摘要： The quasi-Newton family of algorithms for minimizing functions and solving systems of nonlinear equations has achieved a great deal of computational success and forms the core of many software libraries for solving these problems. In this work we extend the theory of the quasi-Newton algorithms to the block case, in which we minimize a collection of functions having a common Hessian matrix, or we solve a collection of nonlinear equations having a common Jacobian matrix. This paper focuses on the linear algebra: update formulas, positive definiteness, least-change secant properties, relation to block conjugate gradient algorithms, finite termination for quadratic function minimization or solving linear systems, and the use of the quasi-Newton matrices as preconditioners.

摘要： Non-numerical problems aand methods are steadily gaining in importance alongside problems and methods of;computational mathematics. Non-numerical problems primarily include symbolic computation, which is the subject of vigorous research [1]. Symbolic computation includes many problems of group theory, in particular problems of the theory of finitely generated free groups. One of the main algorithms of the theory of finitely generated free groups is the algorithm to construct the basis of a subgroup of a free group (Nielsen algorithm). It is described in detail in [2], and its implementation in the algebraic programming system APS-1 is presented in [3]. In this paper we use the Nielsen algorithm as a base for the implementation of algorithms for some problems in the theory of free groups, semigroups, and finite algebras. The paper consists of three parts: the first part presents the necessary information from the theory of groups and finite automata;the second part sets out some problems of the theory of finitely generated free groups (semigroups) with the corresponding solution algorithms;and the third part considers the representation of finite algebras in the APS-1 system and algorithms to evaluate expressions in finite algebras.

摘要： Algorithms for computing integral bases of an algebraic function field are implemented in some computer algebra systems. They are used e.g. for the integration of algebraic functions. The method used by Maple 5.2 and AXIOM is given by Trager in [Trager,1984]. He adapted an algorithm of Ford and Zassenhaus [Ford,1978], that computes the ring of integers in an algebraic number field, to the case of a function field. It turns out that using algebraic geometry one can write a faster algorithm. The method we will give is based on Puiseux expansions. One can see this as a variant on the Coates' algorithm as it is described in [Davenport,1981]. Some difficulties in computing with Puiseux expansions can be avoided using a sharp bound for the number of terms required which will be given in Section 3. In Section 5 we derive which denominator is needed in the integral basis. Using this result 'intermediate expression swell' can be avoided. The Puiseux expansions generally introduce algebraic extensions. These extensions will not appear in the resulting integral basis.

关键词： 铜   电解   助剂   用量   电流密度   最佳搭配   数学模型  

关键词： 算法   数学语言   形象结合   程序设计  

摘要： 算法在程序设计中的地位日益突出，巧妙的构思常会掩盖算法的直观性，使程序变的艰涩难懂．本文利用集合和形象结合等数学语言及手段，联系实例剖析算法，使程序变的通俗直观，避免了因大篇幅的算法说明及步骤叙述不严密而导致的歧义性．

关键词： Algebra   algorithms   Programming  

关键词： INTEGRABLE SYSTEMS   DYNAMICAL SYSTEMS   QUANTIZATION   LIE GROUPS   KORTEWEG?DE VRIES EQUATION   ALGEBRAS   PARTIAL DIFFERENTIAL EQUATIONS  

摘要： The generalized theory of the R-structure on affine operator Lie algebras is used to construct a complete theory of Lax integrable nonlinear dynamical systems in multidimensions. The operator bi-Hamiltonian structures and their functional reductions are discussed in great detail in the examples of operator Korteweg-de Vries and Benney-Kaup dynamical systems. As an important by-product of the developed algebraic theory, the Dirac canonical quantization problem is solved almost completely for the Neumann-Bogoliubov-type oscillatory dynamical system on spheres' associated via Moser with the spectral moment map on an affine associative metrized Lie coalgebra with a one-parameter gauge two-cocycle. Some remarks are given on the problem of extending the developed algebraic theory to the case of Lax integrable dynamical systems on discrete manifolds.

关键词： RECONSTRUCTION   TOMOGRAPHY  

摘要： A tomographic reconstruction algorithm similar to the well-known algebraic reconstruction technique (ART) is presented. Similar to ART, the approximate algebraic reconstruction technique (AART) technique consists of a sequence of displacements of the image vector based on the projection error. AART is a column-relaxation technique that is a series of vector displacements of the image vector parallel to its coordinate axes. AART is compared with ART, a standard conjugate-gradient technique, and a conjugate-gradient technique augmented by nonnegativity. The use of relaxation parameters to improve the performance of both ART and AART in the presence of noise is discussed, and the use of an iteration termination criterion based on random generalized cross validation is illustrated.