Quadrature rules based on step functions

A "brute force" kind of numerical integration can be done, if the integrand is reasonably well-behaved (i.e. piecewise continuous and of bounded variation), by evaluating the integrand with very small increments.

Illustration of the rectangle rule.

This simplest method approximates the function by a step function (a piecewise constant function, or a segmented polynomial of degree zero) that passes through the point ${\textstyle \left({\frac {a+b}{2}},f\left({\frac {a+b}{2}}\right)\right)}$. This is called the midpoint rule or rectangle rule

$${\displaystyle \int _{a}^{b}f(x)\,dx\approx (b-a)f\left({\frac {a+b}{2}}\right).}$$

Quadrature rules based on interpolating functions

A large class of quadrature rules can be derived by constructing interpolating functions that are easy to integrate. Typically these interpolating functions are polynomials. In practice, since polynomials of very high degree tend to oscillate wildly, only polynomials of low degree are used, typically linear and quadratic.

Illustration of the trapezoidal rule.

The interpolating function may be a straight line (an affine function, i.e. a polynomial of degree 1) passing through the points ${\displaystyle \left(a,f(a)\right)}$ and ${\displaystyle \left(b,f(b)\right)}$. This is called the trapezoidal rule

$${\displaystyle \int _{a}^{b}f(x)\,dx\approx (b-a)\left({\frac {f(a)+f(b)}{2}}\right).}$$

Illustration of Simpson's rule.

For either one of these rules, we can make a more accurate approximation by breaking up the interval ${\displaystyle [a,b]}$ into some number ${\displaystyle n}$ 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

$${\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {b-a}{n}}\left({f(a) \over 2}+\sum _{k=1}^{n-1}\left(f\left(a+k{\frac {b-a}{n}}\right)\right)+{f(b) \over 2}\right),}$$

where the subintervals have the form ${\displaystyle [a+kh,a+(k+1)h]\subset [a,b],}$ with ${\textstyle h={\frac {b-a}{n}}}$ and ${\displaystyle k=0,\ldots ,n-1.}$ Here we used subintervals of the same length ${\displaystyle h}$ but one could also use intervals of varying length ${\displaystyle \left(h_{k}\right)_{k}}$.

Interpolation with polynomials evaluated at equally spaced points in ${\displaystyle [a,b]}$ 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.

Quadrature rules with equally spaced points have the very convenient property of nesting. The corresponding rule with each interval subdivided includes all the current points, so those integrand values can be re-used.

If we allow the intervals between interpolation points to vary, we find another group of quadrature formulas, such as the Gaussian quadrature formulas. A Gaussian quadrature rule is typically more accurate than a Newton–Cotes rule that uses the same number of function evaluations, if the integrand is smooth (i.e., if it is sufficiently differentiable). Other quadrature methods with varying intervals include Clenshaw–Curtis quadrature (also called Fejér quadrature) methods, which do nest.

Gaussian quadrature rules do not nest, but the related Gauss–Kronrod quadrature formulas do.

Adaptive algorithms

This section is an excerpt from Adaptive quadrature.[edit]

Adaptive quadrature is a numerical integration method in which the integral of a function ${\displaystyle f(x)}$ is approximated using static quadrature rules on adaptively refined subintervals of the region of integration. Generally, adaptive algorithms are just as efficient and effective as traditional algorithms for "well behaved" integrands, but are also effective for "badly behaved" integrands for which traditional algorithms may fail.

Extrapolation methods

The accuracy of a quadrature rule of the Newton–Cotes type is generally a function of the number of evaluation points. The result is usually more accurate as the number of evaluation points increases, or, equivalently, as the width of the step size between the points decreases. It is natural to ask what the result would be if the step size were allowed to approach zero. This can be answered by extrapolating the result from two or more nonzero step sizes, using series acceleration methods such as Richardson extrapolation. The extrapolation function may be a polynomial or rational function. Extrapolation methods are described in more detail by Stoer and Bulirsch (Section 3.4) and are implemented in many of the routines in the QUADPACK library.

