ma/cs375 fall 2002 1 ma/cs 375 fall 2002 lecture 6

MA/CS375 Fall 2002 1

MA/CS 375

Fall 2002Lecture 6


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.


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.


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

1023( 1) 1. 2s em


A Convergent Binary Representation of Any Number

Between 0 and 1

1

1 where m 1 or 02

n

n nnn

m m

a similar representation in base 10:

1

1 where 0,1,2,3,4,5,6,7,8 or 910

n

n nnn

d d d

Volunteer ?


Finite Binary Approximations of a Real Number

N1

1 +T where 1 or 02

n N

n nnn

m m m

We can easily show that TN is bounded as:(think of Zeno’s paradox)

12N NT


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

:f

0

limdf x f x f xdx


Monster Functions

Due to some of these finite precision effectsthere are some odd behaviors of Matlabdocumented in:

“Numerical Monsters”, Christopher Essex,Matt Davison and Christian Schulzky.


Team Tasks

We are going to investigate several examplesfrom:

“Numerical Monsters”, Christopher Essex,Matt Davison and Christian Schulzky.

A.K.A “When Good Formulas Go Bad in Matlab”

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Teams

• Form teams of four

• Choose one of the following monster exercises

• Remember to put your names in the titles

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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

1 1xf x

x

0x

1,1x

4 ,4

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Monster #2

• Consider:



• Explain what is going on in a text box, label everything, print it out and hand it in.

log(1 )x xf x e e

x

0,50x

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• Consider:



• 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.

22 log (1 2 )x xf x

x

0,60x

Monster #3

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• Consider:


• 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.

sin sincos

x xg x x

0

0,2*x 1 4,1 7,1 12,1 15e e e e

0.1,0.01,0.001,...,1 20e

Monster #4

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Explanations of Team Examples

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• Consider:




• Explain what is going on.

1 1xf x

x

0x

1,1x

4 ,4

Recall Monster #1

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

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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 !.


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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 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.


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Recall Monster #2

• Consider:



• Explain what is going on in a text box, label everything, print it out and hand it in.

log(1 )x xf x e e

x

0,50x

MA/CS375 Fall 2002 21

Limit of log(1 )x xf x e e

log 1lim lim log 1 lim

1lim by l'Hopital's rule

1lim 11

xx x

xx x x

x

x

xx

xx

ef x e e

e

eee

e

MA/CS375 Fall 2002 22

Monster #2(finite precision effects from large*small)

1 as f x x

As x increases past 30 we seethat f deviates from 1 !!

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As x increases past ~=36 we see that f drops to 0 !!

Monster #2 cont(finite precision effects from large*small)

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• Consider:



• Explain what is going on. What happens at x=54?

22 log (1 2 )x xf x

x

0,60x

Recall Monster #3

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

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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)

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• Consider:


• 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.

sin sincos

x xg x x

0

0,2*x 1 4,1 7,1 12,1 15e e e e

0.1,0.01,0.001,...,1 20e

RecallMonster #4

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Monster 4 cont

0 0

0

sin sin sin cos cos( )sin sinlim lim

sin cos 1 cos( )sinlim

cos

x x x x x

x x

x

Behavior as delta 0 :

or if you are feeling lazy use the definition of derivative, and remember: d(sin(x))/dx = cos(x)

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

OK

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

OK

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Worse

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When we make the delta around about machine precision we see O(1) errors !.

Bad

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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.

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Approximate Explanation of Monster #4

2

2

'

ˆˆ ˆ( )

ˆ ˆ ˆ ( )

( )

ˆ ˆˆ ˆ

'

f x f x f x O

y xx O

f y f y O

f x O

f y f x f y f x O

f x O O

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 35

2ˆ ˆˆ ˆ '

ˆ ˆˆ ˆ

f y f x f x O O

f y f xO O

i.e. for or equivalently approximation error decreases as delta decreasesize.

BUT for round off dominates!.

810

810

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.

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..

ma/cs375 fall 2002 1 ma/cs 375 fall 2002 lecture 6

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