Showing papers on "Algebraic expression published in 1983"

Three-dimensional algebraic grid generation

Abstract: One of the major approaches to numerical grid generation is the explicit algebraic expression of a physical grid as a function of a uniform grid in a rectangular computational coordinate system. The algebraic methods are based on mathematical interpolation, and the primary advantages are speed and directness. The relation between interpolation and grid generation is described. For three-dimensional grid generation, transfinite interpolation using the coordinate control processes developed in the multisurface method and two-boundary technique are advocated. Grid singularities encountered in three dimensions are discussed, and the exploration of multiple overlapping grids is proposed. Some aspects of interactive algebraic grid computation in three dimensions are discussed.

A Definition of the ARCA Notation

Abstract: ARCA is a programming notation intended for interactive specification and manipulation of combinatorial graphs. The main body of this report is a technical description of ARCA sufficiently detailed to allow an interpreter to be developed. Some simple illustrative programs are included. ARCA incorporates variables for denoting primitive data elements (essentially vertices, edges and scalars), and diagrams (essentially embedded graphs). A novel feature is the use of two kinds of variable: the one storing values (as in conventional procedural languages), the other functional definitions (as in nonprocedural languages). By means of such variables, algebraic expressions over the algebra of primitive data elements may represent either explicit values or formulae. The potential applications and limitations of ARCA, and more general "algebraic notations" defined using similar principles, are briefly discussed.

A general lexicographic partial enumeration algorithm for the solution of integer nonlinear programming problems

Abstract: This dissertation presents a general lexicographic partial enumeration algorithm for solving discrete nonlinear optimization problems of the form: Minimize g(,0)(x), subject to g(,i) (x) (GREATERTHEQ) b(,i); i = 1, 2, . . . , m, x = (x(,1),x(,2), . . . , x(,n)), 0 (LESSTHEQ) lb(,j) (LESSTHEQ) x(,j) (LESSTHEQ) ub(,j); j = 1, 2, . . . , n. Here each x(,j) is an integer variable with integer lower bound lb(,j) and integer upper bound ub(,j). The algorithm can solve any discrete optimization problem of this form, but it is more efficient if some of the functions g(,i) can be expressed as differences of two isotone nondecreasing functions. It is even more efficient if some of the functions g(,i) are isotone nondecreasing functions, and it is most efficient if the objective function or some of the constraint functions, or both, are linear. Some of the features of this algorithm are as follows: (1) It locates the global constrained discrete optimal solution by function evaluations only and so does not require that the functions by continuous or even defined for noninteger values of the variables. It is not even necessary to have explicit algebraic expressions for the functions. (2) It is easy to program and requires a small amount of computer memory. (3) It is not necessary to transform the variables to weighted sums of binary variables. (4) No extra constraints or transformations are needed to impose the lower and upper bounds on the non-negative variables. (5) It incorporates a special new approach to take advantage of a linear objective function or any linear constraint functions, or both, that may be present. (6) It may be used to obtain approximate solutions to mixed and continuous nonlinear optimization problems. After a review of the literature on discrete nonlinear optimization, the algorithm is presented and its operation is described in detail. It is shown that in this algorithm, we are working in a space in which there is a minimum total number of possible points. Rules for reducing the number of infeasible points to be considered are given and illustrated, and then guidelines for reordering the variables to reduce the overall solution time are developed and illustrated. Illustrative applications are given in the areas of multi-echelon repairable item provisioning and aircraft development and production scheduling, and a number of test problems from the literature are solved.

Computer Algebra and Complex Analysis

Abstract: Taking complex analysis to mean complex numerical analysis, I perceive my mission here to be that of disseminating the algebraic approach taken by computer algebraists to many mathematical problems, which arise from and are important to complex analysis. In turn, complex numerical analysis can be, and have been, providing essential theoretical and computational results for computer algebra. The cross fertilization should and must continue in order that computational mathematics progress with the joint aid of both tools, rather than branching into orthogonal pursuits with disparate approaches. First, we discuss the different issues and principal concerns of computer algebra. Then, the algebraic approach to a long standing problem in calculus or complex analysis, indefinite integration in closed form, will be motivated and derived through examples. Algorithmic solution to the basic, thought provoking, problem of rational function integration as well as theoretical foundation underlying the algorithm for elementary function integration will be discussed. Further issues and approaches will be illustrated through another central (implicitly essential) problem of computer algebra, that is simplification of symbolic and algebraic expressions. We conclude by showing a set of computer executed problems in integration to reveal some of the new capabilities added to the arsenal of a mathematician through the efforts of computer algebra.