1. MA/CS375 Fall 2002 1 MA/CS 375 Fall 2002 Lecture 6

2. MA/CS375 Fall 2002 2 Exercise: Blending Two Images Find one partner to work with (i.e. your neighbor) Write a script which reads in two images. Use the min function to find the minimum number of rows for both pictures (MINrows) Use the min function to find the minimum number of columns for both pictures (MINcolumns) Crop the images using > pic1 = pic1(1:MINrows,1:MINcols,1:3) > pic2 = pic2(1:MINrows,1:MINcols,1:3) Create a third picture (pic3) which is the average of pic1 and pic2 Plot pic1,pic2,pic3 in the same figure using subplot Put your names on the title, print it out and hand it in.

3. MA/CS375 Fall 2002 3 Finite Precision Effects Recall that we are computing in a finite precision computing paradigm. i.e. all non integer numbers are represented by a finite approximation. e.g. 1/2 is represented exactly, but 1/3 is not.

4. MA/CS375 Fall 2002 4 Non-Special IEEE Floating Point Double Precision A double precision (Matlab default) number is represented by a 64 bit binary number: Here s is the sign bit m is the mantissa (length to be determined) e is the exponent and 0 < e < 2047

5. MA/CS375 Fall 2002 5 A Convergent Binary Representation of Any Number Between 0 and 1 a similar representation in base 10: Volunteer ?

6. MA/CS375 Fall 2002 6 Finite Binary Approximations of a Real Number We can easily show that T N is bounded as: (think of Zeno’s paradox)

7. MA/CS375 Fall 2002 7 Definition of a Derivative Suppose we are given a function: – where f is smooth and well defined at every point on the real numbers. –we can define the derivative of f at a point x as: –later on today we will try to use this definition (and see what goes wrong ) Review

8. MA/CS375 Fall 2002 8 Monster Functions Due to some of these finite precision effects there are some odd behaviors of Matlab documented in: “Numerical Monsters”, Christopher Essex, Matt Davison and Christian Schulzky.

9. MA/CS375 Fall 2002 9 Team Tasks We are going to investigate several examples from: “Numerical Monsters”, Christopher Essex, Matt Davison and Christian Schulzky. A.K.A “When Good Formulas Go Bad in Matlab”

10. MA/CS375 Fall 2002 10 Teams Form teams of four Choose one of the following monster exercises Remember to put your names in the titles

11. MA/CS375 Fall 2002 11 Monster #1 Consider: What should its behavior be as: Use subplots for the following 2 plots Plot this function at 1000 points in: Label everything nicely, include your name in the title. In a text box explain what is going on, print it out and hand it in

12. MA/CS375 Fall 2002 12 Monster #2 Consider: What should its behavior be as: Plot this function at 1000 points in: Explain what is going on in a text box, label everything, print it out and hand it in.

13. MA/CS375 Fall 2002 13 Consider: What should its behavior be as: Plot this function at 1000 points in: Explain what is going on in a text box and in particular what happens at x=54, label everything, print it out and hand it in. Monster #3

14. MA/CS375 Fall 2002 14 Consider: What should its behavior be as: Plot four subplots of the function at 1000 points in: for Now fix x=0.5 and plot this as a function of for Explain what is going on, print out and hand in. Monster #4

15. MA/CS375 Fall 2002 15 Explanations of Team Examples

16. MA/CS375 Fall 2002 16 Consider: What should its behavior be as: Plot this function at 1000 points in: Explain what is going on. Recall Monster #1

17. MA/CS375 Fall 2002 17 Monster #1 ((large+small)-large)/small

18. MA/CS375 Fall 2002 18 when we zoom in we see that the large+small operation is introducing order eps errors which we then divide with eps to get O(1) errors !. Monster #1 ((large+small)-large)/small

19. MA/CS375 Fall 2002 19 when we zoom in we see that the large+small operation is introducing order eps errors which we then divide with eps to get O(1) errors !. Each stripe is a region where 1+ x is a constant ( think about the gaps between numbers in finite precision ) Then we divide by x and the stripes look like hyperbola. The formula looks like (c-1)/x with a new c for each stripe. Monster #1 ((large+small)-large)/small

20. MA/CS375 Fall 2002 20 Recall Monster #2 Consider: What should its behavior be as: Plot this function at 1000 points in: Explain what is going on in a text box, label everything, print it out and hand it in.

21. MA/CS375 Fall 2002 21 Limit of

22. MA/CS375 Fall 2002 22 Monster #2 (finite precision effects from large*small) As x increases past 30 we see that f deviates from 1 !!

23. MA/CS375 Fall 2002 23 As x increases past ~=36 we see that f drops to 0 !! Monster #2 cont (finite precision effects from large*small)

24. MA/CS375 Fall 2002 24 Consider: What should its behavior be as: Plot this function at 1000 points in: Explain what is going on. What happens at x=54? Recall Monster #3

25. MA/CS375 Fall 2002 25 Monster 3 (finite precision large*small with binary stripes)

26. MA/CS375 Fall 2002 26 As we require more than 52 bits to represent 1+2^(-x) we see that the log term drops to 0. Monster 3 (finite precision large*small with binary stripes)

27. MA/CS375 Fall 2002 27 Consider: What should its behavior be as: Plot four subplots of the function at 1000 points in: for Now fix x=0.5 and plot this as a function of for Explain what is going on, print out and hand in. Recall Monster #4

28. MA/CS375 Fall 2002 28 Monster 4 cont Behavior as delta  0 : or if you are feeling lazy use the definition of derivative, and remember: d(sin(x))/dx = cos(x)

29. MA/CS375 Fall 2002 29 Monster 4 cont ( parameter differentiation, delta=1e-4) OK

30. MA/CS375 Fall 2002 30 Monster 4 cont ( parameter differentiation, delta=1e-7) OK

31. MA/CS375 Fall 2002 31 Monster 4 cont ( parameter differentiation, delta=1e-12) Worse

32. MA/CS375 Fall 2002 32 Monster 4 cont ( parameter differentiation, delta=1e-15) When we make the delta around about machine precision we see O(1) errors !. Bad

33. MA/CS375 Fall 2002 33 Monster 4 cont ( numerical instablitiy of parameter differentiation) As delta gets smaller we see that the approximation improves, until delta ~= 1e-8 when it gets worse and eventually the approximate derivate becomes zero.

34. MA/CS375 Fall 2002 34 Approximate Explanation of Monster #4 1) Taylor’s thm: 2) Round off errors 3) Round off in computation of f and x+delta 4) Put this together:

35. MA/CS375 Fall 2002 35 i.e. for or equivalently approximation error decreases as delta decreasesize. BUT for round off dominates!.

36. MA/CS375 Fall 2002 36 Summary Ok – so in the far limits of the range of finite precision really dodgy things happen. But recall, the formula for the derivative of sine worked pretty well for a large range of numbers. Try to avoid working with two widely separated numbers.

37. MA/CS375 Fall 2002 37 Summary of Lecture 6 Today’s lecture was all about the way Matlab (and in fact almost all languages) store representations of real numbers We found out: – because the set of possible double precision numbers only covers a discrete subset of points on the real line there are finite gaps between consecutive numbers –If A is “large” and B is very “small” the sum A+B actually results in A+B ~= A –and so on..