Lagrange Interpolation: Difference between revisions
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p = Lagrange(1⋅ℓ_0(x) + 4⋅ℓ_1(x) + 1⋅ℓ_2(x) + 5⋅ℓ_3(x))
Polynomial(p) = Polynomial(-21.0 + 35.83333333333333*x - 16.0*x^2 + 2.1666666666666665*x^3)
</pre>
=== Rational base polynomial answer ===
<syntaxhighlight lang="julia">using Polynomials
function Lagrange(pts::Vector{Vector{Int}})
xs = first.(pts)
cs = last.(pts)
n = length(xs)
n == 1 && return Polynomial(cs[1])
arr = ones(Int, n)
for j in 2:n
for k in 1:j
arr[k] = (xs[k] - xs[j]) * arr[k]
end
arr[j] = prod(xs[j] - xs[i] for i in 1:(j - 1))
end
ws = 1 .// arr
q = Polynomial(0 // 1)
x = variable(q)
for i in eachindex(ws)
m = prod(x - xs[j] for j in eachindex(xs) if j != i)
q += m * ws[i] * cs[i]
end
return q
end
const testpoints = [[1, 1], [2, 4], [3, 1], [4, 5]]
@show Lagrange(testpoints)
</syntaxhighlight>{{out}}
<pre>
Lagrange(testpoints) = Polynomial(-21//1 + 215//6*x - 16//1*x^2 + 13//6*x^3)
</pre>
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