Computer algebra is an alternative and complement to numerical mathematics. Its importance is steadily increasing. This volume is the first systematic and complete treatment of computer algebra. It presents the basic problems of computer algebra and the best algorithms now known for their solution with their mathematical foundations, and complete references to the original literature. The volume follows a top-down structure proceeding from very high-level problems which will be well-motivated for most readers to problems whose solution is needed for solving the problems at the higher level. The volume is written as a supplementary text for a traditional algebra course or for a general algorithms course. It also provides the basis for an independent computer algebra course.
Given two polynomials over an integral domain, the problem is to compute their polynomial remainder sequence (p.LS.) over the same domain. Following Habicht, we show how certain powers of leading coefficients enter systematically all following remainders. The key tool is the subresultant chain of two polynomials. We study the primitive, the reduced and the improved subresultant p.LS. algorithm of Brown and Collins as basis for computing polynomial greatest common divisors, resultants or Sturm sequences. Habicht's subresultant theorem allows new and simple proofs of many results and algorithms found in different ways in computer algebra.
The Problem
Polynomial Division and DeterminantsThe problem we want to study in this chapter is the efficient computation of polynomial sequences based on polynomial division. Let F be a field and A, BE F[x] be polynomials of degree m and n respectively with m ~ n. There exist unique polynomials Q and R such that A = BQ + R with R = 0 or deg R < deg B.Q and R can be computed by the following algorithm
QR(A, B; Q, R).( (m-n+1)
A new algorithm is described, for the isolation of the real roots of a real polynomial. This algorithm utilizes the sequence of derivatives of the given polynomial, relying on Rolle's theorem and a tangent construction to decide whether an interval contains two roots or none. The algorithm is carefully compared, both analytically and empirically, with an algorithm based on Sturm's theorem, and is found to be significantly faster in general. It also requires less memory, and produces isolating intervals for the derivatives as a cost-free by-product.
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