Numerical Analysis Fall. Numerical Differentiation

Transcription

1 Numerical Analysis 5 Fall Numerical Differentiation

2 Differentiation The mathematical definition of a derivative begins with a difference approimation: and as is allowed to approach zero, the difference becomes a derivative: y f i f i dy d lim f i f i

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

4 Forward Finite-Difference

5 Backward Finite-Difference

6 Centered Finite-Difference

7 Accurate Differentiation or Richardson Etrapolation

8 Richardson Etrapolation

9 Richardson Etrapolation As with integration, the Richardson etrapolation 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 ) estimates and the interval is halved (h =h /), an improved O(h 4 ) estimate may be formed using: For the cases where there are two O(h 4 ) estimates and the interval is halved (h =h /), an improved O(h 6 ) estimate may be formed using: D 4 3 D(h ) 3 D(h ) For the cases where there are two O(h 6 ) estimates and the interval is halved (h =h /), an improved O(h 8 ) estimate may be formed using: D 6 5 D(h ) 5 D(h ) D D(h ) 63 D(h )

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 eample, using a second-order Lagrange polynomial to fit three points and taking its derivative yields: f f f f

11 Lagrange Interpolating Polynomials Another method that uses shifted value to epress an interpolating polynomial is the Lagrange interpolating polynomial. The differences between a simply polynomial and Lagrange interpolating polynomials for first and second order polynomials is: Order Simple Lagrange st f () a a f () L f L f nd f () a a a 3 f () L f L f L 3 f 3 where the L i are weighting coefficients that are functions of.

12 Unequally Spaced Data f f f f f f f f

13 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.

14 Partial Derivatives

15 Numerical Differentiation with MATLAB MATLAB has two built-in functions to help take derivatives, diff and gradient: diff() Returns the difference between adjacent elements in diff(y)./diff() Homework Returns the difference between adjacent values in y divided by the corresponding difference in adjacent values of

16 Numerical Differentiation with Homework MATLAB f = 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=. 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: [f, fy] = gradient(f, h)

17 Visualization Homework MATLAB can generate contour plots of functions as well as vector fields. Assuming and y represent a meshgrid of and y values and z represents a function of and y, contour(, y, z) can be used to generate a contour plot [f, fy]=gradient(z,h) can be used to generate partial derivatives and quiver(, y, f, fy) can be used to generate vector fields