Numerical Integration
Numerical integration involves calculation of an integral by numerical techniques. An integral is a mathematical representation of an area or generalization of an area and is one of the fundamental objects of calculus. The word quadrature can mean the numerical computation of an integral containing only one variable. Gaussian quadratures and Newton-Cotes formulas are methods of numerical integration. Cubature means the numerical computation of a multiple integral, a set of integrals taken over multiple variables, and includes Monte Carlo integration.
The main difference between the two classifications of numerical integration methods, Gaussian quadratures and Newton-Cotes formulas, is the way a curve is divided into parts for analysis. Gaussian quadratures provide flexibility and efficiency for known smooth functions but Newton-Cotes formulas are the most widely used numerical integration methods. All Gaussian quadratures involve evaluating an integral by the summation of the product of the function evaluated at optimal points and a weighting function related to that point plus an error function: ab f(x)dx = nk=1 w(xk)f(xk) + Rn(x), where xk are the points at which the function is evaluated, w(x) is the weighting function for each point, and Rn is the error function. Gaussian quadratures allow one to pick the optimal abscissas as which to evaluate the function. The fundamental theorem of Gaussian quadratures shows that the optimal abscissas are the roots of the orthogonal polynomial for the same interval and weighting function. Gaussian quadratures have a higher degree of accuracy than numerical integrations carried out using Newton-Cotes formulas. This is because Gaussian quadratures fit all polynomials up to degree 2n exactly. Some commonly used Gaussian quadratures are the Gauss-Legendre formula and the Gauss-Chebyshev formula, which are used for closed, definite integrals, the Gauss-Hermite formula, which is used for integrals that have - and as the limits of integration, and the Gauss-Laguerre formula, which is used for integrals on the interval [0, ).
The Newton-Cotes formulas are numerical integration methods involve dividing up the interval over which integration is to occur into equal parts. Then polynomials that approximate the function are substituted for the function. The polynomials are calculated at the nodal points and summed with weighting coefficients to find the solution to the original integral. These methods are not as accurate as using the Gaussian quadrature methods but with computers being readily available calculating integrals this way is the most widely used numerical integration technique. Newton-Cotes formulas include the trapezoidal rule, which uses linear equations, Simpson's rule, which uses parabolic equations, Simpson's 3/8 rule, which uses cubic equations, and Bode's rule, which uses fourth degree polynomial equations. The Romberg integration is an extension of the trapezoidal rule that uses refinements to remove some error terms. There are many other Newton-Cotes formulas employed to numerically integrate functions, these are just a few.
Numerical integration methods can be traced back to about 260 B.C. when Archimedes of Syracuse, an Italian mathematician, perfected methods of numerical integration. He employed an early form of integration called the method of exhaustion. This method allows calculation of an area by calculating the areas of a sequence of polygons that approximate the original area. Archimedes gave an accurate approximation of using this method. He published a book, Quadrature of the parabola, in which he determines the area of a segment of a parabola cut off by any chord. Enhancements in methods of numerical integration were achieved by several mathematicians over the next century but Ibrahim, an Arab mathematician, made a special contribution when he introduced a method of integration that was more general than that of Archimedes in the 10th century A.D.. Thomas Simpson, an English mathematician, is also another important contributor to methods of numerical integration. He introduced a Newton-Cotes formula that bears his name in the 18th century. Since then many other methods have been introduced and employed to determine areas by numerical integration.
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