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

MA/CS375 Fall 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.

MA/CS375 Fall 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.

MA/CS375 Fall 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

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

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

MA/CS375 Fall 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

MA/CS375 Fall 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.

MA/CS375 Fall 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”

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

MA/CS375 Fall 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

MA/CS375 Fall 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 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

MA/CS375 Fall 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

MA/CS375 Fall Explanations of Team Examples

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

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

MA/CS375 Fall 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

MA/CS375 Fall 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

MA/CS375 Fall 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 Limit of

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

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

MA/CS375 Fall 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

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

MA/CS375 Fall 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)

MA/CS375 Fall 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

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

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

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

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

MA/CS375 Fall 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.

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

MA/CS375 Fall 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 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..