Papers, Notes, Books & Help For Students

Numerical integration aims at approximating definite integrals using numerical techniques. There are many situations where numerical integration is needed. For example, several well defined functions do not have an anti-derivative, i.e. their anti-derivative cannot be expressed in terms of primitive function. A popular example is the function e��x2 whose anti-derivative does not exist. This function arises in a variety of applications such as those related to probability and statistics analyses. Furthermore, many applications in science and engineering are represented by integral differential equations that require a special treatment for the integral terms (e.g. expansion, liberalization, closure …).

Therefore, numerical integration does not only provide a means for evaluating integrals numerically, but also grants us the ability to approximate special functions that are defined in terms of integrals. Without loss of generality, there are two classes of problems where numerical integration is needed. In the first class, one wishes to evaluate the integral of a well defined function. In this case, the integrand can be evaluated a various points because and numerical integration techniques help define the optimum number of these points as well as their locations. The second class of problems for applying numerical integration is found in differential equations the most common of which are those that express conservation principles. For example, the population balance equation, a well known partial differential equation encountered in process modeling and biological systems, exhibits source terms that are represented as integrals of the solution variable (e.g. the number density function). The most common technique for numerical integration is called quadrature. The recipe for quadrature consists of three steps

1. Approximate the integrand by an interpolating polynomial using a specified number of points or nodes

2. Substitute the interpolating polynomial into the integral

3. Integrate

………………..

Reasons for numerical Integration:

There are several reasons for carrying out numerical integration. The integrand f may be known only at certain points, such as obtained by sampling. Some embedded systems and other computer applications may need numerical integration for this reason.

A formula for the integrand may be known, but it may be difficult or impossible to find an anti derivative. An example of such an integrand is exp(-t^2).

It may be possible to find an anti derivative symbolically, but it may be easier to compute a numerical approximation than to compute the anti derivative. That may be the case if the anti derivative is given as an infinite series or product, or if its evaluation requires a special function which is not available.

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. 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.

It should be noted, however, that a ‘brute force’ kind of numerical integration can always be done, in a very simplistic way, by evaluating the integrand with very small increments.

Quadrature rules based on interpolating functions:

A large class of quadrature rules can be derived by constructing interpolating functions which are easy to integrate. Typically these interpolating functions are polynomials.

The simplest method of this type is to let the interpolating function be a constant function (a polynomial of order zero) which passes through the point ((a+b)/2, f((a+b)/2)). This is called the midpoint rule or rectangle rule.

The interpolating function may be an affine function (a polynomial of degree 1) which passes through the points (a, f(a)) and (b, f(b)). This is called the trapezoidal rule.

For either one of these rules, we can make a more accurate approximation by breaking up the interval [a, b] into some number n of subintervals, computing an approximation for each subinterval, then adding up all the results. This is called a composite rule, extended rule, or iterated rule. For example, the composite trapezoidal rule can be stated as

Interpolation with polynomials evaluated at equally-spaced points in [a, b] yields the Newton-Cotes formulas, of which the rectangle rule and the trapezoidal rule are examples. Simpson’s rule, which is based on a polynomial of order 2, is also a Newton-Cotes formula.

If we allow the intervals between interpolation points to vary, we find another group of quadrature formulas, called Gaussian quadrature formulas. A Gaussian quadrature rule is typically more accurate than a Newton-Cotes rule which requires the same number of function evaluations, if the integrand is smooth (i.e., if it has many derivatives.)

………………

Numerical integration is the approximate computation of an integral using numerical techniques. The numerical computation of an integral is sometimes called quadrature. Ueberhuber (1997, p. 71) uses the word “quadrature” to mean numerical computation of a univariate integral, and “cubature” to mean numerical computation of a multiple integral.

There are a wide range of methods available for numerical integration. A good source for such techniques is Press et al. (1992). Numerical integration is implemented in Mathematica as NIntegrate[f, x, xmin, xmax].

The most straightforward numerical integration technique uses the Newton-Cotes formulas (also called quadrature formulas), which approximate a function tabulated at a sequence of regularly spaced intervals by various degree polynomials. If the endpoints are tabulated, then the 2- and 3-point formulas are called the trapezoidal rule and Simpson’s rule, respectively. The 5-point formula is called Boole’s rule. A generalization of the trapezoidal rule is Romberg integration, which can yield accurate results for many fewer function evaluations.

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

Modern numerical integrations methods based on information theory have been developed to simulate information systems such as computer controlled systems, communication systems, and control systems since in these cases, the classical methods (which are based on approximation theory) are not as efficient (Smith 1974).

Numerical integration aims at approximating definite integrals using numerical techniques. There are many situations where numerical integration is needed. For example, several well defined functions do not have an anti-derivative, i.e. their anti-derivative cannot be expressed in terms of primitive function. A popular example is the function e��x2 whose anti-derivative does not exist. This function arises in a variety of applications such as those related to probability and statistics analyses. Furthermore, many applications in science and engineering are represented by integral differential equations that require a special treatment for the integral terms (e.g. expansion, liberalization, closure …).

Therefore, numerical integration does not only provide a means for evaluating integrals numerically, but also grants us the ability to approximate special functions that are defined in terms of integrals. Without loss of generality, there are two classes of problems where numerical integration is needed. In the first class, one wishes to evaluate the integral of a well defined function. In this case, the integrand can be evaluated a various points because and numerical integration techniques help define the optimum number of these points as well as their locations. The second class of problems for applying numerical integration is found in differential equations the most common of which are those that express conservation principles. For example, the population balance equation, a well known partial