Increasing the Reliability of Adaptive Quadrature Using Explicit Interpolants

We present two new adaptive quadrature routines. Both routines differ from previously published algorithms in many aspects, most significantly in how they represent the integrand, how they treat non-numerical values of the integrand, how they deal with improper divergent integrals and how they estimate the integration error. The main focus of these improvements is to increase the reliability of the algorithms without significantly impacting their efficiency. Both algorithms are implemented in Matlab and tested using both the "families" suggested by Lyness and Kaganove and the battery test used by Gander and Gautschi and Kahaner. They are shown to be more reliable, albeit in some cases less efficient, than other commonly-used adaptive integrators.Comment: 32 pages, submitted to ACM Transactions on Mathematical Softwar

Inner product quadrature formulas

We investigate an approach to approximating the integral (0.1) ⨍[sub]R w(x)f(x)g(x)dx ≡ I (f;g), where R is a region in one-dimensional Euclidean space, and w a weight function. Since (0.1) may be regarded as a continuous bi-linear functional in f and g we approximate it by a discrete bi-linear functional, which we term an Inner Product Quadrature Formula (I.P.Q.F.). (0.2) Q(f;g) ≡ f̲ᵀAg̲, Where f̲ᵀ = (Sₒ(f), . . . , s[sub]m(f))ᵀ g̲ᵀ = (Tₒ(g), . . . , T[sub]n(g)) ᵀ A = (aᵢ[sub]j)ᵐi=o,ⁿj=0, And a[sub]i[sub]j are real numbers, ᵐi=0 ⁿj =0 |aᵢ [sub]j | > 0 The so-called elementary functionals {Sᵢ}ᵐi=0 and {T[sub]j}ⁿj=0 are two sets of linearly independent linear functionals, acting f and g respectively, defined over a certain subspace of functions to which f and g belong. The simplest example of these functionals is function evaluation at a given point. The matrix A is determined by requiring (0.2) to be exact for certain classes of functions f and g, say F ≡ {₀, . . . , ᵧ}, ≥0 G ≡ {₀, . . . , [sub] } ≥0 In Chapter 1 we introduce the concept of I.P.Q.F. in more detail and make some general comments about approaches available when examining numerical integration. After explaining in some detail why we feel I.P.Q.F. are a useful tool in §2.1, we proceed in the remainder of Chapter 2 to investigate various conditions which may be placed on ᵞ, [super] {Sᵢ}ᵐi=0 and {T[sub]j}ⁿj=0 in order to guarantee the existence of I.P.Q.F. exact when F and G. In particular we investigate the question of maximizing + . In the case where ᵢ and [sub]j are the standard monomials of degree i and j respectively, some results have already been published in B.I.T. (1977) p. 392-408. We investigate various choices of ᵢ and [sub]j : (a) {ᵢ}ᵐ⁺¹ I = 0 (i.e. = m+1) and {[sub]j}ᵐ[sub]j = 0 (i.e. = m) being Tchebychev sets (§2.7), (b) {ᵢ}²ᵐ⁺¹ I = 0 (i.e. = 2m+1) being a Tchebychev set and [super] contains only one function (i.e. = 0) (§2.6) (c) ᵢ ≡ ([sub]l)ⁱ, i=0,1, . . . and ᵢ = ᵢ, i= 0, 1, … (§2.8). In Chapter 3 we consider the question of compounding I.P.Q.F. both in the classical sense, and, briefly, by examining spline functions, regarding them as providing a link between an I.P.Q.F on one hand and a compounded I.P.Q.F. on the other. Various methods of theoretically estimating the errors involved are considered in Chapter M-. In the fifth Chapter we examine various ways in which the concept of I.P.Q.F. might (or might not) be extended. Finally, we make some brief comments about the possible applications of I.P.Q.F., and give a few examples