Algebraic Manipulation System Research Articles

An algebraic formula manipulation system for education and its implementation on different PCs

When studying the application of APL to high school education to help students in their understanding of mathematical concepts, many teachers encountered difficulties in that the result of their study could not be used in their classes because schools had provided them with computer models which can not run APL. The authors and their co-workers of the Computer Study Group of MiYagi Prefecture Society of Mathematical Education found a way to solve such a difficulty by porting study files from NiHonGo APL to I-APL. This paper discusses the problems encountered in classrooms, the new system applied to three popular models of personal computers, the actual use of the APL study files in eight schools for learning polynomial factorisation, and evaluation of both teachers and students.

A review of Macsyma

The successes and failures of the Macsyma algebraic manipulation system are reviewed from the point of view of one of the original contributors. A retrospective examination is provided of some of the controversial ideas that worked, and some that did not. Input/output, language semantics, knowledge-adjunction, mathematical semantics, the model of the user, and other issues are considered. Possible future directions for algebraic manipulation system building are discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Using a small algebraic manipulation system to solve differential and integral equations by variational and approximation techniques

The microcomputer algebraic manipulation system MUMATH is used to implement the classical variational, Galerkin and least-squares techniques for solving boundary-value problems in differential equations and also for solving Fredholm integral equations. Examples are given which extend the precision of known results. The technique is presented as a general algorithm which can readily be implemented on other algebraic manipulation systems.

Eleven proofs of sin 2 x+cos 2 x = 1

Several papers in the algebraic manipulation literature have discussed the problem of recognizing, in various guises, the equivalence of sin 2 x+cos 2 x and 1. This is a simplified form of a general substitution problem, and it would seem that a neat, universally applicable technique would be of some interest. This paper does not provide one, but does attempt to show, in various ways, how a particular algebraic manipulation system "knows" this famous identity, in spite of its reluctance to apply it.

Computerized symbolic manipulation in nonlinear finite element analysis

The potential of using computerized symbolic manipulation in the development of nonlinear finite elements is discussed. Three tasks which can be efficiently performed using computerized symbolic manipulation are identified: (1) generation of algebraic expressions for the stiffness coefficients of nonlinear finite elements, (2) generation of FORTRAN source code for numerical evaluation of stiffness coefficients, and (3) checking the correctness of the FORTRAN statements for the arrays of coefficients. The symbolic and algebraic manipulation system MACSYMA is used in the present study. Two sample MACSYMA programs are presented for the development of the nonlinear stiffness coefficients of two-dimensional, shear-flexible, doubly-curved deep shell elements. The first program is for displacement models and the second program is for mixed models with discontinuous stress-resultant fields at interelement boundaries.

Application of REDUCE-2 algebraic manipulation system in calibration problems of track chamber picture processing devices

The problem is considered of correcting nonlinear coordinate distortions generated by cathode ray tube scanner. Orthogonal bivariate polynomials are used to solve the computational part of this problem. The symbolic algebraic algorithm, proposed for the evaluation of orthogonal polynomials, is performed by the REDUCE-2 algebraic manipulation system.

An Algol 68 based algebraic manipulation system

AbstractAlthough the topic of algebraic manipulation by computer is an old one, it is still a characteristic of many algebraic manipulation systems that they often neglect the needs and characteristics of the end user. This paper describes an algebraic manipulation system, written in Algol 68, which attempts to remedy this state of affairs.

Problem 7 and systems of algebraic equations

In [1], Johnson and Graham considered the function[EQUATION]and asked for a method to find the minimal polynomial (of degree 10) of the value of the maximum of this function. They proposed such a method which needs the factorization of a polynomial of degree 40 into four irreductible tenth-degree factors and they set the following questions:1) "it would be of considerable interest if this problem could be solved by some other method which avoids the factorization".2) "it would be particularly interesting if there were an algebraic manipulation system which could do all the steps in an efficient and natural manner".

