初等函数可验证赋值及误差分析

研究了GNU标准下初等函数的赋值原理及算法实现.基于IEEE 754-2008浮点标准,利用误差分析基本结论,对GNU下C语言标准数学函数库中的初等函数赋值程序进行理论误差分析.利用Boost库中提供的区间类,将以浮点数作为基本数据类型的程序重写成以区间作为基本类型的程序,使用区间算术对初等函数进行可验证赋值,从而得到一个包含真实值的区间包络,并由此给出GNU下初等函数的数值误差界.     

Evaluation of functions on microcomputers: square root

This paper treats the evaluation of one of the elementary functions on short wordlength computers. The setting is a binary fixed point short wordlength (8–16 bits) machine where the intent is to suggest improvements in ROM- or microcode-based software which include the square root function as part of a more general mathematical software library or for special computation in real-time applications. This paper focuses on the evaluation of square roots and features a careful treatment of Newton's method with linear initialization. Comparisons with other popular algorithms are made based on storage requirements, speed, and accuracy, with some indication of the effect that special hardware features have on the performance of these routines.     

The design of the Boost interval arithmetic library

We present the design of the Boost interval arithmetic library, a C++$++$ library designed to handle mathematical intervals efficiently and in a generic way. Interval computations are an essential tool for reliable computing. Increasingly a number of mathematical proofs have relied on global optimization problems solved using branch-and-bound algorithms with interval computations; it is therefore extremely important to have a mathematically correct implementation of interval arithmetic. Various implementations exist with diverse semantics. Our design is unique in that it uses policies to specify three independent variable behaviors: rounding, checking, and comparisons. As a result, with the proper policies, our interval library is able to emulate almost any of the specialized libraries available for interval arithmetic, without any loss of performance nor sacrificing the ease of use. This library is openly available at www.boost.org.     

Parameterized floating-point logarithm and exponential functions for FPGAs

As FPGAs are increasingly being used for floating-point computing, the feasibility of a library of floating-point elementary functions for FPGAs is discussed. An initial implementation of such a library contains parameterized operators for the logarithm and exponential functions. In single precision, those operators use a small fraction of the FPGA’s resources, have a smaller latency than their software equivalent on a high-end processor, and provide about ten times the throughput in pipelined version. Previous work had shown that FPGAs could use massive parallelism to balance the poor performance of their basic floating-point operators compared to the equivalent in processors. As this work shows, when evaluating an elementary function, the flexibility of FPGAs provides much better performance than the processor without even resorting to parallelism. The presented library is freely available from http://www.ens-lyon.fr/LIP/Arenaire/.     

基于编译时插桩的浮点异常检测方法

浮点数是实数的有限精度编码,在进行浮点计算时,可能会导致不精确或者异常的结果,因此实现有效的浮点异常检测方法很重要。现有异常检测方法不面向浮点数学函数,由此提出了一种面向浮点数学函数的异常检测方法。该方法依据IEEE-754标准中定义的上溢出、下溢出、被零除、无效操作和不精确5类异常,并结合申威高性能数学函数库中使用的浮点控制寄存器FPCR和IEEE-754标准定义的浮点异常产生条件的相关理论,通过将异常类型和浮点运算指令进行对应分类,在程序编译时进行插桩以检测出浮点数学函数中出现的异常,同时记录代码覆盖率。最后将该方法应用于数学函数库,对库中100多个浮点数学函数进行了测试实验。实验结果表明,该浮点异常检测方法能够有效检测各类异常。     

A structure theorem for the elementary functions and its application to the identity problem

This paper uses elementary algebraic methods to obtain new proofs for theorems on algebraic relationships between the logarithmic and exponential functions. The main result is a multivariate version of a special case of the structure theorem due to Risch that gives, in a very explicit fashion, the possible algebraic relationships between the exponential and logarithm functions. The structure theorem has important applications to symbolic mathematical computation in that it in essence provides a canonical form for the elementary transcendental functions, and hence solves the identity problem for this class of functions. Such applications are discussed in the last section.This research was supported in part by National Science Foundation Grants GJ-30125X and MCS76-23762.     

