09 numerical differentiation
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Interpolation/Curve Fitting
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Objectives
• Understanding the difference between regression and interpolation
• Knowing how to “best fit” a polynomial into a set of data
• Knowing how to use a polynomial to interpolate data
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Measured Data
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Polynomial Fit!
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Line Fit!
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Which is better?
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Curve Fitting
• If the data measured is of high accuracy and it is required to estimate the values of the function between the given points, then, polynomial interpolation is the best choice.
• If the measurements are expected to be of low accuracy, or the number of measured points is too large, regression would be the best choice.
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Interpolation
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Why Interpolation?
• When the accuracy of your measurements are ensured
• When you have discrete values for a function (numerical solutions, digital systems, etc …)
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Acquired Data
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
But, how to get the equation of a function that passes by all the
data you have!
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Equation of a Line: Revision
xaay 21 If you have two points
1211 xaay
2212 xaay
2
1
2
1
2
1
1
1
y
y
a
a
x
x
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Solving for the constants!
12
122
12
21121 &
xx
yya
xx
yxyxa
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
What if I have more than two points?
• We may fit a polynomial of order one less that the number of points we have. i.e. four points give third order polynomial.
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Third-Order Polynomial
34
2321 xaxaxaay
For the four points
314
2131211 xaxaxaay
324
2232212 xaxaxaay
334
2333213 xaxaxaay
344
2434214 xaxaxaay
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
In Matrix Form
4
3
2
1
4
3
2
1
34
224
33
223
32
222
31
211
1
1
1
1
y
y
y
y
a
a
a
a
xxx
xxx
xxx
xxx
Solve the above equation for the constants of the polynomial.
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Newton's Interpolation Polynomial
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Newton’s Method
• In the previous procedure, we needed to solve a system of linear equations for the unknown constants.
• This method suggests that we may just proceed with the values of x & y we have to get the constants without setting a set of equations
• The method is similar to Taylor’s expansion without differentiation!
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Equation of a Line: Revision
xaay 21 If you have two points
1211 xaay
2212 xaay
2
1
2
1
2
1
1
1
y
y
a
a
x
x
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
For the two points
12
12
1
1
xx
yy
xx
yy
12
12
1
1
xx
yy
xx
yxf
112
121 xx
xx
yyyxf
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
For the three points
213
121
xxxxa
xxaaxf
11 ya
12
122 xx
yya
13
12
12
23
23
3 xx
xxyy
xxyy
a
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Using a table
xiyi
x1y1
x2y2
x3y3
13
12
12
23
23
xx
xxyy
xx
yy
12
12
xx
yy
23
23
xx
yy
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
In General
• Newton’s Interpolation is performed for an nth order polynomial as follows
nn xxxxa
xxxxaxxaaxf
...... 11
213121
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Example
• Find a 3rd order polynomial to interpolate the function described by the given points
xY
-11
02
15
216
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Solution: System of equations
• A third order polynomial is given by:
34
2321 xaxaxaaxf
11 4321 aaaaf
20 1 af
51 4321 aaaaf
168422 4321 aaaaf
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
In matrix form
16
5
2
1
8421
1111
0001
1111
4
3
2
1
a
a
a
a
1
1
1
2
4
3
2
1
a
a
a
a
322 xxxxf
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Newton’s Method
• Newton’s methods defines the polynomial in the form:
3214
213121
xxxxxxa
xxxxaxxaaxf
11
11
4
321
xxxa
xxaxaaxf
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Newton’s Method
xY
-11111
0234
1511
216
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Newton’s Method
• Finally:
11
111
xxx
xxxxf
xxxxxxf 3211
322 xxxxf
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Advantage of Newton’s Method
• The main advantage of Newton’s method is that you do not need to invert a matrix!
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ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Homework #6
• Chapter 18, pp. 505-506, numbers:18.1, 18.2, 18.3, 18.5.