Showing papers on "Greatest common divisor published in 1988"
24 Oct 1988
TL;DR: In this article, a black-box representation for multivariate polynomials and rational functions with rational coefficients was developed for their evaluation, and it was shown that within this representation, the polynomial greatest common divisor and factorization problems as well as the problem of extracting the numerator and denominator of a rational function can be solved in random time in the usual parameters.
Abstract: Algorithms are developed that adopt a novel implicit representation for multivariate polynomials and rational functions with rational coefficients, that of black boxes for their evaluation. It is shown that within this evaluation-box representation, the polynomial greatest common divisor and factorization problems as well as the problem of extracting the numerator and denominator of a rational function can be solved in random polynomial time in the usual parameters. Since the resulting evaluation programs for the goal polynomials can be converted efficiently to sparse format, solutions to sparse problems such as the sparse ration interpolation problem follow as a consequence. >
124 citations
Book•
01 Jan 1988
TL;DR: In this article, the authors present a set of algorithms for solving the problem of whole number computations in the Indian-Arabic system, including the following: 1.1 The Problem Solving Process and Strategies.2.
Abstract: Preface.1. Introduction to Problem Solving.1.1 The Problem Solving Process and Strategies.1.2 Three Additional Strategies.2. Sets, Whole Numbers, and Numeration.2.1 Sets as a Basis for Whole Numbers.2.2 Whole Numbers and Numeration.2.3 The Hindu-Arabic System.2.4 Relations and Functions.3. Whole Numbers: Operations and Properties.3.1 Addition and Subtraction.3.2 Multiplication and Division.3.3 Ordering and Exponents.4. Whole Number Computation: Mental, Electronic, and Written.4.1 Mental Math, Estimation, and Calculators.4.2 Written Algorithms for Whole-Number Operations.4.3 Algorithms in Other Bases.5. Number Theory.5.1 Primes, Composites, and Tests for Divisibility.5.2 Counting Factors, Greatest common Factor, and Least Common Multiple.6. Fractions.6.1 The Sets of Fractions.6.2 Fractions: Addition and Subtraction.6.3 Fractions: Multiplication and Division.7. Decimals, Ratio, Proport ion, and Percent.7.1 Decimals.7.2 Operations with decimals.7.3 Ratios and Proportion.7.4 Percent.8. Integers.8.1 Addition and Subtraction.8.2 Multiplication, Division and Order.9. Rational Numbers, Real Numbers and Algebra9.1 The Rational Numbers.9.2 The Real Numbers.9.3 Functions and Their Graphs.10. Statistics.10.1 Organizing and Picturing Information.10.2 Analyzing Data.10.3 Misleading Graphs and Statistics.11. Probability.11.1 Probability and Simple Experiments.11.2 Probability and Complex Experiments.11.3 Additional Counting Techniques.11.4 Simulation, Expected Value, Odds, and Conditional Probability.12. Geometric Shapes.12.1 Recognizing Geometric Shapes.12.2 Analyzing Shapes.12.3 Properties of Geometric Shapes: Lines and Angles.12.4 Regular Polygons and Tessellations.12.5 Describing Three-Dimensional Shapes.13. Measurement.13.1 Measurement with Nonstandard and Standard Units.13.2 Length and Area.13.3 Surface Area.13.4 Volume.14. Geometry Using Triangle Congruence and Similarity.14.1 Congruence of Triangles.14.2 Similarity of Triangles.14.3 Basic Euclidean Constructions.14.4 Additional Euclidean Constructions.14.5 Geometric Problem Solving Using Triangle Congruence and Similarity.15. Geometry Using Coordinates.15.1 Distance and Slope in the Coordinate Plane.15.2 Equations and Coordinates.15.3 Geometric Problem Solving Using Coordinates.16. Geometry Using Transformations.16.1 Transformations.16.2 Congruence and Similarity Using Transformations.16.3 Geometric Problem Solving Using Transformations.Epilogue: An Eclectic Approach to Geometry.Topic 1. Elementary Logic.Topic 2. Clock Arithmetic: A Mathematical System.Topic 3. Introduction to Graph Theory.References.Answers to Exercise/Problem Sets-Part A, Chapter Tests, and Topics.Photograph Credits.Index.
88 citations
TL;DR: In this paper, a new method is presented for the computation of a greatest common divisor (gcd) of two polynomials, along with their polynomial remainder sequence (prs).
