Richardson extrapolation

In numerical analysis, Richardson extrapolation is a sequence acceleration method, used to improve the rate of convergence of a sequence of estimates of some value . In essence, given the value of for several values of , we can estimate by extrapolating the estimates to . It is named after Lewis Fry Richardson, who introduced the technique in the early 20th century.[1][2] In the words of Birkhoff and Rota, "its usefulness for practical computations can hardly be overestimated."[3]

Practical applications of Richardson extrapolation include Romberg integration, which applies Richardson extrapolation to the trapezoid rule, and the Bulirsch–Stoer algorithm for solving ordinary differential equations.

Example of Richardson extrapolationEdit

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 function

  where   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).

General formulaEdit

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

 

where the ai are unknown constants and the ki are known constants such that hki > hki+1.

k0 is the leading order step size behavior of Truncation error as  

The exact value sought can be given by

 

which can be simplified with Big O notation to be

 

Using the step sizes h and   for some constant t, the two formulas for A are:

 
 

Multiplying the second equation by tk0 and subtracting the first equation gives

 

which can be solved for   to give

 

Therefore, using   the truncation error has been reduced to  . This is in contrast to   where the truncation error is   for the same step size  

By this process, we have achieved a better approximation of A by subtracting the largest term in the error which was O(hk0). This process can be repeated to remove more error terms to get even better approximations.

A general recurrence relation beginning with   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 k0 (leading order step size behavior of Truncation error) when neither its value nor A* (exact value) is known a priori. Such a technique can be useful for quantifying an unknown rate of convergence. Given approximations of A from three distinct step sizes h, h / t, and h / s, 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 O terms is approximately valid)

 

which can be solved numerically to estimate k0 for some arbitrary choices of h, s and t.

Example pseudocode code for Richardson extrapolationEdit

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 Trapezoidal(f, tStart, tEnd, h, y0) exists which attempts to compute y(tEnd) by performing the trapezoidal method on the function f, with starting point y0 and tStart and step size h.

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.

tStart = 0          % Starting time
tEnd = 5            % Ending time
f = -y^2            % The derivative of y, so y' = f(t, y(t)) = -y^2
                    % The solution to this ODE is y = 1/(1 + t)
y0 = 1              % The initial position (i.e. y0 = y(tStart) = y(0) = 1)
tolerance = 10^-11  % 10 digit accuracy is desired

maxRows = 20                % Don't allow the iteration to continue indefinitely
initialH = tStart - tEnd    % Pick an initial step size
haveWeFoundSolution = false % Were we able to find the solution to within the desired tolerance? not yet.

h = initialH

% Create a 2D matrix of size maxRows by maxRows to hold the Richardson extrapolates
% Note that this will be a lower triangular matrix and that at most two rows are actually
% needed at any time in the computation.
A = zeroMatrix(maxRows, maxRows)

%Compute the top left element of the matrix
A(1, 1) = Trapezoidal(f, tStart, tEnd, h, y0)

% Each row of the matrix requires one call to Trapezoidal
% This loops starts by filling the second row of the matrix, since the first row was computed above
for i = 1 : maxRows - 1 % Starting at i = 1, iterate at most maxRows - 1 times
    h = h/2             % Half the previous value of h since this is the start of a new row
    
    % Call the Trapezoidal function with this new smaller step size
    A(i + 1, 1) = Trapezoidal(f, tStart, tEnd, h, y0)
    
    for j = 1 : i         % Go across the row until the diagonal is reached
        % Use the value just computed (i.e. A(i + 1, j)) and the element from the
        % row above it (i.e. A(i, j)) to compute the next Richardson extrapolate
     
        A(i + 1, j + 1) = ((4^j).*A(i + 1, j) - A(i, j))/(4^j - 1);
    end
    
    % After leaving the above inner loop, the diagonal element of row i + 1 has been computed
    % This diagonal element is the latest Richardson extrapolate to be computed.
    % The difference between this extrapolate and the last extrapolate of row i is a good
    % indication of the error.
    if (absoluteValue(A(i + 1, i + 1) - A(i, i)) < tolerance)  % If the result is within tolerance
        print("y(5) = ", A(i + 1, i + 1))                      % Display the result of the Richardson extrapolation
        haveWeFoundSolution = true
        break                                                  % Done, so leave the loop
    end
end

if (haveWeFoundSolution == false)   % If we weren't able to find a solution to within the desired tolerance
    print("Warning: Not able to find solution to within the desired tolerance of ", tolerance);
    print("The last computed extrapolate was ", A(maxRows, maxRows))
end

See alsoEdit

ReferencesEdit

  1. ^ Richardson, L. F. (1911). "The approximate arithmetical solution by finite differences of physical problems including differential equations, with an application to the stresses in a masonry dam". Philosophical Transactions of the Royal Society A. 210 (459–470): 307–357. doi:10.1098/rsta.1911.0009.
  2. ^ Richardson, L. F.; Gaunt, J. A. (1927). "The deferred approach to the limit". Philosophical Transactions of the Royal Society A. 226 (636–646): 299–349. doi:10.1098/rsta.1927.0008.
  3. ^ Page 126 of Birkhoff, Garrett; Gian-Carlo Rota (1978). Ordinary differential equations (3rd ed.). John Wiley and sons. ISBN 0-471-07411-X. OCLC 4379402.
  • Extrapolation Methods. Theory and Practice by C. Brezinski and M. Redivo Zaglia, North-Holland, 1991.
  • Ivan Dimov, Zahari Zlatev, Istvan Farago, Agnes Havasi: Richardson Extrapolation: Practical Aspects and Applications, Walter de Gruyter GmbH & Co KG, ISBN 9783110533002 (2017).

External linksEdit