Lagrange and the solution of numerical equations

Abstract

In 1798 J.-L. Lagrange published an extensive book on the solution of numerical equations. Lagrange had developed four versions of a general systematic algorithm for detecting, isolating, and approximating, with arbitrary precision, all real and complex roots of a polynomial equation with real coefficients. In contrast to Newton's method, Lagrange's algorithm is guaranteed to converge. Some of his powerful ideas and techniques foreshadowed methods developed much later in geometry and abstract algebra. For instance, in order to make a more efficient algorithm for isolating roots, Lagrange essentially worked in a quotient ring of a polynomial ring. And to accelerate both the convergence and calculation of his continued fraction expansions of the roots, he employed nonsimple continued fractions and Möbius transformations. © 2001 Academic Press.