Polynomial Interpolation

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Polynomial Interpolation

• By Adam Mallen

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Outline Polynomial Interpolation

• 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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What is it?

• The interpolating polynomial is the polynomial of
  least degree which passes through all the data
  points
• Formally
• A unique solution to this problem is guaranteed

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Example
X Y
0 -10
10 3
20 -30
30 6
40 10
50 -2
60 15
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Example
X Y
0 -10
10 3
20 -30
30 6
40 10
50 -2
60 15
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Is it different from regression?

• Interpolation models must take on the exact
  values of the known data points
• Regression models minimize the residuals
• Given n1 data points
• The best fit polynomials of degree lt n
• form regression models.
• The best fit polynomial of degree n
• is the interpolating polynomial because the sum
  of the residuals is exactly zero.

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Interpolation vs. Regression
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Interpolation vs. Regression
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Interpolation vs. Regression
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When would you use it?

• 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

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Interpolation, not Forecasting
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What can go wrong?

• 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.

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Runge Phenomenon 5 nodes
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Runge Phenomenon 9 nodes
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Runge Phenomenon 17 nodes
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Runge Phenomenon 21 nodes
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How do you solve for it?
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A Solution Method

• We can represent this system of equations as

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Matlab Solution

• function result poly_interp(x, y)
• x and y are column vectors with the x and y
  values of the data points
• there are n1 data points
• n length(x) - 1
• construct the Vandermonde matrix
• V zeros(n1,n1)
• for i1n1
• for j1n1
• 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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What else?

• Lagrange form of the interpolating Polynomial
• Newton form of the interpolating Polynomial
• Chebyshev nodes
• Hermite interpolation problem
• Harmonic function interpolation
• Lebesgue constants

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Questions?