Approximate Integration

Approximate Integration 

the branch of numerical analysis devoted to the development and application of methods for the approximate evaluation of definite integrals.

Let y = f(x) be a continuous function on the closed interval [a, b] and let

If the values of an antiderivative F(x) of f(x) are known at x = a and x = b, then by the Newton-Leibniz formula (the fundamental theorem of the integral calculus)

I(f) = F(b) – F(a)

If F(d) and F(b) are not known, the value of I(f) must be found by other means. One way is to approximate the value of I(f) by means of quadrature formulas, that is, suitable linear combinations of a finite number of values of f(x) and its derivatives. A quadrature formula containing only values of the function f(x) takes the form

The points x_k, k = 1, 2, …, n, x_k∈ [a, b], are called nodes and the coefficients A_k are called weights.

For every continuous function f(x), the value of I(f) can be computed by means of sums S_n (f) with arbitrary accuracy. The choice of quadrature formula is determined by the class Ω to which the given function f(x) belongs, by the method by which it is specified, and by the available computational resources. The error of the quadrature formula is the difference

R_n(f) = I(f) – S_n(f)

A quadrature formula contains 2n + 1 parameters that are independent of f(x): n, and A_k where k = 1, 2, …, n. The parameters are chosen so that for f ∈ Ω the error is acceptably small. The accuracy of a quadrature formula for f ∈ Ω is characterized by the quantity r_n (Ω), defined as the least upper bound of ǀR_n (f)ǀ on Ω:

Let

A quadrature formula for which W_n(Ω) = r_n (Ω) is said to be optimal on the class Ω. The weights and nodes in an optimal quadrature formula may be arbitrary or subject to certain constraints.

Quadrature formulas are divided into elementary and composite. Several methods have been developed for constructing elementary quadrature formulas. Let ω_q((x), q = 0, 1, …, be a complete system of functions in Ω, and let any f(x) ∈ Ω be sufficiently well approximated by linear combinations of the initial functions ω_q(x). Let I (ω_q), q = 0, 1, 2, …, be computable exactly. For every n, the parameters of the quadrature formula can be determined from the requirement that

I (ω_q) = S_n(ω_q) q = 0, 1, …, m

for as large a value of m as possible. In the Newton-Cotes method, the nodes x_k of the quadrature formula are chosen, and the weights A_k must be determined. In the Chebyshev method, certain constraints are imposed in advance on the weights A_k (for example, A_k = (b – a)/n), and the nodes x_k must be determined. In the Gauss method, both the weights A_k and the nodes x_k must be determined. In the Markov method, j nodes, j < n, are fixed in advance, and the weights and remaining nodes must be determined. The accuracy of the quadrature formulas obtained by these methods increases substantially with a felicitous choice of the functions ω_q(x).

The Newton-Cotes formulas are based on the system of functions w_q = x^q, where q= 0, 1, …. The nodes x_k divide the integration interval into equal parts. Examples of these formulas are the rectangular rule, trapezoidal rule, and Simpson rule.

Integration on the interval [a, b] can be reduced to integration on the interval [–1, 1] by a change of variable. Thus the weights and nodes of elementary formulas on [a, b] can easily be determined if they are known for [–1, 1].

In the case of composite formulas, the initial integral can be represented in the form

Elementary quadrature formulas are used to compute the integrals on the intervals [a_i, a_{i + 1}].

In the Gauss formulas, m = 2n – 1. For a = –1 and b = 1, the nodes x_k are the roots of the nth-degree Legendre polynominal P_n (x), and

The Chebyshev quadrature formula exists with A_k = l/n, l = b – a, and x_k ∈ [a, b] only for n = 1, …, 7, 9. In the Chebyshev formula m = n – 1. The use of equal weights minimizes the probability of error if the values of f(x) have independent random errors with the same variance.

In the computation of integrals of functions with period l, Gauss quadrature formulas are used most often. They are of the form

There are quadrature formulas for the computation of integrals of the form

where p(x) is a fixed weight function. It is selected so that for all f∈ Ω the functions f(x) are well approximated by linear combinations of the functions ω_q(x).

For the approximate computation of indefinite integrals, the integrals are represented as definite integrals with a variable upper limit, and the formulas given above are used.

Tables of nodes, weights, and estimates of the error of quadrature formulas are given in special handbooks.

Quadrature formulas for the computation of multiple integrals are sometimes called cubature formulas. Multiple integrals can be computed as iterated integrals, using the quadrature formulas described above. Since the number of nodes increases substantially with the multiplicity, a number of special formulas have been developed for the computation of multiple integrals.

The calculation of integrals by computer is usually accomplished by means of standard programs. In the case of single integrals, standard programs with automatic calling sequences are most often used.

REFERENCES

Krylov, V. I. Priblizhennoe vychislenie integralov, 2nd ed. Moscow, 1967.
Bakhvalov, N. S. Chislennye metody. Moscow, 1973.
Nikol’skii, S. M. Kvadraturnye formuly. Moscow, 1958.
Berezin, I. S., and N. P. Zhidkov. Metody vychisleniia, 3rd ed., part 1. Moscow, 1966.
Sobolev, S. L. Vvedenie v teoriiu kubaturnykh formul. Moscow, 1974.
Korobov, N. M. Teoretikochislovye metody v priblizhennom analize. Moscow, 1963.

V. I. LEBEDEV