Numerical integration

Numerical integration is a way to find the value of a definite integral using special methods and steps. It helps us calculate things that are hard or impossible to solve exactly, like finding the area under a curve. This is very useful in many areas of science and math.

The main idea is to break a complicated problem into smaller, easier pieces that we can add up. There are many different algorithms, or step-by-step plans, to do this. These methods can be used for problems with one or more dimensions, and sometimes they are called quadrature or cubature.

Numerical integration comes from an old problem of finding a square with the same area as another shape. Today, it helps solve many important problems in engineering, physics, and computer science by giving answers that are close enough for practical use. For more information about related topics, see Numerical methods for ordinary differential equations and Numerical methods for partial differential equations.

Motivation and need

There are several reasons why we use numerical integration instead of finding the exact answer by hand. Sometimes, we only know the value of a function at certain points, like when we measure it in real life or in special computer systems. In other cases, we might have a formula, but it’s very hard or impossible to find the exact answer using normal math rules. For example, the function f (x) = exp(−x²) doesn’t have a simple answer we can write down. Finally, even when we can find the exact answer, it might be easier to use a computer to estimate the answer instead.

History

Main article: Quadrature (geometry)

The idea of numerical integration began around 1915 when the term first appeared in a book by David Gibb. Long before that, people have tried to find the area of different shapes, a process called "quadrature." Ancient mathematicians, like those in Greece, thought of finding area as turning a shape into a square of the same size. They could only use a compass and straightedge to do this.

Over time, many smart people worked on finding areas of different shapes. For example, Archimedes found the area of a sphere’s surface and parts of a parabola. Later, during the Middle Ages in Europe, new methods helped find areas of curves like cycloids and hyperbolas. These efforts helped create the idea of integral calculus, which gave us better ways to calculate areas today.

Methods for one-dimensional integrals

A quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration.

Numerical integration methods combine evaluations of the integrand to get an approximation to the integral. The integrand is evaluated at a finite set of points called integration points and a weighted sum of these values is used to approximate the integral. The integration points and weights depend on the specific method used and the accuracy required from the approximation.

One simple method approximates the function by a step function, passing through the midpoint of the interval. This is called the midpoint rule or rectangle rule. Another method uses a straight line passing through the endpoints of the interval, called the trapezoidal rule. Both methods can be made more accurate by breaking the interval into smaller pieces and adding up the results, known as composite rules.

| ∫ a b f ( x ) d x − ( b − a ) f ( a ) | ≤ ( b − a ) 2 2 sup a ≤ x ≤ b | f ′ ( x ) | , {\displaystyle \left|\int _{a}^{b}f(x)\,dx-(b-a)f(a)\right|\leq {(b-a)^{2} \over 2}\sup _{a\leq x\leq b}\left|f'(x)\right|,}

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Multidimensional integrals

When we want to calculate integrals in more than one dimension, one way is to break them down into a series of one-dimensional integrals. However, this can become very difficult as the number of dimensions grows. There are several methods to handle this challenge.

One common method is called Monte Carlo integration, which uses random sampling to estimate the integral. This can often be more accurate than breaking the problem into one-dimensional pieces. Another approach is using sparse grids, which combine one-dimensional results in a clever way. Bayesian quadrature is a statistical method that helps manage the uncertainty in the calculation.

Main article: Monte Carlo integration

Connection with differential equations

The task of finding the value of a definite integral can be linked to solving a special type of equation called an ordinary differential equation. By using a key idea from calculus, we can turn the integral problem into a problem where we need to find a function whose rate of change matches another function.

There are special methods for solving these kinds of equations, and they can help us calculate the integral. One common method, known as the Runge–Kutta method, can be used to find the solution. Sometimes, solving equations using integrals is called "solving by quadrature."