By Adam Mallen
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What is it?
How is it different from regression?
When would you use it?
What can go wrong?
How do we find the interpolating polynomial?
Can you do this in Matlab?
What else?
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The interpolating polynomial is the polynomial of
least degree which passes through all the data points
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Formally:
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A unique solution to this problem is guaranteed
X
0
10
20
30
40
50
60
Y
-10
3
-30
6
10
-2
15
40
30
20
10
0
-10
-20
-30
-40
-10
0
10
20
30
40
50
60
70
X
0
10
20
30
40
50
60
Y
-10
3
-30
6
10
-2
15
40
30
20
10
0
-10
-20
-30
-40
-10
0
10
20
30
40
50
60
70
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Interpolation models must take on the exact
values of the known data points
Regression models minimize the residuals
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Given n+1 data points:
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◦ The “best fit” polynomials of degree < n
form regression models.
◦ The “best fit” polynomial of degree = n
is the interpolating polynomial because the sum of
the residuals is exactly zero.
40
30
20
10
0
-10
-20
-30
-40
-10
0
10
20
30
40
50
60
70
3
2
1
0
-1
-2
-3
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.5
1
0.5
0
-0.5
-1
-1.5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
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Regression models assume that
measurements have noise.
Regression models estimate f(x) and may be
used for forecasting future and past values.
Interpolation models may be suitable when
measurements are believed to be exact.
Interpolation models estimate values between
known data points.
NOT for forecasting
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
2
4
6
8
10
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Runge Phenomenon
Divergence for some selection of nodes
Splines can help solve these problems
However, …
◦ Splines may only be differentiable a certain number of
times at the data points.
◦ Polynomials are infinitely differentiable
◦ Splines can be more complicated to compute and store.
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
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We can represent this system of equations as
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function result = poly_interp(x, y)
% x and y are column vectors with the x and y values of the data points
% there are n+1 data points
n = length(x) - 1;
% construct the Vandermonde matrix
V = zeros(n+1,n+1);
for i=1:n+1
for j=1:n+1
V(i,j) = x(i).^(j-1);
end %for
end %for
% solve the system of equations
alpha = V\y;
result = fliplr(alpha');
end
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Lagrange form of the interpolating Polynomial
Newton form of the interpolating Polynomial
Chebyshev nodes
Hermite interpolation problem
Harmonic function interpolation
Lebesgue constants