# ma/cs375 fall 2002 1 ma/cs 375 fall 2002 lecture 7

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• MA/CS 375Fall 2002Lecture 7

MA/CS375 Fall 2002

• Explanations of Team Examples

MA/CS375 Fall 2002

• Recall Monster #1Consider:

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

• Monster #1((large+small)-large)/small

MA/CS375 Fall 2002

• Monster #1((large+small)-large)/smallwhen 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

• Monster #1((large+small)-large)/smallwhen 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 regionwhere 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.

MA/CS375 Fall 2002

• Recall Monster #2Consider:

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

• Limit of

MA/CS375 Fall 2002

• Monster #2(finite precision effects from large*small)As x increases past 30 we seethat f deviates from 1 !!

MA/CS375 Fall 2002

• 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

• 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 2002

• Monster 3(finite precision large*small with binary stripes)

MA/CS375 Fall 2002

• 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

• RecallMonster #4Consider:

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

• Monster 4 contBehavior 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

• Monster 4 cont(parameter differentiation, delta=1e-4)OK

MA/CS375 Fall 2002

• Monster 4 cont (parameter differentiation, delta=1e-7)OK

MA/CS375 Fall 2002

• Monster 4 cont (parameter differentiation, delta=1e-12)Worse

MA/CS375 Fall 2002

• 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 2002

• 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 2002

• Approximate Explanation of Monster #41) Taylors thm:

2) Round off errors

3) Round off in computation of f and x+delta

4) Put this together:

MA/CS375 Fall 2002

• i.e. for or equivalently approximation error decreases as delta decrease in size.

BUT for round off dominates!.

MA/CS375 Fall 2002

• Ok so these were extreme casesReiteration: the numerical errors were decreasing as delta decreased until delta was approximately 1e-8

MA/CS375 Fall 2002

• Matlab Built-in Derivative Routinesdiff 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

• diffdemo.mUsing diff on

F = x^3diff

MA/CS375 Fall 2002

• MA/CS375 Fall 2002

• diffdiffdemo.mUsing diff on

F = sin(x)

MA/CS375 Fall 2002

• MA/CS375 Fall 2002

F = x^2

MA/CS375 Fall 2002

• MA/CS375 Fall 2002

MA/CS375 Fall 2002

• MA/CS375 Fall 2002

MA/CS375 Fall 2002

• MA/CS375 Fall 2002