Showing papers in "ACM Sigsam Bulletin in 1974"

Sparse polynomial arithmetic

Abstract: Sparse polynomial representations are used in a number of algebraic manipulation systems, including Altran. This paper discusses the arithmetic operations with sparsely represented polynomials; we give particular attention to multiplication and division. We give new algorithms for multiplying two polynomials, with n and m terms, in time mnlogm;, these algorithms have the property that, in the usual univariate dense case, the algorithm is bounded by mn. Division algorithms are discussed which run in comparable time.

Quantifier elimination for real closed fields by cylindrical algebraic decomposition--preliminary report

Abstract: Tarski in 1948, [18] published a quantifier elimination method for the elementary theory of real closed fields (which he had discovered in 1930) As noted by Tarski, any quantifier elimination method for this theory also provides a decision method, which enables one to decide whether any sentence of the theory is true or false Since many important and difficult mathematical problems can be expressed in this theory, any computationally feasible quantifier elimination algorithm would be of utmost significance

Problem #7

Abstract: The function F(x) = (1/2-x) (1-x2)1/2+x(1+(1-(1/2+x)2)1/2) has a maximum at about x = .343771, where it attains the value of approximately .674981. This value is the root of an irreducible polynomial of tenth degree over the integers; the problem is to find this polynomial. The obvious way of proceeding is as follows:(1) Differentiate F(x), set it equal to zero, and clear radicals. The result is a tenth degree polynomial P(x) over the integers which has a root at about x = .343771.

MACSYMA - the fifth year

Abstract: 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.

The automatic construction of pure recurrence relations

Abstract: A method is presented for the construction of pure recurrence relations for functions related to generalized hyper-geometric functions. It is an improvement of a technique by Fasenmeyer, and is suitable for automatic computer programming. Our method has an easy geometrical interpretation and also gives a constructive proof for the existence of a pure recurrence relation.

Computational aspects of Hensel-type univariate polynomial greatest common divisor algorithms

Abstract: Two Hensel-type univariate polynomial Greatest Common Divisor (GCD) algorithms are presented and compared. The regular linear Hensel construction is shown to be generally more efficient than the Zassenhaus quadratic construction. The UNIGCD algorithm for UNIvariate polynomial GCD computations, based on the regular Hensel construction is then presented and compared with the Modular algorithm based on the Chinese Remainder Algorithm. From both an analytical and an experimental point of view, the UNIGCD algorithm is shown to be preferable for many common univariate GCD computations. This is true even for dense polynomials, which was considered to be the most suitable case for the application of the Modular algorithm.

Toward a formal implementation of computer algebra

Abstract: We consider in this paper the task of synthesizing an algebraic system. Today the task is significantly simpler than in the pioneer days of symbol manipulation, mainly because of the work done by the pioneers in our area, but also because of the progress in other areas of Computer Science. There is now a considerable collection of algebraic algorithms at hand and a much better understanding of data structures and programming constructs than only a few years ago.

On computing with factored rational expressions

Abstract: This paper discusses the extension of an efficient polynomial manipulation system to deal with rational expressions. The proposed system is intended for a wide range of practical computations, including problems involving sparse expressions in many indeterminates.

The SCRATCHPAD language

Abstract: SCRATCHPAD is an interactive system for symbolic mathematical computation. Its user language, originally intended as a special-purpose non-procedural language, was designed to capture the style and succinctness of common mathematical notations, and to serve as a useful, effective tool for on-line problem solving. This paper describes extensions to the language which enable it to serve also as a high-level programming language, both for the formal description of mathematical algorithms and their efficient implementation.

Problem #8: random walk

Abstract: This problem was brought to my attention by Professor Violet Cane of Manchester University, and derives from correlated random walks, in statistics. It turned out to be more difficult than I thought, causing difficulties to a number of algebra systems.

Factored rational expressions in ALTRAN

Abstract: In this paper we present detailed algorithms for the basic arithmetic operations on symbolic rational expressions represented by formal quotients of factored polynomials. These algorithms are currently implemented in the ALTRAN system for symbolic algebra but the descriptions given in this paper are system independent.First we describe the representation and examine the need for options to permit control over the amount of effort expended in the search for factors or in canonicalizing results. We then present algorithms in the form of programs for equality-test, exponentiation, multiplication, and addition, and mention the modifications required for division and subtraction.We conclude by presenting the results of several benchmark tests comparing the performance of these algorithms with others previously used.

