Joseph F. Traub

Optimal Order of One-Point and Multipoint Iteration

The problem is to calculate a simple zero of a nonlinear function ƒ by iteration. There is exhibited a family of iterations of order 2<supscrpt><italic>n</italic>-1</supscrpt> which use <italic>n</italic> evaluations of ƒ and no derivative evaluations, as well as a second family of iterations of order 2<supscrpt><italic>n</italic>-1</supscrpt> based on <italic>n</italic> — 1 evaluations of ƒ and one of ƒ′. In particular, with four evaluations an iteration of eighth order is constructed. The best previous result for four evaluations was fifth order. It is proved that the <italic>optimal</italic> order of one general class of multipoint iterations is 2<supscrpt><italic>n</italic>-1</supscrpt> and that an upper bound on the order of a multipoint iteration based on <italic>n</italic> evaluations of ƒ (no derivatives) is 2<supscrpt><italic>n</italic></supscrpt>. It is conjectured that a multipoint iteration without memory based on <italic>n</italic> evaluations has optimal order 2<supscrpt><italic>n</italic>-1</supscrpt>.

On Euclid's Algorithm and the Theory of Subresultants

Abstract : A key ingredient for systems which perform symbolic computer manipulation of multivariate rational functions are efficient algorithms for calculating polynomial greatest common divisors. Euclids algorithm cannot be used directly because of problems with coefficient growth. The search for better methods leads naturally to the theory of subresultants. This paper presents an elementary treatment of the theory of subresultants, and examines the relationship of the subresultants of a given pair of polynomials to their polynomial remainder sequence as determined by Euclids algorithm. This relation is expressed in the fundamental theorem of this paper. The results are essentially the same as those of Collins but the presentation is briefer, simpler, and somewhat more general. The fundamental theorem finds further applications in the proof that the modular algorithm for polynomial GCD terminates. (Author)

A three-stage variable-shift iteration for polynomial zeros and its relation to generalized Rayleigh iteration

SummaryWe introduce a new three-stage process for calculating the zeros of a polynomial with complex coefficients. The algorithm is similar in spirit to the two stage algorithms studied by Traub in a series of papers. We prove that the mathematical algorithm always converges and show that the rate of convergence of the third stage is faster than second order. To obtain additional insight we recast the problem and algorithm into matrix form. The third stage is inverse iteration with the companion matrix, followed by generalized Rayleigh iteration.

A Three-Stage Algorithm for Real Polynomials Using Quadratic Iteration

We introduce a new three-stage process for calculating the zeros of a polynomial with real coefficients. The algorithm finds either a linear or quadratic factor, working completely in real arithmet...

The Algebraic Theory of Matrix Polynomials

A matrix S is a solvent of the matrix polynomial

Convergence and Complexity of Newton Iteration for Operator Equations

M(X) = A_0 X^m + \cdots + A_m

Faster evaluation of multidimensional integrals

if

ALGORITHMS FOR SOLVENTS OF MATRIX POLYNOMIALS

M(S) = 0

Path Integration on a Quantum Computer

where

A discipline in crisis

A_i ,X,S