# Accuracy of Richardson's error estimate in the presence of rounding errors

Richardson extrapolation is a well known technique in scientific computing. It is used to compute error estimates when our approximations can be viewed as a function of a parameter $$h$$. Common applications are numerical differentiation, integration, and the solution of ordinary as well as partial differential equations. Typically, we seek to compute a target value $$T$$ and we have the ability to compute approximations $$A = A_h$$ for many or all $$h>0$$. Now, if there exists an asymptotic error expansion of the form $$T - A_h = \alpha h^p + \beta h^q + O(h^r), \quad h \rightarrow 0, \quad h > 0,$$ where $$0 < p < q < r, \quad \alpha \not = 0, \quad \beta \not = 0,$$ then Richardson's error estimate is given by $$E_h^{est} = \frac{A_h - A_{2h}}{2^p-1}$$ and $$T - A_h = E_h^{est} + O(h^q), \quad h \rightarrow 0, \quad h >0.$$ This is true in exact arithmetic only. In practice, Richardson's error estimate suffers from subtractive cancellation when $$h$$ is sufficiently small and $$A_h \approx A_{2h}$$.

Is there any published literature on the subject of estimating the accuracy of Richardson's error estimate in presence of rounding errors?

Restrictions:

1. The typical paper will compare Richardson's error estimate to the actual error for problems where the exact target value is known. I am not interested in such papers. It is the ability to accurately estimate the accuracy of Richardson's error estimate in the general case which is of interest to me.
2. In the general case, we do not have a formula for the coefficients, i.e., $$\alpha$$ and $$\beta$$, which define the asymptotic error expansion.
3. I am not interested in using the method $$M_h$$ given by $$M_h = A_h + E_h^{est}.$$ It is known that this method satisfies an asymptotic error expansion of the form $$T-M_h = \alpha' h^q + O(h^r), \quad h \rightarrow 0, \quad h>0.$$ Therefore, the error associate with $$M_h$$ can be estimated as $$T - M_h \approx \frac{M_h - M_{2h}}{2^q-1},$$ amd this would, in principle, allow us to gauge the accuracy of the original error estimate. However, this derived method is even more vulnerable to subtractive cancellation than the parent $$A_h$$.

The very practical problem of computing the values of $$p$$ and $$q$$ is not an issue. It is straightforward to verify that Richardson's fraction $$F_h$$ given by $$F_h = \frac{A_{2h} - A_{4h}}{A_h - A_{2h}}$$ satisfies $$F_h - 2^p = \Theta(h^{q-p}), \quad h \rightarrow 0, \quad h > 0.$$ Therefore, the values of $$p$$ and $$q$$ can (frequently, but not universally) be obtained by observing the computed values of Richardson's fraction for several different values of $$h$$.

• Not exactly focused on Richardson extrapolation, but a general high level theoretical analysis of rounding errors as well as other kind of error involved in the analysis of performances of numerical algorithms is the monograph by Solomon Mikhlin Error Analysis in Numerical Processes, published by Wiley in 1991. Feb 8, 2020 at 11:36
• Mikhlin subdivides the general error involved in numerical processes in four classes, jointly present (loc. cit. §1.4 pp. 21-22): 1. Approximation errors 2. Perturbation errors 3. Algorithm errors 4. Rounding errors and analyzes throughly each of them. He also reviews the literature on such kind of errors: in particular, at page 23 there's a brief survey of literature pertaining the problem of rounding errors. Feb 8, 2020 at 11:43
• @DanieleTampieri. Thank you for the reference to Mikhlin's excellent textbook. Unfortunately, I need something much more specific. The accuracy of Richardson's error estimate hinges on the relative size of the different terms on the asymptotics error expansion as well as the rounding errors. Standard analysis of the rounding error produces bounds which tend to pessimistic. This is not sufficient here. Feb 8, 2020 at 13:39