In numerical analysis, Richardson extrapolation is a method used to estimate some value if the truncation error is known to have a (possibly 1-sided) power-series expansion at . The method is most often applied as a sequence acceleration method to improve the rate of convergence of iterative methods. It is named after Lewis Fry Richardson, who introduced the technique in the early 20th century, though a form of the idea was already known to Christiaan Huygens in the 17th century and used his calculation of . Practical applications of Richardson extrapolation include Romberg integration, which applies Richardson extrapolation to the trapezoid rule, and the BulirschâÂÂStoer algorithm for solving ordinary differential equations. In the words of Birkhoff and Rota, "its usefulness for practical computations can hardly be overestimated."
Let be an approximation of '(exact value) that depends on a step size (where ) with an error formula of the form
where the are unknown constants and the are known constants such that . Furthermore, represents the truncation error of the approximation such that Note that by simplifying with Big O notation, the following asymptotic formulae are implied:
Richardson extrapolation is a process that finds a better approximation of by changing the error formula from to Therefore, by replacing with the truncation error has reduced from to for the same step size . The general pattern occurs in which is a more accurate estimate than when . By this process, we have achieved a better approximation of by subtracting the largest term in the error which was . This process can be repeated to remove more error terms to get even better approximations.
Using the step sizes and for some constant , the two formulas for are:
To improve our approximation from to by removing the first error term, we multiply by and subtract to give us
This multiplication and subtraction was performed because is an approximation of . We can solve our current formula for to give
which can be written as by setting
A general recurrence relation can be defined for the approximations by
where satisfies
The Richardson extrapolation can be considered as a linear sequence transformation.
Additionally, the general formula can be used to estimate (leading order step size behavior of Truncation error) when neither its value nor is known a priori. Such a technique can be useful for quantifying an unknown rate of convergence. Given approximations of from three distinct step sizes , , and , the exact relationship yields an approximate relationship (please note that the notation here may cause a bit of confusion, the two O appearing in the equation above only indicates the leading order step size behavior but their explicit forms are different and hence cancelling out of the two terms is only approximately valid)
which can be solved numerically to estimate for some arbitrary valid choices of , , and .
As , if and is chosen so that , this approximate relation reduces to a quadratic equation in , which is readily solved for in terms of and .
Suppose that we wish to approximate , and we have a method that depends on a small parameter in such a way that
Let us define a new functionwhere and are two distinct step sizes.
Then
is called the Richardson extrapolation of A(h), and has a higher-order error estimate compared to .
Very often, it is much easier to obtain a given precision by using R(h) rather than A(h′) with a much smaller h′. Where A(h′) can cause problems due to limited precision (rounding errors) and/or due to the increasing number of calculations needed (see examples below).
The following pseudocode in MATLAB style demonstrates Richardson extrapolation to help solve the ODE , with the Trapezoidal method. In this example we halve the step size each iteration and so in the discussion above we'd have that . The error of the Trapezoidal method can be expressed in terms of odd powers so that the error over multiple steps can be expressed in even powers; this leads us to raise to the second power and to take powers of in the pseudocode. We want to find the value of , which has the exact solution of since the exact solution of the ODE is . This pseudocode assumes that a function called <code>Trapezoidal(f, tStart, tEnd, h, y0)</code> exists which attempts to compute <code>y(tEnd)</code> by performing the trapezoidal method on the function <code>f</code>, with starting point <code>y0</code> and <code>tStart</code> and step size <code>h</code>.
Note that starting with too small an initial step size can potentially introduce error into the final solution. Although there are methods designed to help pick the best initial step size, one option is to start with a large step size and then to allow the Richardson extrapolation to reduce the step size each iteration until the error reaches the desired tolerance.