XXV.—On Bernoulli's Numerical Solution of Algebraic Equations

Abstract

The aim of the present paper is to extend Daniel Bernoulli's method of approximating to the numerically greatest root of an algebraic equation. On the basis of the extension here given it now becomes possible to make Bernoulli's method a means of evaluating not merely the greatest root, but all the roots of an equation, whether real, complex, or repeated, by an arithmetical process well adapted to mechanical computation, and without any preliminary determination of the nature or position of the roots. In particular, the evaluation of complex roots is extremely simple, whatever the number of pairs of such roots. There is also a way of deriving from a sequence of approximations to a root successive sequences of ever-increasing rapidity of convergence.

From qd to LR, or, how were the qd and LR algorithms discovered?

“…Our main conclusion is that he found the qd algorithm by studying previous work of Hadamard (1892), Aitken (1926Aitken ( , 1931 and Lanczos (1950, Chapter VI) and by improving on it. The insight that was truly impressive was to see that a step of the progressive qd algorithm (see below) can be interpreted as the LR transform on a tridiagonal matrix.…”

“…This is Bernoulli's (1732, p. 92) method for finding such a greatest root (see Aitken, 1926) . König (1884) established more than 150 years later that the analogous result holds for any power series of an analytic function with a single simple pole on the boundary of the disk of convergence.…”

“…Note that it expresses the square of H (ν) k as a difference of products of next neighbours. It was the New Zealander Alexander Craig Aitken (1895Aitken ( -1967, in Scotland, who in 1926 came up with the now obvious algorithmic conclusion (see Aitken, 1926Aitken, , 1931. He was unaware of Hadamard's work but rediscovered Theorem 3.1 and knew Jacobi's identity (3.8), which he considered as a special case of a 'theorem of compound determinants'.…”

“…The Chebyshev algorithm was later revived, analysed and modified by Sack & Donovan (1972), Wheeler (1974) and in a series of papers by Gautschi (the first of these being Gautschi, 1970), who also came up with the name modified Chebyshev algorithm for the more stable version using modified moments. Rutishauser (1954b) was aware of the work of Hadamard (1892), Aitken (1926Aitken ( , 1931 and Lanczos (1950) when he worked on Stiefel's problem. It seems that, in the second half of 1952 or early in 1953, he took Aitken's work, improved it in a significant way and made the connections to a number of related topics and applications.…”

An efficient one-point extrapolation method for linear convergence

“…The 'ô-process', developed by Aitken [1] and applied to <j> by Steffensen [7] (see [3, pp. 135-139] or [6, Appendix E]), is one such extrapolation method.…”