1. MA/CS375 Fall 2002 1 MA/CS 375 Fall 2002 Lecture 8

2. MA/CS375 Fall 2002 2 Matlab Built-in Derivative Routines diff takes the derivative of a function of one variable sampled at a set of discrete points gradient takes the x and y derivatives of a function of two variables

3. MA/CS375 Fall 2002 3 diffdemo.m Using diff on F = x^3 diff

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5. 5 diff diffdemo.m Using diff on F = sin(x)

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7. 7 Explanation of Error Curve We can find good reason for the shape of the error curve..

8. MA/CS375 Fall 2002 8 Recall: Taylor’s Theorem With Cauchy Remainder 1) Taylor’s theorem with Cauchy Remainder: Here R n is the Cauchy Remainder

9. MA/CS375 Fall 2002 9 Recall: Taylor’s Theorem With Cauchy Remainder 2) Using a first order expansion (n=1)

10. MA/CS375 Fall 2002 10 Introducing Finite Precision Errors 3) round off due to finite precision representation of x and delta 4) round off due to finite precision computation of f

11. MA/CS375 Fall 2002 11 Putting It All Together

12. MA/CS375 Fall 2002 12 Final Result

13. MA/CS375 Fall 2002 13 Analysis of Final Result In words: the error created by approximating the derivative of f by the first order formula is given by two terms First term proportional to delta times the second derivative of f at some point between x and (x plus delta) Second term is due to finite precision representation of f, x, delta

14. MA/CS375 Fall 2002 14 Finding the Gradient of Two- dimensional Functions

15. MA/CS375 Fall 2002 15 Definition of Partial Derivatives Given a function f of two variables x,y We define the two partial derivatives by

16. MA/CS375 Fall 2002 16 Using gradient on F = x^2 gradientdemo.m gradient

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18. MA/CS375 Fall 2002 18 Using gradient on F = x^2+y^2 gradientdemo1.m

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20. MA/CS375 Fall 2002 20 Using gradient on F = (x^2)*(y^2) gradientdemo2.m

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22. MA/CS375 Fall 2002 22 Using gradient on F = (sin(pi*x))*(cos(pi*y)) gradientdemo3.m

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24. MA/CS375 Fall 2002 24 Individual Class Exercise Part 1 1) Using the following formula compute the approximate derivative of: f=x.^5 at 1000 points between x=-1 and 1 with delta = 1e-4 Do not use diff

25. MA/CS375 Fall 2002 25 Individual Class Exercise Part 2 2) Plot the error defined by: actualerror = abs(dfdx-5*x.^4); 3) On the same graph plot the error defined by: guesserror = abs(delta*(5*4*x.^3)/2); 4) Write comments on the graph about what you see. 5) Label the axes. Add a title to the graph. Add a legend to the graph.

26. MA/CS375 Fall 2002 26 Individual Class Exercise Part 3 Repeat on separate graphs for: delta = 1e-6, 1e-8, 1e-10, 1e-15 Explain what you see on each graph. Hand this in at the start of next lecture (Monday 09/09/02). This will be graded. Remember to include your name and staple.

27. MA/CS375 Fall 2002 27 Summary We have narrowed down the error term to two concretely defined terms Next time we will use these ideas to find edges in images.