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MA/CS 375

MA/CS 375. Fall 2002 Lecture 7. Explanations of Team Examples. Recall Monster #1. Consider: What should its behavior be as: Plot this function at 1000 points in: Plot this function at 1000 points in: Explain what is going on. Monster #1 ((large+small)-large)/small.

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MA/CS 375

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  1. MA/CS 375 Fall 2002 Lecture 7 MA/CS375 Fall 2002

  2. Explanations of Team Examples MA/CS375 Fall 2002

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

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

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

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

  7. 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. MA/CS375 Fall 2002

  8. Limit of MA/CS375 Fall 2002

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

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

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

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

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

  14. RecallMonster #4 • 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. MA/CS375 Fall 2002

  15. 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) MA/CS375 Fall 2002

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

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

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

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

  20. Monster 4cont(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. MA/CS375 Fall 2002

  21. 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: MA/CS375 Fall 2002

  22. i.e. for or equivalently approximation error decreases as delta decrease in size. BUT for round off dominates!. MA/CS375 Fall 2002

  23. Ok – so these were extreme cases • Reiteration: the numerical errors were decreasing as delta decreased until delta was approximately 1e-8 MA/CS375 Fall 2002

  24. 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 MA/CS375 Fall 2002

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

  26. MA/CS375 Fall 2002

  27. diff Using diff on F = sin(x) diffdemo.m MA/CS375 Fall 2002

  28. MA/CS375 Fall 2002

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

  30. MA/CS375 Fall 2002

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

  32. MA/CS375 Fall 2002

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

  34. MA/CS375 Fall 2002

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

  36. MA/CS375 Fall 2002

  37. Comments • Next time we will revisit the accuracy of taking derivatives… MA/CS375 Fall 2002

  38. 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. MA/CS375 Fall 2002

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