science and Technology Center funded under contract
STC-91-19999; DIMACS is a cooperative project of Rutgers University, Princeton University, AT&T Bell Laboratories and Bellcore,
NJ, USA. 相似文献
11.
Orthogonal expansions in product Jacobi polynomials with respect to the weight function Wα, β(x)=∏dj=1 (1−xj)αj (1+xj)βj on [−1, 1]d are studied. For αj, βj>−1 and αj+βj−1, the Cesàro (C, δ) means of the product Jacobi expansion converge in the norm of Lp(Wα, β, [−1, 1]d), 1p<∞, and C([−1, 1]d) if
Moreover, for αj, βj−1/2, the (C, δ) means define a positive linear operator if and only if δ∑di=1 (αi+βi)+3d−1. 相似文献
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12.
Rong-Qing Jia 《Journal of Approximation Theory》1983,37(4):293-310
For an integer k 1 and a geometric mesh (qi)−∞∞ with q ε (0, ∞), let Mi,k(x): = k[qi + k](· − x)+k − 1, Ni,k(x): = (qi + k − qiMi,k(x)/k, and let Ak(q) be the Gram matrix (∝Mi,kNj,k)i,jεz. It is known that Ak(q)−1∞ is bounded independently of q. In this paper it is shown that Ak(q)−1∞ is strictly decreasing for q in [1, ∞). In particular, the sharp upper bound and lower bound for Ak (q)−1 are obtained: for all q ε (0, ∞). 相似文献
13.
Jaromír Šimša 《Aequationes Mathematicae》1992,44(1):35-41
Summary We consider the problem when a scalar function ofn variables can be represented in the form of a determinant det(f
i
(x
j
)), the so-called Casorati determinant off
1,f
2,,f
n
. The result is applied to the solution of some functional equations with unknown functionsH of two variables that involve determinants det(H(x
i
,x
j
)). 相似文献
14.
Wolfram Luther 《BIT Numerical Mathematics》1995,35(3):352-360
In this article we describe a fast method to obtain highly accurate tables for all elementary functions by using Bresenham's algorithm. For nearly equally spaced table-points {x
i
} we construct pairs {f(x
i
),g(x
i
)} such thatf(x
i
) is a machine number andg(x
i
) is very close to an exactly representable number. By a random sampling in an interval centered onx
i
we can even find a triplet
of nearly machine numbers. The table method together with a polynomial approximation of the function near a table value provides last bit accuracy for more than 99.8% of the argument values without using extended precision calculations [3, 4, 10, 11]. 相似文献
15.
We consider the pseudo-Euclidean space (R
n
, g), with n ≥ 3 and g
ij
= δ
ij
ε
i
, ε
i
= ±1, where at least one ε
i
= 1 and nondiagonal tensors of the form T = Σ
ij
f
ij
dx
i
dx
j
such that, for i ≠ j, f
ij
(x
i
, x
j
) depends on x
i
and x
j
. We provide necessary and sufficient conditions for such a tensor to admit a metric ḡ, conformal to g, that solves the Ricci tensor equation or the Einstein equation. Similar problems are considered for locally conformally
flat manifolds. Examples are provided of complete metrics on R
n
, on the n-dimensional torus T
n
and on cylinders T
k
×R
n-k
, that solve the Ricci equation or the Einstein equation.
Partially supported by CAPES/PROCAD.
Partially Supported By Cnpq, Capes/Procad. 相似文献
16.
For a stationary autoregressive process of order p and disturbance variance σ2 it is shown that the determinant of the covariance of T (≥p) consecutive random variables of the process is (σ2)T Πi,j=1p (1 − wiwj)−1, where w1, …, wp are the roots of the associated polynomial equation. 相似文献
17.
Liang Hanying Zhang Dongxia Lu Baoxian Dept.of Appl.Math. Tongji Univ. Shanghai China. 《高校应用数学学报(英文版)》2004,19(3):302-310
§1 IntroductionConsider the following heteroscedastic regression model:Yi =g(xi) +σiei, 1≤i≤n,(1.1)whereσ2i=f(ui) ,(xi,ui) are nonrandom design points,0≤x0 ≤x1 ≤...≤xn=1and0≤u0≤u1 ≤...≤un=1,Yi are the response variables,ei are random errors,and f(·) andg(·) are unknown functions defined on closed interval[0 ,1] .It is well known thatregression model has many applications in practical problems,sothe model (1.1) and its special cases have been studied extensively. For instance,… 相似文献
18.
Ronald DeVore Guergana Petrova Przemyslaw Wojtaszczyk 《Constructive Approximation》2011,33(1):125-143
Let f be a continuous function defined on Ω:=[0,1]
N
which depends on only ℓ coordinate variables, f(x1,?,xN)=g(xi1,?,xil)f(x_{1},\ldots,x_{N})=g(x_{i_{1}},\ldots,x_{i_{\ell}}). We assume that we are given m and are allowed to ask for the values of f at m points in Ω. If g is in Lip1 and the coordinates i
1,…,i
ℓ
are known to us, then by asking for the values of f at m=L
ℓ
uniformly spaced points, we could recover f to the accuracy |g|Lip1
L
−1 in the norm of C(Ω). This paper studies whether we can obtain similar results when the coordinates i
1,…,i
ℓ
are not known to us. A prototypical result of this paper is that by asking for C(ℓ)L
ℓ
(log 2
N) adaptively chosen point values of f, we can recover f in the uniform norm to accuracy |g|Lip1
L
−1 when g∈Lip1. Similar results are proven for more general smoothness conditions on g. Results are also proven under the assumption that f can be approximated to some tolerance ε (which is not known) by functions of ℓ variables. 相似文献
19.
We consider boolean circuits C over the basis Ω={,} with inputs x1, x2,…,xn for which arrival times are given. For 1in we define the delay of xi in C as the sum of ti and the number of gates on a longest directed path in C starting at xi. The delay of C is defined as the maximum delay of an input.Given a function of the form
f(x1,x2,…,xn)=gn−1(gn−2(…g3(g2(g1(x1,x2),x3),x4)…,xn−1),xn)