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Numerical integration (quadrature): methods, errors, and applications

Overview of numerical integration (quadrature): common algorithms, historical notes, error behavior, multidimensional extensions and practical guidance for approximating definite integrals.

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Numerical integration, often called quadrature, denotes a collection of techniques for approximating the value of a definite integral when an exact antiderivative is unavailable or impractical to obtain. In many practical problems the integrand is known only through measurements or evaluations at a finite set of points, so analytic integration is impossible; in those situations one replaces the true integral by a computable approximation plus an estimate of the error. For general introductions and examples see reference 1 and reference 2.

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Common methods

Several families of rules are widely used, each with different tradeoffs between accuracy, cost and robustness. Newton–Cotes formulas (including the trapezoidal and Simpson's rules) approximate the integrand by low-degree polynomials on subintervals and then integrate those polynomials exactly; older introductions are documented at reference 3. Gaussian quadrature chooses both evaluation points and weights to make the rule exact for polynomials up to as high a degree as possible for a given number of points; historical and computational notes are available at reference 4.

Other useful approaches include adaptive composite rules, which subdivide the interval until a tolerance is met; Romberg integration, which applies Richardson extrapolation to the trapezoidal rule; and stochastic methods such as Monte Carlo integration, suitable for high-dimensional or noisy problems. Interpolation-based schemes typically use polynomials or splines to represent the integrand between known sample points — a topic described in more detail at reference 5 and reference 6.

Error, convergence and practical considerations

Understanding the error behavior of a quadrature rule is central to choosing and tuning a method. Error estimates often depend on derivatives of the integrand: for example, the error term for simple Newton–Cotes rules involves higher-order derivatives evaluated somewhere in the interval. The Euler–Maclaurin formula connects sums and integrals and explains how endpoint corrections improve accuracy; see reference 7. Gaussian rules achieve high algebraic accuracy for smooth functions but require computation of orthogonal polynomials and their roots.

In practice, integrands may be singular, highly oscillatory, or defined by discrete data; each case favors different strategies. Adaptive subdivision helps with localized irregularities, variable transformations can tame endpoint singularities, and Monte Carlo or quasi-Monte Carlo methods are often preferable in many dimensions. Algorithmic references and implementations can be found at reference 8, reference 9, and reference 10.

Historically, names attached to key formulas reflect their development: Newton and Cotes for polynomial-based composite rules, Gauss for optimal node selection, and Euler and Maclaurin for the asymptotic connection between sums and integrals — historical discussions are available at reference 11 and reference 12. Distinctions worth noting include quadrature (one-dimensional numerical integration) versus cubature (higher-dimensional integration) and deterministic versus stochastic methods. Together these tools make numerical integration an essential component in scientific computing, engineering simulation and data analysis.

Numerical integration seeks the simplest possible approximation for the surface {\displaystyle S=\int _{x_{u}}^{x_{o}}f(x)\,dx}

Quadrature Procedure

Graphical methods

In graphical methods, the graph of the integrand is plotted in a coordinate system with linear axes and the area between the graph and the abscissa is determined.

Counting method

A particularly simple procedure is to plot the graph on graph paper and then determine the number of "square millimetre boxes" (area elements) covered by the area S. In this case, only half of the area elements through which the graph passes are counted. Here, area elements through which the graph passes are only counted by half. The approximation then results with the number of square millimeters Nand the scale parts Δ \Delta xand Δ \Delta yto:

{\displaystyle \int _{a}^{b}f(x)\,dx\approx Nmm^{2}\cdot {\tfrac {\Delta x}{mm}}\cdot {\tfrac {\Delta y}{mm}}}

Measurement

Another graphical method is the measurement of the area by means of a planimeter.

quadrature formula calculation

In general, a quadrature formula consists of a weighted sum of function values

{\displaystyle Q(f)=(x_{o}-x_{u})\sum _{i=0}^{n}w_{i}f(x_{i}).}

The places x_0,\ldots, x_ncalled supporting places and the numbers w_0,\ldots,w_nweights. The weights here depend on the distances of a grid point to the neighboring grid points. There are different approaches how to choose grid points and weights in a way that the quadrature error as small as possible. E(f)

A quadrature formula has the degree of precision (or exactness) nif it integrates all polynomial functions up to the maximum degree nexactly, and is nthe largest possible natural number with this property.

