On a certain adaptive method of approximate integration and its stopping criterion
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Aequationes Mathematicae
On a certain adaptive method of approximate integration and its stopping criterion Szymon Wa˛sowicz
Abstract. We introduce a new quadrature rule based on Chebyshev’s and Simpson’s rules. The corresponding composite rule induces the adaptive method of approximate integration. We propose a stopping criterion for this method and we prove that if it is satisfied for a function which is either 3-convex or 3-concave, then the integral is approximated with the prescribed tolerance. Nevertheless, we give an example of a function which does satisfy our criterion, but the approximation error exceeds the assumed tolerance. The numerical experiments (performed by a computer program created by the author) show that integration of 3-convex functions with our method requires considerably fewer steps than the adaptive Simpson’s method with a classical stopping criterion. As a tool in our investigations we present a certain inequality of Hermite–Hadamard type. Mathematics Subject Classification. Primary 65D30; Secondary 26A51, 26D15, 41A55, 41A80, 65D32. Keywords. Approximate integration, Quadratures, Adaptive methods, Chebyshev’s rule, Simpson’s rule, Higher-order convexity, Hermite–Hadamard type inequality.
1. Introduction 1.1. Quadratures and adaptive methods of numerical integration The numerical integration of a function f : [a, b] → R is often performed by applying the quadrature rule: b f (x) dx ≈ Q[f ; a, b] :=
I[f ; a, b] := a
m
wk f λk a + (1 − λk )b ,
k=1
where the coefficients λ1 , . . . , λm ∈ [0, 1] and the weights w1 , . . . , wm ∈ R (usually positive) are fixed. For example, the following assignment is the familiar
S. Wa˛sowicz
simple Simpson’s rule: S[f ; a, b] =
AEM
a+b b−a f (a) + 4f + f (b) . 6 2
In this paper we also deal with the simple Chebyshev’s rule √ √ a+b 2+ 2 2− 2 b−a a+ b +f C[f ; a, b] = f 3 4 4 2 √ √ 2− 2 2+ 2 a+ b . +f 4 4 For the sake of the record, let us also list simple two-point Gauss’ rule √ √ √ √ 3− 3 b−a 3+ 3 3− 3 3+ 3 G[f ; a, b] = a+ b +f a+ b . f 2 6 6 6 6 Every quadrature rule induces the so-called composite rule, which is created by subdividing the interval [a, b] into n subintervals with equally spaced endpoints: b−a , k = 0, 1, . . . , n n and if I[f ; xk , xk+1 ] ≈ Q[f ; xk , xk+1 ], then we sum over k ∈ {0, . . . , n − 1} to arrive at the rule: a = x0 < x1 < · · · < xn = b
with
xk = a + k ·
I[f ; a, b] ≈ Qn [f ; a, b] :=
n−1
Q[f ; xk , xk+1 ].
(1)
k=0
In this way we get Simpson’s and Chebyshev’s composite rules, denoted respectively Sn and Cn . Let ε > 0 be a fixed tolerance. In most cases it is the integral by a simple quadrature rule, since
not enough to approximate
Q[f ; a, b] − I[f ; a, b] > ε. In order to achieve the desired precision we apply the composite rule Qn with n large enough to have
Qn [f ; a, b] − I[f ; a, b] < ε. Such a method of approximation is called adaptive. How to find such n? If the function f is regular enough, the error terms of the simple quadratures are known and the
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