Section 6.3 Richardson s Extrapolation. Extrapolation (To infer or estimate by extending or projecting known information.)

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1 Section 6.3 Richardson s Extrapolation Key Terms: Extrapolation (To infer or estimate by extending or projecting known information.) Illustrated using Finite Differences The difference between Interpolation and Extrapolation: Interpolation is filling in data points between the data that you already have. For example - drawing a line (fitting a curve) from the first data point you have to the last allows you to estimate data points between those two extremes (or between any data points that you have). ie. 'filling in between'. Extrapolation is filling in data points beyond the data that you have (extending the data). For example fitting a curve to the data that you have using an equation, then extending that curve beyond the first and last points enables you to estimate values (or extrapolate them) beyond the measured data. Source: answers.yahoo.com

2 In a Section 6.2 first- and second-order finite difference approximation formulas for first and second derivatives were obtained. Higher order formulas can of course be derived by interpolating more data points, but an alternative for obtaining higher-order approximations is to use a procedure known as extrapolation. The basic idea behind extrapolation is that whenever the leading term in the error for an approximation formula is known, we can combine two approximations obtained from that formula using different values of the parameter h to obtain a higher-order approximation. This process will be illustrated in this section for finite difference approximations to derivatives ; the technique is known as Richardson extrapolation. In a later section, we will apply extrapolation to numerical integration formulas. In numerical analysis, Richardson extrapolation is a sequence acceleration method, used to improve the rate of convergence of a sequence. It is named after Lewis Fry Richardson, who introduced the technique in the early 20th century. It has been said that "... its usefulness for practical computations can hardly be overestimated. Ref:

3 Recall in Section 2.6 we used the Aitken Δ 2 process to accelerate a linearly convergent sequence. We couldn t claim that new sequence was quadratically convergent. The best we could say was new sequence would converge faster to the limit. The Aitken Δ 2 process accelerates convergence by reducing the asymptotic error constant. That is, the value of λ shrinks in the limit of the ratio of errors; en 1 lim e h 0 n We also had the special case of Steffensen s Method which applies the Aitken Δ 2 process to a convergent fixed point method by using three iterations to form a new estimate which is a linear combination of the three successive estimates. The result is that the new sequence was quadratically convergent. This computation basically jumped ahead of terms of the original sequence. It is an extrapolation.

4 Two important ideas: (i) (ii) Error estimation has a large heuristic component. Selecting how to obtain and combine two estimates V 1 and V 2 of an exact value Q is as much an art as a science. It costs significantly more to estimate Q and produce an error estimate than just to estimate Q. Extrapolation can be applied whenever it is known that an approximation technique has an error term with a predictable form, one that depends on a parameter, usually the step size h. The objective of extrapolation is to find an easy way to combine low order approximations in an appropriate way to produce formulas with a higher-order truncation error. Ref: B & F 8 th.

5 Here we consider a particular example, namely the second order centered difference formula for f ꞌ: For notational convenience, let D denote the true value of the derivative, and let D h denote the approximation obtained using a step size of h. That is, From our previous arguments we know that x 0 - h < ξ < x 0 + h; hence, as h 0 we have that ξ x 0. Therefore as h 0 Let's now look at the error term more precisely. Since we see that

6 What does the expression imply? Since that ratio goes to zero as h goes to zero it must be that the numerator goes to zero faster than h 2 goes to zero. Thus we have D = D h + K 1 h 2 + terms that go to zero faster than h 2 goes to zero. (Here K 1 = (1/6) f ꞌꞌꞌ(x 0 ).) Rather than this phrase we use the Small-O Notation o(h 2 ).

7 As stated earlier, the process of extrapolation uses two approximations computed from the same formula, but with different values of h, to obtain a higher order approximation. For the second-order central difference approximation to f ꞌ, we have just established Approximation #1 Now we get a second approximation by using h/2 instead of h, thus we have Next we form the difference of the two approximations to obtain Approximation #2 Recall that the leading error term in approximation #1 is K 1 h 2 so we solve the preceding expression for this term and obtain Replacing the leading term in approximation by this expression and using algebra gives The formula is a higher order approximation for the first derivative in the sense that it converges to zero faster than h 2 converges to zero as h 0.

8 The expression D h or D h/2. is an improved estimate of f ꞌ(x 0 ) than either the estimates Example: Extrapolating the Derivative of the Natural Logarithm Let's consider the approximation of the derivative of the function f(x) = ln x using the second-order central difference approximation formula. With x 0 = 2, we find Applying the extrapolation formula to these results produces: Not only has the approximation error been significantly reduced as a result of extrapolation, the cut in error due to the cut in the step size is also larger. In particular, it appears as if the extrapolated values are fourth-order approximations - having cut h by a factor of 2, the error has dropped by a factor of 16! 1.58E-7 divided by 16 is approximately 1.10E-8

9 We can extrapolate the extrapolated approximations. We call this repeated extrapolation. Note in the development of we assumed that f ꞌꞌꞌ existed and was continuous. For repeated extrapolation we make similar assumptions. In order to keep track of the various approximations that will be generated, let's modify our notation slightly. Let D (1) h denote the original approximation, D (2) h denote the first extrapolation, D (3) h denote the next extrapolation and so on. Furthermore, we will adopt the convention that the step size associated with an extrapolated value is the larger of the two step sizes used to calculate the extrapolated value. For example, for the second-order central difference approximation to f ꞌ, we would write It follows that

10 Using arguments similar to those given in the development of the first extrapolation we can show that or Example : Continuing to extrapolation the Derivative of the Natural Logarithm The first extrapolation gives: Next extrapolation is: The error in this final approximation is

11 Note that the formula for D (3) h has a structure similar to the formula for D (2) h. In particular, the lower order approximation with the larger step size is subtracted from a multiple of the lower order approximation with the smaller step size. This result is then divided by the sum of the coefficients in the numerator. The coefficient of 16 arises from the fact that the step size was cut by a factor of 2 and the lower order approximations were fourth-order; i.e., 16 = 2 4. In general, if a p-th order formula is extrapolated by cutting the step size by a factor of b, the extrapolation formula will take the form For our example we used centered differences where for the first extrapolation p = 2 and b = 2 thus we got For the next extrapolation p = 4 and b = 2 thus we got

12 When performing repeated extrapolations, it is convenient to organize the calculations into an extrapolation table like the one below. Listing the order of approximation associated with each column helps to keep track of the weights needed to compute each successive column. For the centered difference approximations p 1 = 2, p 2 = 4, p 3 = 8