Computational Science in the 17th Century. Numerical Solution of Algebraic Equations: Digit–by–Digit Computation

Abstract

In this paper we give a complete overview of test–problems by Viète from 1600, Harriot from 1631 and Oughtred from 1647. The original material is not easily accessible due to archaic language and lack of conciseness. Viéte’s method was gradually elucidated by the subsequent writers Harriot and Oughtred using symbols and being more concise. However, the method is presented in tables and from the layout of the tables it is difficult to find the general principle. Many authors have therefore described Viète’s process inaccurately and in this paper we give a precise description of the divisor used in the process which has been verified on all the test–problems. The process of Viète is an iterative method computing one digit of the root in each iteration and has a linear rate of convergence and we argue that the digit–by–digit process lost its attractiveness with the publications in 1685 and 1690 of the Newton-Raphson method which doubles the number of digits for each iteration.