Lagrange Interpolation Formula
For any distinct complex numbers and any complex numbers
, there exists a unique polynomial
of degree less than or equal to
such that for all integers
,
, and this polynomial is
.
While this formula may appear intimidating, it's actually not so difficult to see what is going on: for each term in the sum, we are finding a polynomial of degree that goes through the points
and
for
. When we add them all together, we end up with a polynomial that interpolates the desired points.
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