Abstract: A new method is presented for the computation of a greatest common divisor (gcd) of two polynomials, along with their polynomial remainder sequence (prs). This method is based on our generalization of a theorem by Van Vleck [12] and uniformly treats both normal and abnormal prs's, making use of Bareiss's [3] integer-preserving transformation algorithm for Gaussian elimination. Moreover, for the polynomials of the prs's, this method provides the smallest coefficients that can be expected without coefficient ged computations (as in Bareiss [3]) and it clearly demonstrates the divisibility properties; hence, it combines the best of both the reduced and the subresultant prs algorithms.
21 citations
Proceedings Article•
01 Jan 1988
TL;DR: An Omega (log log n) lower bound is proved on the depth of any computation tree with operations (+, -, /, mod,
15 citations
24 Oct 1988
TL;DR: In this paper, an Omega lower bound on the depth of any computation tree with operations (+, -, /, mod, >) was proved, and a lower bound of O(log log n) was also proved.
Abstract: An Omega (log log n) lower bound is proved on the depth of any computation tree with operations (+, -, /, mod, >
13 citations
24 May 1988
Abstract: Many-valued logic symmetric functions appearing in various applications are investigated from the standpoint of determining the number of n-ary functions belonging to a considered set (called the spectrum of the set). Respective spectra are given of k-valued functions that are p-symmetric, self-dual, and self-dual p-symmetric, where p is a partition of (1,. . .,n). It is proved that there exist self-dual totally symmetric n-ary k-valued logic functions if and only if the greatest common divisor of k and n is equal to one. A test for detecting the self-dual symmetry property is described. Respective spectra are also given of k-valued symmetric functions that are threshold, multithreshold, monotone, and unate (for the monotone and unate functions k=3 only). >
6 citations
04 Jul 1988
6 citations
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TL;DR: In this paper, the authors follow the steps of development of a simple program and explain some of the fundamental concepts of programming and of the basic facilities of Modula, where the task is to compute the greatest common divisor (gcd).
Abstract: Let us follow the steps of development of a simple program and thereby explain some of the fundamental concepts of programming and of the basic facilities of Modula. The task shall be, given two natural numbers x and y, to compute their greatest common divisor (gcd).
5 citations
TL;DR: Two new binary Euclidean algorithms to calculate the greatest common divisor are given and an exhaustive search for all odd integers of moderate length shows that these algorithms use fewer iterations on the average than the two presently known algorithms.
Abstract: Two new binary Euclidean algorithms to calculate the greatest common divisor are given. An exhaustive search for all odd integers of moderate length shows that these algorithms use fewer iterations on the average than that the two presently known algorithms.
3 citations
Patent•
08 Jul 1988
TL;DR: In this article, the authors proposed to obtain large scale circuit integration with small size by connecting plural identical arithmetic circuits each comprising a multiplier on the Galois field, an adder, and m-stage of register arrays storing its addition output and selector output and using polynomials A, B to obtain its greatest common divisor polynomial GCD.
Abstract: PURPOSE:To attain large scale circuit integration with small size by connecting plural identical arithmetic circuits each comprising a multiplier on the Galois field, an adder, and m-stage of register arrays storing its addition output and selector output and using polynomials A, B to obtain its greatest common divisor polynomial GCD [A, B]. CONSTITUTION:In obtaining polynomials A, B and L, M, two independent Process sections are to be provided or one Process section is to be used twice. In using one Process section twice, the processing speed is halved and in providing two Process sections independently, number of required PEs is doubled. For example, the relation of selector selection signals S1, 2=11 is selected only at the input of syndrome polynomials Sx, x , selectors outputting D, E inputs at the X, Y outputs are used. Moreover, in processing A, B and L, M by one processing section, the processing element PE shown in figure is used to control S1-S4 thereby inputting A=x , B=Sx, L=0, M=1.
1 citations
Patent•
30 Jun 1988
TL;DR: In this article, the greatest common divisor polynomials of A and B are inputted, whereby the greatest-common-divisor GCB [A, B] is generated.
Abstract: PURPOSE:To increase processing capacity by connecting plural and same type arithmetic circuits consisting of selector circuits, multipliers, adders and registers CONSTITUTION:A processing element (PE) consists of the selector circuit of multiple-input multiple-output or multiple-input single output, the multipliers 2 and 3 on a Galois field having an output from the selector circuit 1 as an input, an adder 4 on the Galois field adding the outputs from the multipliers 2 and 3, and the registers 5-7 accumulating the output from the adder 4, and the output from the selector circuit 1 Plural and same type processing elements (PE) are connected and polynomials A and B are inputted, whereby the greatest common divisor polynomials GCB [A, B] of A and B is generated
TL;DR: The mutual-subtraction algorithm and the Euclidean algorithm are well matched and can be used to find the greatest common factor and least common multiple of integers and to calculate the common period of fractional periods.