Abstract: A unique code (called Hensel's code) is derived for a rational number by truncating its infinite p-adic expansion. The four basic arithmetic algorithms for these codes are described and their application to rational matrix computations is demonstrated by solving a system of linear equations exactly, using the Gaussian elimination procedure.

Abstract: Let D be a Euclidean domain which is a subring of the field of complex numbers and in which the arithmetic operations can be algorithmically performed. Examples of D are Z , the integers, Q , the rational numbers, G , the Gaussian integers, and P , the real algebraic numbers. We are concerned with algorithms which, given any polynomial A(x)∈D[x] and any positive rational number ∈, computes a sequence of disjoint intervals (rectangles) of length (width) ∈ or less, each containing exactly one real (complex) zero of A, and together containing all real (complex) zeros. The algorithms must also compute the multiplicity of each zero. Any algorithm strictly fulfilling all of these specification may truly be described as infallible. While retaining these lofty goals we are nevertheless concerned with algorithms which are as efficient as possible, in practice as well as theory. Such infallible algorithms can be derived quite readily from several diverse mathematical theorems using “exact” arithmetic in the domain D, but these algorithms may differ significantly in their time complexities. Also, one may further improve practical efficiency by appropriate substitution of “approximate” arithmetic (e.g., interval arithmetic) for exact arithmetic in certain contexts without sacrificing infallibility (using exact arithmetic as backup). This paper surveys recent progress on this problem, starting from Heindel's 1970 implementation and analysis of Sturm's theorem, including Pinkert's 1973 application of Sturm sequences to complex zeros, the 1975 Collins-Loos algorithm based on Rolle's theorem, the 1976 Collins-Akritas modification of Uspensky's method based on Descartes' theorem, and concluding with a brief report on current research by Collins and Chou relating to use of approximate arithmetic and the “principle of argument”. Both theoretical time bounds and empirical times are presented.

Abstract: We show how techniques of interval arithmetic can be used to give bounds on the global minima of unconstrained optimization problems. We illustrate the techniques using the design of a hypothetical chemical plant.

Abstract: PURPOSE: To execute real-time arithmetic such as vector conversion and display its results in a display system, by performing various arithmetic processes in the process of reading from a screen memory and displaying. COPYRIGHT: (C)1978,JPO&Japio

Abstract: This paper describes the operation and implementation of the arithmetic proof rule for the quantifier free integer arithmetic used in the PL/CV 2 program verification system. The general arithmetic satisfiability problem underlying the rule is shown to be NP complete.

Abstract: In this paper the congruence (f ∘ g)(n) = 0 (mod n) and the functional equation f ∘ f ∘ … ∘ f = g, are studied, where ∘ is an exponential regular convolution.

Abstract: The structure for hardware realisation of Fast Walsh-Fourier Transform (FWFT) is presented in this paper. The development of the approach is based on Shanks' algorithm for FWFT and use is made in the realisation of serial storage and simple arithmetic. Two alternative structures are proposed one based on PCM encoding and serial two's complement arithmetic, whilst the other is based on delta-sigma encoding with appropriate arithmetic operations. Comparison of these alternative solutions is given in terms of hardware requirements and the mean square error produced.

Abstract: During recent years a number of papers concerning a mathematical foundation of computer arithmetic have been written. Some of these papers are still unpublished. The papers consider the spaces which occur in numerical computations on computers depending on a properly defined computer arithmetic. The following treatment gives a summary of the main ideas of these papers. Many of the proofs had to be sketched or completely omitted. In such cases the full information can be found in the references.

Abstract: Algorithms for general partial fraction decomposition are obtained by using modular polynomial arithmetic. An algorithm is presented to compute inverses modulo a power of a polynomial in terms of inverses modulo that polynomial. This algorithm is used to make an improvement in the Kung-Tong partial fraction decomposition algorithm.

Abstract: This paper describes a combined arithmetic unit and language support system which allows user specifications of the arithmetic. Limited extensions to a high-level language, in connection with a generalized underlying arithmetic unit, allow a single skeletal unified numeric operand type to be refined into a variety of data types. The interpretation of operands by the operators is based on type descriptors, allowing one set of polymorphic arithmetic operators to be defined across all combinations of user-defined operand types. The arithmetic unit is realized in microcode to achieve efficiency.

Abstract: THe Collins-Loos algorithm for computing isolating intervals for the zeros of an integer polynomial requires the evaluation of polynomials at rational points. This implies the use of arbitrary precision integer arithmetic. It is shown how careful use of single precision, floating point arithmetic within the context of a slightly modified algorithm can make the calculation considerably faster and no less exact. Typically, 95% or more of the evaluations can be done without exact arithmetic. The precise speedup depends on the relative costs of the arithmetic in a given implementation. The implementation on DEC KL-10 computer is some 5 to 10 times faster than the original Univac 1110 implementation in SAC-I.

Abstract: Embedding, a topic discussed sporadically in the literature [l,2] most recently by APL 8nthusiasts [3,4], is the extension o f the functional domain of a programming language, to includ e d a t a structures and operations not previously supported, by writin ~! subroutines only in the language being extended. This latter s e r ves tO distinguish embedding from the prOCeSS Of exten d in g a lang ua CJC' by re-writing parts of the processor for that lan gu a ge .

Abstract: Sussman (1973) has shown that strategies could be reordered according to experience. In his examples no conflicts were experienced after reordering. We shall illustrate that conflicts do arise and present a way of trying to overcome them. The program written (ELS) accepts a set of partially ordered Horn clauses as input and a sequence of problems to solve. The search tree obtained is examined in order to generate a priority ordering for the clauses used. If conflicts are detected, ELS tries to determine the conditions under which the clauses involved should be applied. For example , suppose the problem 3+2=X is to be solved and we are given the clauses: assoc: subs: subz: sue : pred: Generally, the problems are solved in a breadth first manner, but priority orderings between individual clauses are respected. First all clauses that match the given subgoal are chosen. In our case both the clauses 'subs' and 'subz' match the initial goal 3+2-X, as this goal matches X1+X2=X3. Both clauses are actually tried in the search, since initially there is no priority ordering specified between them. If it had been specified that 'subz