Like the integral, quadrature formulas are linear operators.

interpolation points
interpolation points

Interpolatory quadrature formula

An important class of quadrature formulas arises from the idea of approximating the function f(x)by an interpolation polynomial p_n(x) of degree n and then integrating it. The weights are then obtained as the integrals of the Lagrange polynomials to the given support points. By construction, these quadrature formulas have at least degree of accuracy n. The quadrature formula is therefore

{\displaystyle \int _{a}^{b}f(x)\,dx\approx \int _{a}^{b}p_{n}(x)\,dx=(b-a)\sum _{i=0}^{n}w_{i}f(x_{i})}

with the weights

{\displaystyle w_{i}={\frac {1}{b-a}}\int _{a}^{b}L_{in}(x)\,dx}

and the Lagrange polynomials

{\displaystyle L_{in}(x)={\frac {(x-x_{0})\dotsm (x-x_{i-1})(x-x_{i+1})\cdots (x-x_{n})}{(x_{i}-x_{0})\dotsm (x_{i}-x_{i-1})(x_{i}-x_{i+1})\dotsm (x_{i}-x_{n})}}.}

If the integration limits are support points, one speaks of closed quadrature formulas, otherwise of open ones. If the support points are chosen equidistantly, the Newton-Cotes formulas result among others. Among the closed Newton-Cotes formulas are the chordal trapezoid rule and Simpson's rule, among the open ones is the tangent trapezoid rule. The Newton-Cotes formulas for even neven have the degree of precision n+1. The open quadrature formulas also include the Gauss quadrature formulas.

Error estimation

Let [c,d] denote the smallest interval [a,b]containing the interpolation points x_{i} and the interval Furthermore, let f(n+1)-times continuously differentiable on [c,d]. According to the interpolation quality of the interpolation polynomial, there exists a ξ \xi(x) \in [c,d] , such that holds:

f(x) - p_n(x) = \frac{f^{(n+1)}(\xi(x))}{(n+1)!} \prod_{i=0}^n (x-x_i).

By integration, the error formula for numerical quadrature is obtained

{\displaystyle E(f)=\int _{a}^{b}f(x)\,dx-\int _{a}^{b}p_{n}(x)\,dx={\frac {1}{(n+1)!}}\int _{a}^{b}f^{(n+1)}(\xi (x))\prod _{i=0}^{n}(x-x_{i})\,dx}.

If f^{(n+1)}(x)=0 for all x \in [c,d] , the quadrature error is 0. Since this is the case for all polynomials up to degree n , the degree of accuracy of these quadrature formulas is at least n.

From this error formula follows the error estimate

{\displaystyle |E(f)|\leq {\frac {1}{(n+1)!}}\ \max _{c\leq x\leq d}{\left|f^{(n+1)}(x)\right|}\int _{a}^{b}\left|\prod _{i=0}^{n}(x-x_{i})\right|\,dx}.

If the function \prod_{i=0}^n (x-x_i)in the interval not change [a,b]its sign, i.e., if no support point (a,b)lies in the interval , one can derive the following representation for the remainder element using the mean value theorem of integral calculus:

{\displaystyle E(f)={\frac {f^{(n+1)}(\zeta )}{(n+1)!}}\int _{a}^{b}\prod _{i=0}^{n}(x-x_{i})\,dx}.

With an intermediate {\displaystyle \zeta \in [c,d]}.

Similar formulas for the quadrature error are also obtained for special distributions of the grid points in the interval [a,b]for example for the Newton-Cotes formulas or the Gauss quadrature formulas.

If the function is fonly continuous, the above statements do not hold, the error can become very large.

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