共查询到20条相似文献,搜索用时 15 毫秒
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通过引进新的参数,将对称型插值的一般框架作进一步推广和改进,新的插值框架包含更为丰富的插值格式;给出几种新形式的对称型有理插值格式;最后,将结果推广到向量值及矩阵值情形. 相似文献
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The paper describes and studies an iterative algorithm for finding small values of a set of linear forms over vectors of integers. The algorithm uses a linear recurrence relation to generate a vector sequence, the basic idea being to choose the integral coefficients in the recurrence relation in such a way that the linear forms take small values, subject to the requirement that the integers should not become too large. The problem of choosing good coefficients for the recurrence relation is thus related to the problem of finding a good approximation of a given vector by a vector in a certain one-parameter family of lattices; the novel feature of our approach is that practical formulae for the coefficients are obtained by considering the limit as the parameter tends to zero. The paper discusses two rounding procedures to solve the underlying inhomogeneous Diophantine approximation problem: the first, which we call ``naive rounding' leads to a multidimensional continued fraction algorithm with suboptimal asymptotic convergence properties; in particular, when it is applied to the familiar problem of simultaneous rational approximation, the algorithm reduces to the classical Jacobi-Perron algorithm. The second rounding procedure is Babai's nearest-plane procedure. We compare the two rounding procedures numerically; our experiments suggest that the multidimensional continued fraction corresponding to nearest-plane rounding converges at an optimal asymptotic rate.
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William B. Jones Olav Njåstad W.J. Thron 《Journal of Computational and Applied Mathematics》1983,9(2):105-123
Each member G(z) of a family of analytic functions defined by Stieltjes transforms is shown to be represented by a positive T-fraction, the approximants of which form the main diagonal in the two-point Padé table of G(z). The positive T-fraction is shown to converge to G(z) throughout a domain D(a, b) = [z: z?[?b, ?a]], uniformly on compact subsets. In addition, truncation error bounds are given for the approximants of the continued function; these bounds supplement previously known bounds and apply in part of the domain of G(z) not covered by other bounds. The proofs of our results employ properties of orthogonal -polynomials (Laurent polynomials) and -Gaussian quadrature which are of some interest in themselves. A number of examples are considered. 相似文献
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In this paper we are concerned with a non-negative integer and irreducible matrix . The main contribution is to prove that if the matrix satisfies certain spectral and algebraic constraints, the cone:is defined by linear maps , in the sense that v ∈ C is equivalent to, ϕl(v) ⩾ 0 for all l = 0, … , k − 1 (where k is the index of cyclicity of the irreducible matrix). This result allows us to characterize the dimension group generated by the matrix, it is a subgroup of endowed with an order induced by the positive cone of . 相似文献
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Zhu Cao 《Journal of Mathematical Analysis and Applications》2010,365(2):659-667
We give proofs of a list of M. Somos' dissection identities. An eta function identity presented by B.C. Berndt and W.B. Hart, a theorem by H.-C. Chan on the congruence property of a(n) with generating function , and a theorem by G.E. Andrews, A. Schilling, and S.O. Warnaar are shown to be related to dissection identities. Several new corollaries are also presented as applications. 相似文献
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Peter Borwein Kevin G. Hare 《Transactions of the American Mathematical Society》2003,355(12):4767-4779
This paper gives exact rates of quadratic approximations to an infinite class of cubic Pisot numbers. We show that for any cubic Pisot number , with minimal polynomial , such that , and where has only one real root, then there exists a , explicitly given here, such that:
- (1)
- For all 0$">, all but finitely many integer quadratics satisfy
where is the height function. - (2)
- For all 0$"> there exists a sequence of integer quadratics such that
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Khrystyna Kuchmins"ka 《Acta Appl Math》2000,61(1-3):175-183
By the method of majorant fractions and equivalent transformations, the analogies of leszyski–Pringsheim criteria for two-dimensional continued fractions are obtained. 相似文献
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本文建立了Clifford连分式的三项递推关系和Pincherle's定理,并给出了它们的应用,也获得了关于Clifford连分式的矩阵递推关系的最小解的几个性质. 相似文献
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J.H. McCabe 《Journal of Computational and Applied Mathematics》1983,9(2):125-130
Recurrence relations for the coefficients in the asymptotic expansion of a gamma function ratio are derived and a property of these coefficients is proved. The Stieltjes fraction for the series is given and a characteristic of the partial numerators is explained. A connection between the continued fraction and the error of a particular least squares approximation problem is discussed. 相似文献
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Haipeng QU 《Frontiers of Mathematics in China》2010,5(1):117-122
For a positive integer n, a finite p-group G is called an ℳ
n
-group, if all subgroups of index p
n
of G are metacyclic, but there is at least one subgroup of index p
n−1 that is not. A classical result in p-group theory is the classification of ℳ1-groups by Blackburn. In this paper, we give a slightly shorter and more elementary proof of this result. 相似文献
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Let z∊ C be imaginary quadratic in the upper half plane. Then the Rogers-Ramanujan continued fraction evaluated at q = e 2π iz is contained in a class field of Q(z). Ramanujan showed that for certain values of z, one can write these continued fractions as nested radicals. We use the Shimura reciprocity law to obtain such nested radicals
whenever z is imaginary quadratic.
2000 Mathematics Subject Classification Primary—11Y65; Secondary—11Y40 相似文献
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Nayandeep Deka Baruah Nipen Saikia 《Journal of Computational and Applied Mathematics》2003,160(1-2):37-51
In 2001, Jinhee Yi found many explicit values of the famous Rogers–Ramanujan continued fraction by using modular equations and transformation formulas for theta-functions. In this paper, we use her method to find some general theorems for the explicit evaluations of Ramanujan's cubic continued fraction. 相似文献
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A. J. van der Poorten H. J. J. te Riele H. C. Williams. 《Mathematics of Computation》2001,70(235):1311-1328
Let be a prime congruent to 1 modulo 4, and let be rational integers such that is the fundamental unit of the real quadratic field . The Ankeny-Artin-Chowla conjecture (AAC conjecture) asserts that will not divide . This is equivalent to the assertion that will not divide , where denotes the th Bernoulli number. Although first published in 1952, this conjecture still remains unproved today. Indeed, it appears to be most difficult to prove. Even testing the conjecture can be quite challenging because of the size of the numbers ; for example, when , then both and exceed . In 1988 the AAC conjecture was verified by computer for all . In this paper we describe a new technique for testing the AAC conjecture and we provide some results of a computer run of the method for all primes up to .
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Stefan Paszkowski 《Numerical Algorithms》2006,41(4):353-358
Two new analytic expressions for the Harrison integral (1) are given as well as efficient algorithms of its computation resulting from them. 相似文献