一元函数表达式的类设计

在一些数学软件、统计软件和数值模拟软件中需要的一元函数表达式并不确定。因此，需要一种方法能够动态生成任意复杂的一元函数表达式。本文提出了一种构造函数表达式的动态组合法，该方法将四则运算符、基本初等函数、复合函数和常数项看成是对象，并采用C＋＋实现了这些对象的类定义。通过这些对象的动态创建与组合可以复合成任意的一元函数表达式。结果表明,一元函数表达式对象能够通过接口向其调用者输出正确的表达式计算功能。     

可重用的安全特性研究

提出采用面向特性编程技术解决安全功能代码可重用问题。介绍了一个提供安全功能的可重用特性库的基本实现。该特性库基于AspectJ和Java的安全包实现,包含了典型的安全机制。介绍了安全特性库的原理、结构和使用,并给出了一个基于安全特性库实现的实例。讨论了安全特性库在重用性和通用性方面的优缺点,以及下一步的研究方向。     

One function of ambiguities from the sets generated by the genetic code

The specificity of the sets of elements generated by the genetic code is considered. The sets are calculated for unusual ways of recording genetic information on overlapping genes, when one and the same DNA segment encodes two protein sequences. The concept of elementary overlapping is introduced, which is overlapping for individual amino acids. The mathematical ambiguity among the components of the set of elementary overlappings is established. One of the ambiguity functions is investigated in a new model proposed earlier by the author. Its states that overlappings of pairs of genes, belonging to different DNA chains, are mathematical analogs of the stems of the matrix RNA’s secondary structure. It is shown that, due to ambiguities, it is possible to regulate the value of free energy of the stem, which is a functionally significant biochemical characteristic.     

Automatic Forward Error Analysis for Floating Point Algorithms

We investigate absolute and relative error bounds for floating point calculations determined by means of sequences of instructions (as, for example, given by a computer program). We get rigorous error bounds on the round-off or generated error due to the actual machine floating point operations, as well as the propagated error from one sequence to the next in a very convenient way by the computer itself. The results stated in the theorems can be used to implement software tools for the automatic computation of a priori worst case error bounds for floating point computations. These automatically computed bounds are valid simultaneously for all data vectors varying in the domain specified and their corresponding machine vectors fulfilling a maximum prescribed error bound.With great success we have used our method in the past to implement a fast interval library for elementary functions called FI_LIB [12]. Further numerical examples often show a high quality of the computed a priori bounds.     

A learning procedure for mathematics

This paper introduces a learning procedure whose input is a set of data and a set of axioms giving the domains and ranges of elementary functions including predicates. It repeatedly applies these axioms to the input data, which yields more and more complex compositions of the functions. These compositions are used to form quantified propositions, set constructors, and programs which are composed of the elementary functions in the input. The procedure is guided by partial results such as partial programs which are evaluated against the input data. In computer experiments, the procedure constructed mathematical conjectures such as Goldbach's conjecture and developed powerful theorem provers from simple proofs.This work, in whole or in part, describes components of machines or processes protected by one or more patents or patent applications in Europe, the United States of America, or elsewhere. Further information is available from the author.     

基于个人超出值的区间值合作博弈新的求解模型

从个人超出值的视角研究特征函数为区间值的合作博弈和联盟为模糊集的无限模糊联盟区间值合作博弈.首先,利用区间值距离公式定义个人超出值;然后,建立最小化所有局中人个人超出值的最优化模型,进一步得到两类区间值合作博弈的显式解析解,并证明该解的性质;最后,通过数值实例验证所提出区间值合作博弈求解模型的实用性与有效性,为区间值合作博弈提供一种新的求解思路.     

Gaussian integration with rescaling of abscissas and weights

An algorithm for integration of the polynomial functions with a variable weight is considered. It provides an extension of the Gaussian integration, with appropriate scaling of the abscissas and weights. In a first step, orthogonal polynomials are computed for a fixed a=1. Then, using approximate scaling, the initial guess is constructed for a≠1. Finally, numerical values of the abscissas and weights are refined, solving polynomial system using Newton–Raphson method. The final form of the algorithm provides good alternative to usually adopted interval splitting, automatically avoiding problems with limiting values of parameter present in the weight function. Construction of the method requires arbitrary precision arithmetic and special functions, polylogarithms in particular. The final form of the algorithm can be coded using machine precision floating point numbers and standard mathematical library.     

