Methods For One-dimensional Integrals
Numerical integration methods can generally be described as combining evaluations of the integrand to get an approximation to the integral. The integrand is evaluated at a finite set of points called integration points and a weighted sum of these values is used to approximate the integral. The integration points and weights depend on the specific method used and the accuracy required from the approximation.
An important part of the analysis of any numerical integration method is to study the behavior of the approximation error as a function of the number of integrand evaluations. A method which yields a small error for a small number of evaluations is usually considered superior. Reducing the number of evaluations of the integrand reduces the number of arithmetic operations involved, and therefore reduces the total round-off error. Also, each evaluation takes time, and the integrand may be arbitrarily complicated.
A 'brute force' kind of numerical integration can be done, if the integrand is reasonably well-behaved (i.e. piecewise continuous and of bounded variation), by evaluating the integrand with very small increments.
Read more about this topic: Numerical Integration (quadrature)
Famous quotes containing the word methods:
“The comparison between Coleridge and Johnson is obvious in so far as each held sway chiefly by the power of his tongue. The difference between their methods is so marked that it is tempting, but also unnecessary, to judge one to be inferior to the other. Johnson was robust, combative, and concrete; Coleridge was the opposite. The contrast was perhaps in his mind when he said of Johnson: “his bow-wow manner must have had a good deal to do with the effect produced.””
—Virginia Woolf (1882–1941)