Lagrange interpolation

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Lagrange interpolation Gives the same results as
Newton, but different method
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Formal definition for nth order interpolating
polynomial
L is defined as
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Example interpolate exp(2) using exp(1),
exp(1.5) and exp(2.5)
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Same as with Newton interpolation
Matlab code for it
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How to get the coefficients of the interpolating
polynomial (not just single points)? i.e. what
are the as in
Use second order example
with data points
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Substitute the data points into the equation
or
a Vandermonde matrix
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• Can solve with any matrix method
• - but-
• matrix is often ill-conditioned, esp. for large
  n
• not fastest means of getting as (special
  methods for Vandermonde matrices)
• Getting coefficients is generally not a good idea

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Inverse interpolation Given xs, f(x)s -
interpolation lets us get new f(x) for new
x. What about new x from new f(x)?
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A number of methods 1) Switch x and f(x) and do
new interpolation I.e. f(x)1/x
Instead of using
use
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• This may not be best way
• ill conditioned problem

Another method - fit low-order polynomial to a
few points and use root finding Still beware of
ill-conditioning!
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Example
Given f(x)1.4, what is x?
Find interpolating polynomial (quadratic)
Gauss elimination
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After backsubstitution
Use a root finding method for f(x)1.4
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Actually, in this case cant use quadratic formula
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Equally spaced data There are special algorithms
for equally spaced data If
then
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So
If
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then
Why bother?
Useful later for numerical integration Just use
Newton or Lagrange interpolation now
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Extrapolation
Basically dont Example
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Use x2,4,6 and estimate f(3) by interpolatio n
and f(8) by extrapolation
Interpolating function
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