Topic 12.3: Richardson Extrapolation (Examples)

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Example 1

Approximate the derivative of the function f(x) = e-x sin(x) at the point x = 1.0 using the centred divided-difference formula and Richardson extrapolation starting with h = 0.5 and continuing using εstep = 0.0001.

We calculate the derivative using h = 0.5 and h = 0.25:

-0.06821507210
-0.1001894112

Neither approximation is very close to the correct answer of -0.1107937653, however, one step of Richardson extrapolation yields:

-0.06821507210
-0.1001894112 -0.1108475242

Next, we approximate the derivative using h = 0.125, that is, -0.1081453352, and we fill in the third row of the table:

-0.06821507210
-0.1001894112 -0.1108475242
-0.1081453352 -0.1107973099 -0.1107939623

The difference |-0.1108475242 − (-0.1107939623)| = 0.000053 < εstep, and thus -0.1107939623 is our approximation.

The relative error of our approximation is 1.8 × 10-5.

Example 2

Repeat Exercise 1, but use the backward divided-difference formula.

Again, using h = 0.5 and h = 0.25 and applying one step of Richardson extrapolation, we get:

-0.2344655259
-0.1369349794 -0.1044247972

We cannot continue because the error for the backward-difference formula is 1/4 f(4)(ξ)h3, and therefore we cannot resort to using more than one step of Richardson extrapolation. However, we can still generate a table using only one step:

-0.2344655259
-0.1369349794 -0.1044247972
-0.1167038388 -0.1099601253
-0.1121938272 -0.1106904900
-0.1111341968 -0.1107809867

The difference between the last two, |R4,1 − R3,1| = 0.000090 < εstep, so we are finished and we use -0.1107809867 as our approximation.

The relative error of our last answer is 1.2 × 10-4.

Exercise 3

Use Richardson extrapolation to help approximate the 2nd derivative of the function given in Exercise 1, using the same values.

Using the formula for the second derivative, we calculate:

-0.4230489884 -0.4039640736 -0.3976024353 -0.3991434368 -0.3975365579 -0.3975321661

The first column is (f(x+h) - 2 f(x) + f(x-h))/h2 for h = 0.5, 0.25, and 0.125. The next two columns are Richard extrapolations and the difference between the last two values |R2,2 − R1,1| = 0.000070 < εstep.

Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.