Richardson extrapolation

In numerical analysis, Richardson extrapolation is a method used to improve the rate of convergence of a limit typically depending on a parameter going to zero. It is named after Lewis Fry Richardson, who introduced the technique in the early 20th century. In the words of Birkhoff and Rota, "... its usefulness for practical computations can hardly be overestimated."

Example of Richardson extrapolation

Suppose that A(h) is an estimation of order n for $A_{0}=\lim _{h\to 0}A(h)$, i.e. $A_{0}-A(h)\sim h^{n}\,$ as $h\to 0$. Then

$R(h)=A(h/2)+{\frac {A(h/2)-A(h)}{2^{n}-1}}={\frac {2^{n}\,A(h/2)-A(h)}{2^{n}-1}}$

is called the Richardson extrapolate of A(h).

If $A(h)-A_{0}=a_{n}h^{n}+O(h^{m}),~a_{n}\neq 0,~m>n$, then R(h) is an estimate of order greater than or equal to m for A.

More generally, the factor 2 can be replaced by any other factor, as shown below.

Very often, it is much easier to obtain a given precision by using R(h) rather than A(h') with a much smaller h' , which can cause problems due to limited precision (rounding errors) and/or due to the increasing number of calculations needed (see examples below).

General formula

Let A(h) be an approximation of A that depends on a positive step size h with an error formula of the form

$A_{0}-A(h)=a_{1}h^{k_{1}}+a_{2}h^{k_{2}}+\cdots$

where the a_i are nonzero constants and the k_i are positive constants such that k_i < k_i+1.

Using big O notation, one has

$A_{0}=A(h)+a_{1}h^{k_{1}}+O(h^{k_{2}}).\,$

For a different step size h / t with some t>1 (to fix ideas), one gets a second formula for A₀,

$A_{0}=A\!\left({\tfrac {h}{t}}\right)+a_{1}\left({\tfrac {h}{t}}\right)^{k_{1}}+O(h^{k_{2}}).$

Multiplying the second equation by t^k₁ and subtracting the first equation gives

$(t^{k_{1}}-1)A_{0}=t^{k_{1}}A\left({\tfrac {h}{t}}\right)-A(h)+O(h^{k_{1}})$

which can be solved for A to give

$A_{0}={\frac {t^{k_{1}}A\left({\frac {h}{t}}\right)-A(h)}{t^{k_{1}}-1}}+O(h^{k_{2}}).$

By this process, we have achieved a better approximation of A₀ by subtracting the largest term in the error which was O(h^k₁). This process can be repeated to remove more error terms to get even better approximations. It should be noted, however, that the above expression is not necessarily an estimation of order k₂, since several error terms could have disappeared at once. Also, a new error term of order larger than k₂, could have been introduced. Thus, in order to re-iterate the method, the true order of the new estimate needs to be determined first.

A well-known practical use of Richardson extrapolation is Romberg integration, which applies Richardson extrapolation to the trapezium rule.

It should be noted that the Richardson extrapolation can be considered as a linear sequence transformation.

Example

Using Taylor's theorem,

$f(x+h)=f(x)+f'(x)h+{\frac {f''(x)}{2}}h^{2}+\cdots$

the derivative of f(x) is given by

$f'(x)={\frac {f(x+h)-f(x)}{h}}-{\frac {f''(x)}{2}}h+\cdots .$

If the initial approximations of the derivative are chosen to be

$A_{1}(h)={\frac {f(x+h)-f(x)}{h}}$

then k₁ = 1, unless f"(x) = 0.

For t = 2, the first formula extrapolated for A would be

$A=2A_{1}\!\left({\tfrac {h}{2}}\right)-A_{1}(h)+O(h^{2}).$

For the new approximation

$A_{2}(h)=2A_{1}\!\left({\tfrac {h}{2}}\right)-A_{1}(h)={\frac {4f(x+h/2)-f(x+h)-3f(x)}{h}}$

one could extrapolate again to obtain

$A_{0}={\frac {4A_{1}\!\left({\frac {h}{2}}\right)-A_{1}(h)}{3}}+O(h^{3}).$

References

• Extrapolation Methods. Theory and Practice by C. Brezinski and M. Redivo Zaglia, North-Holland, 1991.

External links

• Module for Richardson's Extrapolation
• Richardson Extrapolation Source-Code (C++) exemplified in Romberg integration, and documentation

See also

• Aitken's delta-squared process