×

Sigmoidal transformations and the trapezoidal rule. (English) Zbl 0928.65033

One of the early techniques learned in a first year calculus course is that of transforming an integrand \[ f(x)\text{ to }F(x)= f(g(x)) g'(x). \] In that context this is usually done because a primitive of \(F(x)\) in analytic form is more readily available to the student than one of \(f(x)\). The function \(g(x)\) is termed the substitution function. This process must be as old as the integral calculus itself.
During the latter half of the twentieth century, in numerical quadrature, many such transforms have been suggested, but now with the overall view of arranging that \(F(x)\) be more amenable to numerical integration than is \(f(x)\). These suggestions occur in the literature on an ad hoc basis. The numerical method in mind is almost invariably the trapezoidal rule. This is easy to program, and the sort of integrand functions for which it is particularly efficient are well understood.
In the case that both the original and the transformed interval of integration are finite, no serious loss in generality is incurred by restricting both integration intervals to be \([0,1]\). In this case, one seeks a situation in which many early derivatives of \(F(x)\) vanish at \(x= 0\) and at \(x=1\). To this end, one uses a “sigmoidal” function \(g(x)\). Shaped like a very italic \(S\), this function satisfies \(g(0)= 0\), \(g(1)= 1\), is strictly increasing and infinitely smooth in \((0,1)\); in addition, \(g'(x)\) is symmetric in the interval, is strictly increasing in \((0,1/2)\), and vanishes at the end points.
In the first half of this long paper, the author presents a dictionary of sigmoidal functions \(g(x)\), describing in detail how some are related to others. These include those associated with Korobov, and many others, going back to Sag-Szekeres, and forward to the recent family introduced by Sidi.
In the second half of the paper, asymptotic expressions are derived for the discretization error associated with the use of the offset trapezoidal rule for the numerical integration.
This article is concerned exclusively with one-dimensional finite interval integration, and almost exclusively with the case in which \(f(x)\) is analytic. An interesting short section at the end discusses the abscissa distribution when the result of applying the trapezoidal rule to the transformed function is interpreted as a more sophisticated quadrature rule applied to the original function.
This reviewer finds this paper to be of exceptional value both to scientists who use numerical quadrature, and to scientists who study the theory and the design of quadrature rules.

MSC:

65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures

Citations:

Zbl 0928.65032
PDFBibTeX XMLCite