零误差计算

研究采用有误差的数值计算来获得无误差的准确值具有重要的理论价值和应用价值.这种通过近似的数值方法获得准确结果的计算被称为零误差计算.本文首先指出,只有一致离散集合中的数才能够开展零误差计算,即有非零隔离界的数集,这也是数可以进行零误差计算的一个充要条件.以此为基本出发点,本文分析代数数零误差计算的最低理论精度,该精度对应于恢复近似代数数的准确值时必要的误差控制条件,但由于所采用恢复算法的局限性,这一理论精度往往不能保证成功恢复出代数数的准确值.为此,本文给出采用PSLQ (partial-sum-LQ-decomposition)算法进行代数数零误差计算所需的精度控制条件,与基于LLL (Lenstra-Lenstra-Lovász)算法相比,该精度控制条件关于代数数次数的依赖程度由二次降为拟线性,从而可降低相应算法的复杂度.最后探讨零误差计算未来的发展趋势.     

On the computation of the polar FFT

We show that the polar as well as the pseudo-polar FFT can be computed very accurately and efficiently by the well-known nonequispaced FFT. Furthermore, we discuss the reconstruction of a 2d signal from its Fourier transform samples on a (pseudo)polar grid by means of the inverse nonequispaced FFT.     

On the numerical computation of the Mittag-Leffler function

Recently simple limiting functions establishing upper and lower bounds on the Mittag-Leffler function were found. This paper follows those expressions to design an efficient algorithm for the approximate calculation of expressions usual in fractional-order control systems. The numerical experiments demonstrate the superior efficiency of the proposed method.     

On the computation of modules of long quadrilaterals

We study quadrilateralsQ which are given by two intervals on {:Im = 0} and {:Im = 1}, and two Jordan arcs ₁, ₂, in {:0 Im 1} connecting these two intervals. Many practical problems require the determination of the modulem(Q) ofQ, but ifQ is long, i.e., if

On the computation of linearly constrained stationary points

We study a numerical method for the computation of linearly constrained stationary points. The proposed method can be interpreted as a projected gradient method with constant stepsize in which one allows perturbations in the admissible set and controls these perturbations in each iteration. The method is applicable to some classes of overdetermined problems to which the projected gradient method may not be directly applicable. Illustrative numerical examples are given.     

On the computation of resolvents and Galois groups

A large class of algorithms for computing resolvents of algebraic equations — so called rational transformations — is investigated and characterized group theoretically. The concept of rational transformations implies a program how to develop good methods to determine the Galois group of an equation. It is shown that some known methods are special cases of rational transformations, and a new procedure to find the group of a sextic equation is given. Moreover, all cases in which Galois resolvents can be found by means of rational transformations are classified.     

On the computation of non-normal Padé approximants

Several basic techniques are described for computing Padé approximants which are not normal. These techniques are modifications of some known methods for the computation of normal Padé approximants.     

On the computation of the jdeg of blowup algebras

For a graded algebra , its is a global degree that can be used to study issues of complexity of the normalization . Here some techniques grounded on Rees algebra theory are used to estimate . A closely related notion, of divisorial generation, is introduced to count numbers of generators of .     

On the rapid computation of various polylogarithmic constants

We give algorithms for the computation of the -th digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired. They make it feasible to compute, for example, the billionth binary digit of or on a modest work station in a few hours run time. We demonstrate this technique by computing the ten billionth hexadecimal digit of , the billionth hexadecimal digits of and , and the ten billionth decimal digit of . These calculations rest on the observation that very special types of identities exist for certain numbers like , , and . These are essentially polylogarithmic ladders in an integer base. A number of these identities that we derive in this work appear to be new, for example the critical identity for :

     

On the computation of the projective surgery obstruction groups

The computation of the projective surgery obstruction groupsL _n ^p (ZG), forG a hyperelementary finite group, is reduced to standard calculations in number theory, mostly involving class groups. Both the exponent of the torsion subgroup and the precise divisibility of the signatures are determined. ForG a 2-hyperelementary group, theL _n ^p (ZG) are detected by restriction to certain subquotients ofG, and a complete set of invariants is given for oriented surgery obstructions.Partially supported by NSERC grant A4000.Partially supported by NSF grant DMS-8610730(1) and the Danish Research Council.     

On search by address computation

We study the effect of data distribution on address computation data structures for searching, as typified by the priority queue problem. We compare several techniques showing that, in contrast to sorting, neither one nor multilevel bucket methods are uniformly efficient for this task. However, an enhancement of order preserving extendible hashing is shown to behave asymptotically independently of the amount of data and its distribution. Also conclusions regarding multiattribute file structures are presented.     

On the computation of scaling coefficients of Daubechies' wavelets

In the present paper, Daubechies' wavelets and the computation of their scaling coefficients are briefly reviewed. Then a new method of computation is proposed. This method is based on the work [7] concerning a new orthonormality condition and relations among scaling moments, respectively. For filter lengths up to 16, the arising system can be explicitly solved with algebraic methods like Gröbner bases. Its simple structure allows one to find quickly all possible solutions.     

On the computation of invariant manifolds of fixed points

We present a method for the numerical computation of invariant manifoids of hyperbolic and pseudohyperbolic fixed points of diffeomorphisms. The derivation of this algorithm is based on well-known properties of (almost) invariant foliations. Numerical results illustrate the performance of our method.Supported by NWO grant 611-307-018.Supported by NWO grant 611-306-523.     

On the computation of Lauricella functions of the fourth kind

An identity is proved which simplifies the computation of Lauricella functions of the fourth kind. An application to a problem in mathematical statistics is briefly discussed.