A simple method of taking nth roots of integers

Abstract: When performing algebraic calculations by computer one is occasionally faced with expressions like 4(1/2). A particular example of this is in the statistics calculation described in [1], where the expressions involve[EQUATION]When all we require is the principal (positive) root the expression collapses. Dealing with (q2)(1/2) for an explicit q is not difficult; this note is concerned with the treatment of the 4(1/2).

Symbolic integration of a class of algebraic functions

Abstract: In this presentation we describe the outline of an algorithmic approach to handle a class of algebraic integrands. (It is important to stress that for an extended abstract of the present form, we can at best convey the flavor of the approach, with numerous details missing.) We shall label this approach Carlson's algorithm because it is based on a series of analyses rendered by Carlson and his associates in the last ten years (Refs. 2, 3, 4, 8, and 12). The class of integrands is of the form r(x, y), where y2 is a polynomial in x, and r a rational function in x and y. This is the type of integrand that classically led to the study of elliptic integrals. At first glance this is a rather restricted class of algebraic functions. But in fact many trigonometric and hyperbolic integrands reduce to this form. The richness of this class of integrands is exemplified by a recently published handbook of 3000 integral formulas (Ref. 1). Our proposed approach will cover fifty to seventy percent of the items in the handbook. Furthermore the non-classical approach we shall describe holds great promise of developing to the case where definite integrals can be evaluated in terms of a host of other well-known functions (e.g., Bessel and Legendre).

A p-adic division with remainder algorithm

Abstract: A new algorithm for division with remainder of univariate and multivariate polynomials over the integers is reported. This division algorithm relies on a p-adic construction which is closely related to the Hensel-type constructions used for polynomial factorization and greatest common divisor computations. It furnishes a new and systematic way of looking at the classical problem of division (with or without remainder). Due to the sparseness-preserving property of p-adic constructions, it appears useful as an alternative division algorithm in suitable cases when the polynomials are sparse. Detailed discussion and a more complete computing time analysis will be deferred until a later time as the work progresses further. An hope, in the meantime, is to attract comments and criticism on the algorithm and its significance.

SAC-1 solution of problem #7

Abstract: This note reports on various aspects of our experience in using the SAC-1 system to solve Problem #7, submitted by S. C. Johnson and R. L. Graham, [1]. In solving the problem, several separate SAC-1 main programs were written for the various parts of the problem. These separate programs could, in principle, have been combined into a single main program. The SAC-1 solution involved the use of seven of the ten currently released SAC-1 subsystems. With reference to Loos' bibliography, [2], the seven subsystems used were CO-12, CO-13, CO-14, CO-17, CO-15, CO-16 and CO-19; CO-18, CO-31 and CO-32 were not needed.

Tuning an algebraic manipulation system through measurements

Abstract: 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.

Symbolic mathematical systems now and in the future

Abstract: A survey of existing algebraic systems, from batch-oriented "formula-crunchers" to dialog systems with sophisticated mathematical knowledge is given. Results from a comparison of some systems from aspects such as availability, user-orientedness and runtime-efficiency are presented.Finally possible trends in hardware, firmware and software that would make algebraic systems svailable for the wide community of "small" users in science and technology are discussed.

ALTRAN programs for SIGSAM problem #6

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

Computer recognition of divergences in Feynman diagrams

Abstract: Symbolic and Algebraic Manipulations techniques are applied to a problem of quantum electrodynamics: to find the divergences inclosed in a Feynman graph. These divergences are connected to the topology of the graph. No calculations are performed.

Experience with truncated power series

Abstract: The truncated power series package in ALTRAN has been available for over a year now, and it has proved itself to be a useful and exciting addition to the armoury of symbolic algebra. A wide variety of problems have been attacked with this tool: moreover, through use in the classroom, we have had the opportunity to observe how a large number of people react to the availability of this tool.

