Lagrange interpolation
The polynomial of degree at most n through n + 1 points: existence, uniqueness, basis polynomials, computing small examples, interpolation error and Runge's phenomenon.
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Polynomial interpolation: what is it?
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Finding a polynomial that takes exactly given values at given abscissas.
n + 1 points with distinct abscissas determine a unique polynomial of degree at most n. It goes through every point, even noisy measurements: it is not a fit.
How many polynomials of degree at most 2 pass through three points with distinct abscissas?
- Exactly one
- Infinitely many, with one free coefficient to choose
- Two, depending on whether the parabola opens upward or downward
- None if the three points are not collinear
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Exactly one
Three conditions for three coefficients: the system has a unique solution as soon as the abscissas are distinct. Three collinear points give a degree-1 polynomial, which is indeed “of degree at most 2”.
The abscissas x₀, …, xₙ where the polynomial's value is imposed are the …; through n + 1 distinct nodes there goes a … polynomial of degree at most ….
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The abscissas x₀, …, xₙ where the polynomial's value is imposed are the nodes; through n + 1 distinct nodes there goes a unique polynomial of degree at most n.
Two polynomials of degree at most n that agree at n + 1 points have a difference of degree at most n with n + 1 roots: it is zero.
Lagrange formula: what is it?
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P(x) = y₀·L₀(x) + y₁·L₁(x) + … + yₙ·Lₙ(x): the polynomial through the points (xᵢ; yᵢ), written as a weighted sum of the basis polynomials.
At a node xⱼ, every term vanishes except yⱼ·Lⱼ(xⱼ) = yⱼ: that is what makes P go through each point, without solving any system.