ma/cs375 fall 2002 1 ma/cs 375 fall 2002 lecture 7
TRANSCRIPT
MA/CS375 Fall 2002 1
MA/CS 375
Fall 2002
Lecture 7
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Explanations of Team Examples
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• 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.
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 !.
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 !.
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.
Monster #1((large+small)-large)/small
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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.
log(1 )x xf x e e
x
0,50x
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Limit of log(1 )x xf x e e
log 1lim lim log 1 lim
1lim by l'Hopital's rule
1lim 1
1
x
x xxx x x
x
x
xx
xx
ef x e e
e
eee
e
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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:
• What should its behavior be as:
• Plot this function at 1000 points in:
• 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:
• 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.
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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Monster 4 cont (parameter differentiation, delta=1e-12)
Worse
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Monster 4 cont (parameter differentiation, delta=1e-15)
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 x
x 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:
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2ˆ ˆˆ ˆ '
ˆ ˆˆ ˆ
f y f x f x O O
df x f y f xO O
dx
i.e. for or equivalently approximation error decreases as delta decrease in size.
BUT for round off dominates!.
810
810
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Ok – so these were extreme cases
• Reiteration: the numerical errors were decreasing as delta decreased until delta was approximately 1e-8
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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
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diffdemo.m
Using diff on
F = x^3
diff
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diff
diffdemo.m
Using diff on
F = sin(x)
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gradient
gradientdemo.m
Using gradient on
F = x^2
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Using gradient on
F = x^2+y^2
gradientdemo1.m
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Using gradient on
F = (x^2)*(y^2)
gradientdemo2.m
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Using gradient on
F = (sin(pi*x))*(cos(pi*y))
gradientdemo3.m
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Comments
• Next time we will revisit the accuracy of taking derivatives…
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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.