Conservative (a priori) error estimation

Let ${\displaystyle f}$ have a bounded first derivative over ${\displaystyle [a,b],}$ i.e. ${\displaystyle f\in C^{1}([a,b]).}$ The mean value theorem for ${\displaystyle f,}$ where ${\displaystyle x\in [a,b),}$ gives

$${\displaystyle (x-a)f'(\xi _{x})=f(x)-f(a),}$$

for some ${\displaystyle \xi _{x}\in (a,x]}$ depending on ${\displaystyle x}$.

If we integrate in ${\displaystyle x}$ from ${\displaystyle a}$ to ${\displaystyle b}$ on both sides and take the absolute values, we obtain

$${\displaystyle \left|\int _{a}^{b}f(x)\,dx-(b-a)f(a)\right|=\left|\int _{a}^{b}(x-a)f'(\xi _{x})\,dx\right|.}$$

We can further approximate the integral on the right-hand side by bringing the absolute value into the integrand, and replacing the term in ${\displaystyle f'}$ by an upper bound

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

(1)

where the supremum was used to approximate.

Hence, if we approximate the integral ${\textstyle \int _{a}^{b}f(x)\,dx}$ by the quadrature rule ${\displaystyle (b-a)f(a)}$ our error is no greater than the right hand side of 1. We can convert this into an error analysis for the Riemann sum, giving an upper bound of

$${\displaystyle {\frac {n^{-1}}{2}}\sup _{0\leq x\leq 1}\left|f'(x)\right|}$$

for the error term of that particular approximation. (Note that this is precisely the error we calculated for the example ${\displaystyle f(x)=x}$.) Using more derivatives, and by tweaking the quadrature, we can do a similar error analysis using a Taylor series (using a partial sum with remainder term) for f. This error analysis gives a strict upper bound on the error, if the derivatives of f are available.

This integration method can be combined with interval arithmetic to produce computer proofs and verified calculations.

Integrals over infinite intervals

Several methods exist for approximate integration over unbounded intervals. The standard technique involves specially derived quadrature rules, such as Gauss-Hermite quadrature for integrals on the whole real line and Gauss-Laguerre quadrature for integrals on the positive reals.^[4] Monte Carlo methods can also be used, or a change of variables to a finite interval; e.g., for the whole line one could use

$${\displaystyle \int _{-\infty }^{\infty }f(x)\,dx=\int _{-1}^{+1}f\left({\frac {t}{1-t^{2}}}\right){\frac {1+t^{2}}{\left(1-t^{2}\right)^{2}}}\,dt,}$$

and for semi-infinite intervals one could use

$${\displaystyle {\begin{aligned}\int _{a}^{\infty }f(x)\,dx&=\int _{0}^{1}f\left(a+{\frac {t}{1-t}}\right){\frac {dt}{(1-t)^{2}}},\\\int _{-\infty }^{a}f(x)\,dx&=\int _{0}^{1}f\left(a-{\frac {1-t}{t}}\right){\frac {dt}{t^{2}}},\end{aligned}}}$$

as possible transformations.

Monte Carlo

Monte Carlo methods and quasi-Monte Carlo methods are easy to apply to multi-dimensional integrals. They may yield greater accuracy for the same number of function evaluations than repeated integrations using one-dimensional methods.

A large class of useful Monte Carlo methods are the so-called Markov chain Monte Carlo algorithms, which include the Metropolis–Hastings algorithm and Gibbs sampling.

Sparse grids

Sparse grids were originally developed by Smolyak for the quadrature of high-dimensional functions. The method is always based on a one-dimensional quadrature rule, but performs a more sophisticated combination of univariate results. However, whereas the tensor product rule guarantees that the weights of all of the cubature points will be positive if the weights of the quadrature points were positive, Smolyak's rule does not guarantee that the weights will all be positive.

Bayesian Quadrature

Bayesian quadrature is a statistical approach to the numerical problem of computing integrals and falls under the field of probabilistic numerics. It can provide a full handling of the uncertainty over the solution of the integral expressed as a Gaussian process posterior variance.