Abstract
Numerical integration (quadrature) of proper integrals is characterized by a big variety of methods. This chapter discusses the rectangular rules (based on the forward, backward, and central difference approximation), the trapezoidal rule, and the Simpson rule as a multi-point integration method. It moves on to a more general description – the Newton-Cotes rules and, in particular, to the Romberg method. The Gauss-Legendre quadrature is introduced as a very efficient alternative. Finally, the treatment of improper integrals and of multiple integrals is discussed. Particular emphasis is on the errors involved.
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Notes
- 1.
In this context the intermediate position \(x_{i+1/2}\) is understood as a true grid-point. If, on the other hand, the function value \(f_{i+1/2}\) is approximated by \(\mu f_{i+1/2}\), Eq. (2.29), the method is referred to as the trapezoidal rule.
- 2.
The Lagrange polynomial p n−1(x) to the function f(x) is the polynomial of degree n − 1 that satisfies the n equations \(p_{n-1}(x_{j}) = f(x_{j})\) for \(j = 1,\ldots,n\), where x j denotes arbitrary but distinct grid-points.
- 3.
Particular care is required when dealing with periodic functions!
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Stickler, B.A., Schachinger, E. (2016). Numerical Integration. In: Basic Concepts in Computational Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-27265-8_3
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DOI: https://doi.org/10.1007/978-3-319-27265-8_3
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