Thomas Simpson and ‘Newton's method of approximation’: an enduring myth

Abstract

A resurgence of interest has occurred in ‘Newton's method of approximation’ for deriving the roots of equations, as its repetitive and mechanical character permits ready computer use. If x = α is an approximate root of the equation f(x) = 0, then the method will in most cases give a better approximation aswhere f′(x) is the derivative of the function into which α has been substituted. Older books sometimes called it ‘the Newton–Raphson method’, although the method was invented essentially in the above form by Thomas Simpson, who published his account of the method in 1740. However, as if through a time-warp, this invention has migrated back in time and is now matter-of-factly placed by historians in Newton's De analysi of 1669. This paper will describe the steps of this curious historical transposition, and speculate as to its cause.

Volatile Organic Compounds in the Unsaturated Zone from Radioactive Wastes

“…A unique value for D EN was obtained from the integrated form of Eq. [5] by applying Newton‐Raphson iteration (Kollerstrom, 1992). The linear nature of concentration profiles, even to depths of 25 m (data shown in the following section), suggests that the effective diffusion constants were relatively constant to at least that depth.…”

“…where X E and X N are mole fractions of ethane and nitrogen (dimensionless), J N and J E are mass-fl uxes of nitrogen and ethane (g cm −2 s −1 ), D EN is the eff ective diff usion constant of ethane in nitrogen (cm 2 s −1 ), ω E and ω N are molecular weights of ethane and nitrogen (g mol −1 ), and z is distance along the core from the ethane source reservoir (cm) A unique value for D EN was obtained from the integrated form of Eq. [5] by applying Newton-Raphson iteration (Kollerstrom, 1992). Th e linear nature of concentration profi les, even to depths of 25 m (data shown in the following section), suggests that the eff ective diff usion constants were relatively constant to at least that depth.…”

FastChem: A computer program for efficient complex chemical equilibrium calculations in the neutral/ionized gas phase with applications to stellar and planetary atmospheres

“…A method, known as the (original) NASA method, not to be confused with the algorithm implemented in the NASA-CEA code, was developed by Huff et al (1951). Zeleznik & Gordon (1960 showed the computationally equivalence between Brinkley's 1 For a historical note, see (Kollerstrom 1992) method, the NASA method and the RAND method. Therefore, the three methods together are sometimes called BNR method (Smith & Missen 1982).…”

Newton Methods for Nonlinear Problems

“…Strictly speaking, Newton's method could as well be named as Newton-Raphson-Simpson method-as elaborated in recent articles by N. Kollerstrom [134] or T.J. Ypma [203]. Looking at the graph of f (x)-as depicted in Figure 1.1-any root can be interpreted as the intersection of this graph with the real axis.…”