On Stieltjes’ continued fraction for the gamma function

The first forty-one coefficients of a continued fraction for ln ⁡ Γ ( z ) + z − ( z − 1 / 2 ) ln ⁡ z − ln ⁡ 2 π \ln \Gamma (z) + z - (z - 1/2) \ln z - \ln \sqrt {2\pi } , are given. The computation, based on Wall’s algorithm for converting a function’s power series representation to a continued fraction representation, was run on the algebraic manipulation system MACSYMA. ∗ ∗ ^{\ast \ast }

Symbolic Vector and Dyadic Analysis

A computer program is described which performs symbolic algebra and calculus with vectors and dyadics. Implemented on the MACSYMA algebraic manipulation system, it is intended to be a reasonably complete analysis system for applications such as plasma physics and fluid dynamics. It includes manipulations of dot and cross products; gradient, divergence, curl and Laplacian operators; directional derivatives and outer products. Vector and dyadic equations can be automatically expanded into components for arbitrary orthogonal coordinate systems. Designed primarily for 2-dimensional and 3-dimensional use, the program also has some capability for higher dimensions. The internal structure of the code is discussed with regard to efficiency considerations. Example calculations and execution times are presented.

Factoring Multivariate Polynomials over Algebraic Number Fields

The algorithm for factoring polynomials over the integers by Wang and Rothschild is generalized to an algorithm for the irreducible factorization of multivariate polynomials over any given algebraic number field. The extended method makes use of recent ideas in factoring univariate polynomials over large finite fields due to Berlekamp and Zassenhaus. The procedure described has been implemented in the algebraic manipulation system MACSYMA. $^{\ast \ast }$ Some machine examples with timing are included.

Factoring multivariate polynomials over algebraic number fields

The algorithm for factoring polynomials over the integers by Wang and Rothschild is generalized to an algorithm for the irreducible factorization of multivariate polynomials over any given algebraic number field. The extended method makes use of recent ideas in factoring univariate polynomials over large finite fields due to Berlekamp and Zassenhaus. The procedure described has been implemented in the algebraic manipulation system MACSYMA. ∗ ∗ ^{\ast \ast } Some machine examples with timing are included.

Theory of Enceladus and Dione

A new theory of the motion of Enceladus and Dione is under development. The theory is in literal form, with all constants and parameters appearing explicitly, and is being computed with the aid of the algebraic manipulation system TRIGMAN. The development of the disturbing function; the elimination of the short-period terms; and the resonant Hamiltonian and the constants of integration are discussed. An important feature is that the programs have been written in such a manner that, with only minor modification, they should be able to develop the theories of Mimas-Tethys and Titan-Hyperion as well.

Factoring multivariate polynomials over algebraic number fields in MACSYMA

The algorithm for factoring polynomials over the integers by Wang and Rothschild is generalized to an algorithm for the irreducible factorization of multivariate polynomials over any given algebraic number field. The extended method makes use of recent ideas in factoring univariate polynomials over large finite fields due to Berlekamp and Zassenhaus. The procedure has been implemented in the algebraic manipulation system MACSYMA. Some machine examples with timing are presented.

MACSYMA - the fifth year

MACSYMA is a large symbolic and algebraic manipulation system which has been under development at Project MAC, M.I.T. since 1969. We first discuss some of the design decisions, such as multiple representations, that led to the current system. We then give a brief description of the facilities of MACSYMA by dividing the system into seven packages: language and interactive facilities, general representation, rational function representation and related algorithms, the integration subsystem, the power series subsystem, the MACLISP system, and miscellaneous facilities.

Tuning an algebraic manipulation system through measurements

This paper describes a measurement tool that has been used to analyze, tune, and redesign in part the PL/I-FORMAC Symbolic Mathematics Interpreter. In a number of examples, details are given both on the FORMAC system and on the application of the measurement tool. The basic tool, called invocation count measurement, is simple but quite effective. Its application enabled us to improve FORMAC considerably.

ALTRAN programs for SIGSAM problem #6

In [1], it is suggested that an algebraic manipulation system be used to gain some insight into the following problem:[EQUATION]

Algebraic simplification

Algebraic simplification is examined first from the point of view of a user who needs to comprehend a large expression, and second from the point of view of a designer who wants to construct a useful and efficient system. First we describe various techniques akin to substitution. These techniques can be used to decrease the size of an expression and make it more intelligible to a user. Then we delineate the spectrum of approaches to the design of automatic simplification capabilities in an algebraic manipulation system. Systems are divided into five types. Each type provides different facilities for the manipulation and simplification of expressions. Finally we discuss some of the theoretical results related to algebraic simplification. We describe several positive results about the existence of powerful simplification algorithms and the number-theoretic conjectures on which they rely. Results about the nonexistence of algorithms for certain classes of expressions are included.