Numerical Integration

The approximate computation of an Integral. The numerical computation of an Integral is sometimes calledQuadrature. There are a wide range of methods available for numerical integration. A good source for such techniquesis Press et al. (1992).

The most straightforward numerical integration technique uses the Newton-Cotes Formulas (also called QuadratureFormulas), which approximate a function tabulated at a sequent of regularly spaced Intervals by variousdegree Polynomials. If the endpoints are tabulated, then the 2- and 3-point formulas are called theTrapezoidal Rule and Simpson's Rule, respectively. The 5-point formula is called Bode's Rule. Ageneralization of the Trapezoidal Rule is Romberg Integration, which can yield accurate results for many fewerfunction evaluations.

If the functions are known analytically instead of being tabulated at equally spaced intervals, the best numerical method ofintegration is called Gaussian Quadrature. By picking the abscissas at which to evaluate the function, Gaussianquadrature produces the most accurate approximations possible. However, given the speed of modern computers, the additionalcomplication of the Gaussian Quadrature formalism often makes it less desirable than simply brute-force calculatingtwice as many points on a regular grid (which also permits the already computed values of the function to be re-used). Anexcellent reference for Gaussian Quadrature is Hildebrand (1956).

References

Numerical Methods

Davis, P. J. and Rabinowitz, P. Methods of Numerical Integration, 2nd ed. New York: Academic Press, 1984.

Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 319-323, 1956.

Milne, W. E. Numerical Calculus: Approximations, Interpolation, Finite Differences, Numerical Integration and Curve Fitting. Princeton, NJ: Princeton University Press, 1949.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, 1992.

• Fermat Equation
• Fermat-Euler Theorem
• Fermatian
• Fermat-Lucas Number
• Fermat Number
• Fermat Number (Lucas)
• Fermat Point
• Fermat Polynomial
• Fermat Prime
• Fermat Pseudoprime
• Fermat Quotient
• Fermat's Algorithm
• Fermat's Congruence
• Fermat's Conjecture
• Fermat's Factorization Method
• Fermat's Last Theorem
• Fermat's Lesser Theorem
• Fermat's Little Theorem
• Fermat's Little Theorem Converse
• Fermat Spiral Inverse Curve
• Fermat's Polygonal Number Theorem
• Fermat's Primality Test
• Fermat's Problem
• Fermat's Right Triangle Theorem
• Fermat's Sigma Problem