Corpus ID: 16295237

# Lecture Ii the Gcd

```@inproceedings{LectureIT,
title={Lecture Ii the Gcd},
author={}
}```
Next to the four arithmetic operations, the greatest common denominator (GCD) is perhaps the most basic operation in algebraic computing. The proper setting for discussing GCD's is in a unique factorization domain (UFD). For most common UFDs, that venerable algorithm of Euclid is available. In the domains Z and F [X], an efficient method for implementing Euclid's algorithm is available. It is the so-called half-GCD approach, originating in ideas of Lehmer, Knuth and Schönhage. The presentation… Expand

#### References

SHOWING 1-10 OF 225 REFERENCES
An Algorithm for Exact Division
• T. Jebelean
• Computer Science, Mathematics
• J. Symb. Comput.
• 1993
An algorithm which computes the quotient of two long integers in this particular situation, starting from the least-significant digits of the operands, which is better suited for systolic parallelization in a "least-significant digit first" pipelined manner. Expand
The Subresultant PRS Algorithm
A corollary of the fundamental theorem of subresuitants is given here, which leads to a simple derivation and deeper understanding of the subresultant PRS algorithm and converts a conjecture mentioned in the earher papers into an elementary remark. Expand
Factoring Polynomials Over Algebraic Number Fields
• Mathematics, Computer Science
• TOMS
• 1976
This algorithm has the advantage of factoring nonmonic polynomials without inordinately increasing the amount of work, essentially by allowing denominators in the coefficients of the polynomial and its factors. Expand
Ideals, varieties, and algorithms - an introduction to computational algebraic geometry and commutative algebra (2. ed.)
• Computer Science
• 1997
The algorithmic roots of algebraic object, called a close relationship between ideals, many of polynomial equations in geometric, object called a more than you, for teaching purposes and varieties, and the solutions and reduce even without copy. Expand
Fast computation of GCDs
• R. Moenck
• Mathematics, Computer Science
• STOC
• 1973
An integer greatest common divisor (GCD) algorithm due to Schönhage is generalized to hold in all euclidean domains which possess a fast multiplication algorithm and a new faster algorithm for multivariate polynomial GCD's can be derived. Expand
Computing greatest common divisors and factorizations in quadratic number fields
• Mathematics
• 1989
In a quadratic number field Q(√D), D a squarefree integer, with class number 1, any algebraic integer can be decomposed uniquely into primes, but for only 21 domains Euclidean algorithms are known.Expand
Infallible Calculation of Polynomial Zeros to Specified Precision
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, QExpand
An Exact Method for Finding the Roots of a Complex Polynomial
A new SAC-1 module is described which uses algebraic algorithms based on the classical theorems of Sturm and Routh to isolate the unique roots of G into disjoint squares m the complex plane which can then be refined to any prespecified rational width. Expand
Gaussian elimination is not optimal
t. Below we will give an algorithm which computes the coefficients of the product of two square matrices A and B of order n from the coefficients of A and B with tess than 4 . 7 n l°g7 arithmeticalExpand
Generalized Polynomial Remainder Sequences
Given two polynomials over an integral domain, the problem is to compute their polynomial remainder sequence (p.r.s.) over the same domain. Following Habicht, we show how certain powers of leadingExpand