Introduction to NETFORM

Abstract: Netform is an easy to use, interactive, language for symbolic mathematical computation. The well known, Formac package, which uses the normal representation of formula's, forms the basis of the system. This basis is extended by means of:1° A linear equations solver, which offers two different possibility's to describe a system of linear equations.2° The description of linear (electrical) networks based on the network structure.In this paper a more detailed description of the just mentioned extensions is given. It will be shown that symbolic mathematical computation can be a very attractive and powerful instrument for the analysis of electrical network.At this moment the system is implemented on an IBM 360-50 computer under the MVT operating system and serves 3 2260 display's in a 240k region. Batch processing is possible in order to be independent of the display's. A file mechanism makes off line preparation of a session possible.

Formal representations for algebraic numbers

Abstract: Recent literature (e.g., [12]) shows a growing interest in the representation of algebraic numbers in terms of the polynomials they satisfy. The basis of this interest is the observation that for every algebraic number α there exists a unique polynomial p(x) having the following properties:

ANALITIK: principal features of the language and its implementation

Abstract: A short discussion is given on the programming language ANALITIK available on the Soviet MIR-2 computer designed for description of numerical and analytical mathematical computations. Some properties and principles of the system are presented. Design objectives and their implementation are reviewed.The programming language ANALITIK available on the MIR-2 computer is a special purpose language designed for numerical and analytical computations. The language and the computer have been developed at the Institute of Cybernetics in Kiev under direction of the well known Soviet cybernetician, academician V. M. Glushkov. It represents a continuation of a product line of special purpose computers for mathematical problems. ANALITIK is a compatible extension of the programming language MIR, which was an Algol-like language but more specifically designed for the solution of numerical problems.

FORMAC today, or what can happen to an orphan

Abstract: What Xenakis did not tell about FORMAC in Los Angeles, [30].

Solution to problem #7 using MACSYMA

Abstract: The solution to problem 7 [SIGSAM Bulletin vol 8, no. 1 (issue no. 29), Feb., 1974] was obtained in its entirety with the use of MACSYMA [1]. No hand calculations were required. The factorization step, the major cost of the solution, uncovered a poor design decision in the existing factorization program [2]. This program was modified, and the factorization was accomplished in 602 seconds of PDP-10 machine time. Since the modified factorization program produces shorter run-times on almost all of our standard test cases. it is now the standard routine.

Toward a revision of FORMAC

Abstract: A survey is given of some of the new facilities that have been implemented in FORMAC in an attempt to overcome current deficiencies and move towards a more powerful system.

The use of symbolic computation in solving free boundary problems by the Ritz-Galerkin method

Abstract: This paper presents a particular method of solving an ordinary linear differential free boundary problem by using the methods of Ritz and of Galerkin. The method described benefits from symbolic manipulation because of the parameter derived from the free boundary condition.It is interesting to note how the using of the symbolic manipulation still offers some different solution strategies. This open choice gets our problem solvable with enough generality and flexibility by any of the present systems for symbolic manipulation.The main demand for symbolic computation is actually in performing symbolic integrations. Hence, according to the present situation of the symbolic integration algorithms, the ability of handling our kind of problem by a now-a-day system is discussed. The alternative of using pattern matching techniques is also presented and discussed.The solution of a particular free boundary problem, by the MACSYMA system is presented.

A solution of problem #3 using CAMAL

Abstract: This short note is a response from CAMAL to a paper on REDUCE and MACSYMA [2], in which it was suggested that one should compare different algorithms on each system. Here CAMAL is used on a PDPIO to calculate the functions Urs (r+s≤4) by Fitch's repeated approximation algorithm [1] and Crs (r+s≤4) by Hall's method 2 [4], both solutions of SIGSAM problem 3 [3]. These figures can be compared with a similar comparison for REDUCE [2].

Different polynomial representations and their interaction in the portable algebra system PORT-ALG

Abstract: In recent years a system for the handling of large multivariate polynomials has been developed at the University of Alabama in Birmingham. The coding is generally done in Fortran IV; some special features are coded in PL/1, for the IBM systems 360 or 370. No machine coding is involved.

Presentation of the SCHOONSCHIP system

Abstract: SCHOONSCHIP is the major algebra system in use at CERN. It was designed ten years ago by M. Veltman. It is set up to do long -- but in principle straightforward -- analytic calculations. It is very fast in execution and very economical in storage. It can easily deal with expressions of the size of 104 to 105 terms. This is achieved by writing it almost entirely in (CDC 6000/7000) machine code.