Errors in vector processing and the library libavi.a

In this paper, we describe the results of several tests that check the accuracy of numerical computation on the Cray supercomputer in vector and scalar modes. The known tests were modified to identify the critical point where roundings start causing problems. After describing the tests, we present an interval library called libavi.a. It was developed in Fortran 90 on the Cray Y-MP2E supercomputer of UFRGS-Brazil. This library makes interval mathematics accessible to the Cray supercomputers users. It works with real and complex intervals and intervals matrices and vectors. The library allows overloading of operators and functions. It is organized in four modules: real intervals, interval vectors and matrices, complex intervals, and linear algebra applications.     

Algebraic theory of recombination spaces

A new mathematical representation is proposed for the configuration space structure induced by recombination, which we call "P-structure." It consists of a mapping of pairs of objects to the power set of all objects in the search space. The mapping assigns to each pair of parental "genotypes" the set of all recombinant genotypes obtainable from the parental ones. It is shown that this construction allows a Fourier decomposition of fitness landscapes into a superposition of "elementary landscapes." This decomposition is analogous to the Fourier decomposition of fitness landscapes on mutation spaces. The elementary landscapes are obtained as eigenfunctions of a Laplacian operator defined for P-structures. For binary string recombination, the elementary landscapes are exactly the p-spin functions (Walsh functions), that is, the same as the elementary landscapes of the string point mutation spaces (i.e., the hypercube). This supports the notion of a strong homomorphism between string mutation and recombination spaces. However, the effective nearest neighbor correlations on these elementary landscapes differ between mutation and recombination and among different recombination operators. On average, the nearest neighbor correlation is higher for one-point recombination than for uniform recombination. For one-point recombination, the correlations are higher for elementary landscapes with fewer interacting sites as well as for sites that have closer linkage, confirming the qualitative predictions of the Schema Theorem. We conclude that the algebraic approach to fitness landscape analysis can be extended to recombination spaces and provides an effective way to analyze the relative hardness of a landscape for a given recombination operator.     

A remark on the numerical computation of improper integrals

In this note a transformation is given reducing improper integrals to integrals with smooth integrands. The transformation may be represented in terms of elementary functions and is easily applicable in connection with standard mathematical software for the numerical evaluation of integrals with smooth integrands.     

Hidden verification for computational mathematics

We present hidden verification as a means to make the power of computational logic available to users of computer algebra systems while shielding them from its complexity. We have implemented in PVS a library of facts about elementary and transcendental functions, and automatic procedures to attempt proofs of continuity, convergence and differentiability for functions in this class. These are called directly from Maple by a simple pipe-lined interface. Hence we are able to support the analysis of differential equations in Maple by direct calls to PVS for: result refinement and verification, discharge of verification conditions, harnesses to ensure more reliable differential equation solvers, and verifiable look-up tables.     

一种改进的超越函数通用算法

提出一种适用于高性能计算的基础数学库超越函数通用算法。基于转换、近似、重建等技术,对级数法和迭代法进行改进,运用函数收敛性最好的一段区间进行函数算法设计,通过减少运算次数来降低因多次运算导致的累积误差,保证函数算法的精度。利用查表法构造辅助表参与运算,减少算法开销。测试结果表明,与GNU数学库函数相比,基于该算法的数学库函数运行效率平均提升50.91%。     

Calculus for interval functions of a real variable

Some properties of the algebraical system <I(R),+,o,?>, where <I(R),+,o> is the well known quasinear interval space and “?” is a nonstandard operation such thata?a=o, are given in this paper. The an elementary calculus for interval functions using this nonstandard arithmetic is discussed.     

Approximation of Planck distribution

Calculation of thermal radiation transport frequently requires calculating the integral of the Planck distribution density with respect to a finite interval of spectrum. An analytical approximation of this integral and its related integrals in elementary functions is constructed.