PowerPoints organized by Dr. Michael R. Gustafson II, Duke University

Size: px

Start display at page:

Download "PowerPoints organized by Dr. Michael R. Gustafson II, Duke University"

Karen Crawford
5 years ago
Views:

1 Part 5 Chapter 21 Numerical Differentiation PowerPoints organized by Dr. Michael R. Gustafson II, Duke University 1 All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

2 Chapter Objectives Understanding the application of high-accuracy numerical differentiation formulas for equispaced data. Knowing how to evaluate derivatives for unequally spaced data. Understanding how Richardson extrapolation is applied for numerical differentiation. Recognizing the sensitivity of numerical differentiation to data error. Knowing how to evaluate derivatives in MATLAB with the diff and gradient functions. Knowing how to generate contour plots and vector fields with MATLAB. 2

3 Differentiation The mathematical definition of a derivative begins with a difference approximation: y x f x x i x fx i and as x is allowed to approach zero, the difference becomes a derivative: dy dx lim fx i x fx i x0 x 3

4 High-Accuracy Differentiation Formulas Taylor series expansion can be used to generate highaccuracy formulas for derivatives by using linear algebra to combine the expansion around several points. Three categories for the formula include forward finitedifference, backward finite-difference, and centered finitedifference. 4

5 Forward Finite-Difference

6 6

7 Backward Finite-Difference 7

8 Centered Finite-Difference 8

9 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 ) 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 ) D(h ) 1 9

10 Unequally Spaced Data One way to calculated derivatives of unequally spaced data is to determine a polynomial fit and take its derivative at a point. As an example, using a second-order Lagrange polynomial to fit three points and taking its derivative yields: xx xx xx xx xx xx f x f x f x f x f x fx 0 2x x 1 x 2 x 0 x 1 x 0 x 2 2x x 0 x x0 x1 x0 x2 x1x0 x1x2 x2 x0 x2 x1 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 10

Derivatives and Integrals for Data with Errors A shortcoming of numerical differentiation is that it tends to amplify errors in data, whereas integration tends to

11 Derivatives and Integrals for Data with Errors A shortcoming of numerical differentiation is that it tends to amplify errors in data, whereas integration tends to smooth data errors. One approach for taking derivatives of data with errors is to fit a smooth, differentiable function to the data and take the derivative of the function. 11

12 Numerical Differentiation with MATLAB MATLAB has two built-in functions to help take derivatives, diff and gradient: diff(x) 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 12

13 Numerical Differentiation with MATLAB fx = gradient(f, h) Determines the derivative of the data in f at each of the points. The program 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. 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) 13

14 Visualization MATLAB can generate contour plots of functions as well as vector fields. Assuming x and y represent a meshgrid of x and y values and z represents a function of x and y, 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 14

15 15

16 Part 6 Chapter 22 Initial-Value Problems PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

17 Chapter Objectives Understanding the meaning of local and global truncation errors and their relationship to step size for one-step methods for solving ODEs. Knowing how to implement the following Runge-Kutta (RK) methods for a single ODE: Euler Heun Midpoint Fourth-Order RK Knowing how to iterate the corrector of Heun s method. Knowing how to implement the following Runge-Kutta methods for systems of ODEs: Euler -- Fourth-order RK 17

18 Ordinary Differential Equations Methods described here are for solving differential equations of the form: dy dt f t, y The methods in this chapter are all one-step methods and have the general format: y i1 y i h where is called an increment function, and is used to extrapolate from an old value y i to a new value y i+1. 18

19 Euler s Method The first derivative provides a direct estimate of the slope at t i : dy dt ti ft i, y i and the Euler method uses that estimate as the increment function: ft i, y i y i1 y i f t i, y i h 19

20 20

21 Error Analysis for Euler s Method The numerical solution of ODEs involves two types of error: Truncation errors, caused by the nature of the techniques employed Roundoff errors, caused by the limited numbers of significant digits that can be retained The total, or global truncation error can be further split into: local truncation error that results from an application method in question over a single step, and propagated truncation error that results from the approximations produced during previous steps. 21

22 Error Analysis for Euler s Method The local truncation error for Euler s method is O(h 2 ) and proportional to the derivative of f(t,y) while the global truncation error is O(h). This means: The global error can be reduced by decreasing the step size, and Euler s method will provide error-free predictions if the underlying function is linear. Euler s method is conditionally stable, depending on the size of h. 22

23 23

24 MATLAB Code for Euler s Method f t, y i y y f t, y h i1 i i i i 24

25 >> 4*exp(0.8*t) - 0.5*y; >> [t,y] = eulode(dydt,[0 4],2,1); >> disp([t,y]) 25

making a prediction of the new value of y, then correcting it based on the slope

26 Heun s Method One method to improve Euler s method is to determine derivatives at the beginning and predicted ending of the interval and average them: This process relies on making a prediction of the new value of y, then correcting it based on the slope calculated at that new value. This predictor-corrector approach can be iterated to convergence: 26

27 27

28 28

29 Midpoint Method Another improvement to Euler s method is similar to Heun s method, but predicts the slope at the midpoint of an interval rather than at the end: This method has a local truncation error of O(h 3 ) and global error of O(h 2 ) 29

30 Runge-Kutta Methods Runge-Kutta (RK) methods achieve the accuracy of a Taylor series approach without requiring the calculation of higher derivatives. For RK methods, the increment function can be generally written as: a 1 k 1 a 2 k 2 a n k n where the a s are constants and the k s are k1 f ti, yi k2 f ti ph 1, yi q11k1h k f t p h, y q k hq k h 3 i 2 i k f t p h, y q k hq k hq k h n i n1 i n1,1 1 n1,2 2 n1, n1 n1 where the p s and q s are constants. f t, y i y y f t, y h i1 i i i i 30

31 Classical Fourth-Order Runge-Kutta Method The most popular RK methods are fourth-order, and the most commonly used form is: where: k 1 f t i, y i k 2 f t i 1 2 h, y i 1 2 k 1h k 3 f t i 1 2 h, y 1 i 2 k h 2 k 4 ft i h, y i k 3 h y i1 y i 1 6 k 1 2k 2 2k 3 k 4 h 31

32 y i1 y i 1 6 k 1 2k 2 2k 3 k 4 h k 1 ft i, y i k 2 f t i 1 2 h, y 1 i 2 k h 1 k 3 f t i 1 2 h, y 1 i 2 k h 2 k 4 f t i h, y i k 3 h 32

PowerPoints organized by Dr. Michael R. Gustafson II, Duke University

Part 5 Chapter 19 Numerical Differentiation PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction