NUMERICAL MATHEMATICS & COMPUTING, 7 Ed. 4.3 Estimating Derivatives and Richardson Extrapolation

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1 NUMERICAL MATHEMATICS & COMPUTING, 7 Ed. 4.3 Estimating Derivatives and Richardson Extrapolation Ward Cheney/David Kincaid c UT Austin Engage Learning: Thomson-Brooks/Cole November 7, 2011 Derivatives and / 14Ric

2 4.3 Estimating Derivatives and Richardson Extrapolation Determining the derivative of a function f at a point x is not a trivial numerical problem. Specifically, if f (x) can be computed with only n digits of precision, it is difficult to calculate f (x) numerically with n digits of precision. This difficulty may be traced to the subtraction between quantities that are nearly equal. Here several alternatives are offered for the numerical computation of the first and second derivatives: f (x) and f (x) Derivatives and / 14Ric

3 First-Derivative Formulas via Taylor Series First, we consider the obvious method based on the definition of f (x). It consists of selecting one or more small values of h and writing What error is involved in this formula? f (x) 1 [f (x + h) f (x)] (1) h Derivatives and / 14Ric

4 To find out, use Taylor s Theorem f (x + h) = f (x) + hf (x) h2 f (ξ) Rearranging this equation gives f (x) = 1 h [f (x + h) f (x)] 1 2 hf (ξ) (2) Derivatives and / 14Ric

5 Hence, we see that approximation (1) has error term 1 2 hf (ξ) = O(h) where ξ is in the interval having endpoints x and x + h. Equation (2) shows that in general, as h 0, the difference between the derivative f (x) and the estimate [f (x + h) f (x)]/h approaches zero at the same rate that h does that is, O(h). Derivatives and / 14Ric

6 Equation (2) gives the truncation error for this numerical procedure, namely, 1 2 hf (ξ) Additional (and worse) errors must be expected when calculations are performed on a computer with finite word length. Derivatives and / 14Ric

7 Example 1 Example The program named First used the one-sided rule (1) to approximate the first derivative of the function f (x) = sin x at x = 0.5. Explain what happens when a large number of iterations are performed, say n = 50. There was a total loss of all significant digits! When we examine the computer output closely, we find that, in fact, a good approximation f (0.5) was found, but it deteriorated as the process continued. Derivatives and / 14Ric

8 This was caused by the subtraction of two nearly equal quantities f (x + h) f (x) resulting in a loss of significant digits as well as a magnification of this effect from dividing by a small value of h. We need to stop the iterations sooner! Derivatives and / 14Ric

9 Example (cont.) When to stop an iterative process is a common question in numerical algorithms. In this case, one can monitor the iterations to determine when they settle down, namely, when two successive ones are within a prescribed tolerance. Derivatives and / 14Ric

10 Alternatively, we can use the truncation error term. If we want six significant digits of accuracy in the results, we set since f (x) < 1 and h = 1/4 n. We find 1 2 hf (ξ) n < n > 6/ log So we should stop after about ten steps in the process. The least error of was found at iteration 14. Ward Cheney/David Kincaid c (UT Austin[10pt] NUMERICAL Engage Learning: MATHEMATICS Thomson-Brooks/Cole & COMPUTING, 7 Ed. 4.3 November Estimating 7, Derivatives 10 and / 14Ric

11 With the Romberg method it is advantageous to have the convergence of a numerical processes occur with higher powers of some quantity approaching zero. In the present situation, we want an approximation to f (x) in which the error behaves like O(h 2 ) Ward Cheney/David Kincaid c (UT Austin[10pt] NUMERICAL Engage Learning: MATHEMATICS Thomson-Brooks/Cole & COMPUTING, 7 Ed. 4.3 November Estimating 7, Derivatives 11 and / 14Ric

12 One such method is easily obtained with the aid of the following two Taylor series: f (x + h) = f (x) + hf (x) + 1 2! h2 f (x) + 1 3! h3 f (x) + 1 4! h4 f (4) (x) + f (x h) = f (x) hf (x) + 1 2! h2 f (x) 1 3! h3 f (x) + 1 4! h4 f (4) (x) By subtraction, we obtain f (x + h) f (x h) = 2hf (x) + 2 3! h3 f (x) + 2 5! h5 f (5) (x) + (3) Ward Cheney/David Kincaid c (UT Austin[10pt] NUMERICAL Engage Learning: MATHEMATICS Thomson-Brooks/Cole & COMPUTING, 7 Ed. 4.3 November Estimating 7, Derivatives 12 and / 14Ric

13 This leads to a very important formula for approximating f (x): f (x) = 1 h2 [f (x + h) f (x h)] 2h 3! f (x) h4 5! f (5) (x) (4) Expressed otherwise, with an error whose leading term is which makes it O(h 2 ). f (x) 1 [f (x + h) f (x h)] (5) 2h 1 6 h2 f (x) Ward Cheney/David Kincaid c (UT Austin[10pt] NUMERICAL Engage Learning: MATHEMATICS Thomson-Brooks/Cole & COMPUTING, 7 Ed. 4.3 November Estimating 7, Derivatives 13 and / 14Ric

14 Example Example Modify program First so that it uses the central difference formula (5) to approximate the first derivative of the function f (x) = sin x at x = 0.5. Using the truncation error term for the central difference formula (5), we set 1 6 h2 f (ξ) n < or n > (6 log 3)/ log We obtain a good approximation after about five iterations with this higher-order formula. The least error of was at step 9. Ward Cheney/David Kincaid c (UT Austin[10pt] NUMERICAL Engage Learning: MATHEMATICS Thomson-Brooks/Cole & COMPUTING, 7 Ed. 4.3 November Estimating 7, Derivatives 14 and / 14Ric