Chap. 19: Numerical Differentiation

Transcription

1 Chap. 19: Numerical Differentiation

2 Differentiation Definition of difference: y x f x x i x f x i As x is approaching zero, the difference becomes a derivative: dy dx lim x 0 f x i x f x i x 2

3 High-Accuracy Differentiation Formulas Taylor series expansion can be used to generate high-accuracy formulas for derivatives. f(x)'f(x 0 )%f(x 0 ) ) (x&x 0 )% 1 2! f(x 0 ))) (x&x 0 ) 2 % 1 3! f(x 0 )))) (x&x 0 ) 3 %@@@ Three categories: forward finite-difference backward finite-difference centered finite-difference. 3

4 Numerical Differentiation by Finite Difference Forward: Backward: f(x i ) ) ' f(x i%1 )&f(x i ) %O(h) h f(x i ) ) ' f(x i )&f(x i&1 ) %O(h) h Mid value: f(x i ) ) ' f(x i%1 )&f(x i&1 ) %O(h 2 ) 2h 4

5 Forward Finite-Difference 5

6 Example Derive the central difference differentiation formula. Consider the forward and backward schemes x i%1 &x i 'h x i&1 &x i '&h f(x i%1 )'f(x i )%f(x i ) ) h% 1 2! f(x i ))) h 2 % 1 3! f(x i )))) h 3 % 1 4! f(x i ))))) h 4 %@@@ f(x i&1 )'f(x i )&f(x i ) ) h% 1 2! f(x i ))) h 2 & 1 3! f(x i )))) h 3 % 1 4! f(x i ))))) h 4 %@@@ Y f(x i%1 )&f(x i&1 )'2f(x i ) ) h% 2 3! f(x i )))) h 3 %@@@ ˆ f(x i ) ) ' f(x i%1 )&f(x i&1 ) & 1 2h 3! f(x i )))) h 2 %@@@ 6

7 Backward Finite-Difference 7

8 Centered Finite-Difference 8

9 Example 19.1 f(x)= x-0.5x x 3-0.1x 4 Find f (0.5) with h=0.25 Exact: = Back Center Forward Forward Back Center O(h) O(h 2 ) O(h) O(h 2 ) O(h 2 ) O(h 4 ) f (0.5) t 21.7% -2.4% % 3.77% 0 9

10 Richardson Extrapolation As with integration, the Richardson extrapolation can be used to combine two lower-accuracy estimates of the derivative to produce a higher-accuracy estimate. For the cases where there are two O(h 2 ) estimates and the interval is halved (h 2 =h 1 /2), an improved O(h 4 ) estimate may be formed using: D 4 3 D(h 2 ) 1 3 D(h 1 ) For the cases where there are two O(h 4 ) estimates and the interval is halved (h 2 =h 1 /2), an improved O(h 6 ) estimate may be formed using: D D(h 2 ) 1 15 D(h 1 ) For the cases where there are two O(h 6 ) estimates and the interval is halved (h 2 =h 1 /2), an improved O(h 8 ) estimate may be formed using: D D(h 2 ) 1 63 D(h 1 ) 10

11 Example 19.2 Use Richardson Extrapolation to estimate the first derivative of the function in Example 19.1 at x=0.5 with step sizes of h 1 =0.5 and h 2 =0.25. D(0.5)' 0.2&1.2 '&1.0, ε 1 t '&9.6% D(0.25)' & '& , ε 0.5 t '&2.4 D' 4 3 (& )& 1 3 (&1)'&0.9125, ε '0% t 11

12 Unequally Spaced Data Determine a polynomial fit and take its derivative at a point. Example: using a second-order Lagrange polynomial to fit three points and taking its derivative yields: f x f x 0 2x x 1 x 2 x 0 x 1 x 0 x 2 2x x 0 x 2 2x x 0 x 1 f x 1 x 1 x 0 x 1 x 2 f x 2 x 2 x 0 x 2 x 1 12

13 Derivatives and Integrals for Data Numerical differentiation tends to amplify errors in data. Use regression to fit a smooth, differentiable function to the data and take the derivative of the function. with Errors Numerical integration tends to smooth data errors. Positive and negative error cancel each other. 13

14 Numerical Differentiation with diff(x) MATLAB Returns the difference between adjacent elements in x diff(y)./diff(x) Returns the difference between adjacent values in y divided by the corresponding difference in adjacent values of x. Gives central difference differentiation. Error: O([h/2] 2 ) 14

15 Numerical Differentiation with MATLAB (cont.) fx = gradient(f, h) Determines the derivative of the data in f at each of the points. Uses forward difference for the first point, backward difference for the last point, and centered difference for the interior points. h is the spacing between points; if omitted h=1. Error: Error: O(h 2 ) The major advantage of gradient over diff is gradient s result is the same size as the original data. Gradient can also be used to find partial derivatives for matrices: [fx, fy] = gradient(f, h) 15

16 Example 19.4 and 19.5 f(x)=0.2+25x-200x x 3-900x x 5 f (x)=25-400x+2025x x x 4 Use diff to find the derivatives. Use gradient to find the derivatives. 16

17 Example 19.4 and 19.5 (cont.) diff gradient 17

18 Visualization contour(x, y, z) can be used to generate a contour plot [fx, fy]=gradient(z,h) can be used to generate partial derivatives and quiver(x, y, fx, fy) can be used to generate vector fields 18

19 Example: HW# Voltage 19

20 Example: HW#8 (cont.) Voltage contour and E-field vector 20