7. Interpolasi Chevbyshev

Interpolasi Chebyshev melibatkan penggunaan Chebyshev nodes dan polynomial untuk membangun interpolasi polynomial. Interpolasi ini dapat meminimalkan Runge’s phenomenon.

7.1. Chebyshev nodes

Idenya adalah mendistribusikan error secara seragam. Chebyshev nodes dapat dipilih di dalam interval \([a,b]\) dengan

\[\bar{x}_i = \frac{1}{2} (a + b) + \frac{1}{2} (b - a) \cos \left( \frac{2i + 1}{2n + 2} \pi \right), \hspace{1em} i=0,1, \cdots, n\]

using Polynomials
using Plots

function chebyshev_nodes(n, a, b)
    return [(a + b) / 2 + (b - a) / 2 * cos(π * (2*i + 1) / (2 * n + 2) * π) for i in 1:n]
end

chebyshev_nodes (generic function with 1 method)

function poly_Lagrange(f, nodes, x)
    n = length(nodes)
    return sum(f(nodes[i])* prod([(x - nodes[j]) / (nodes[i] - nodes[j]) for j in 1:n if j ≠ i]) for i in 1:n)
end

poly_Lagrange (generic function with 1 method)

runge(z) = 1 / (1 + 25z^2)

runge (generic function with 1 method)

N = 30
x = range(-1, 1, length=N+1)
y = runge.(x);

scatter(x, y, label="titik data",
    xlabel="x",
    ylabel="y"
)

7.2. Pengujian

xx = range(-1, 1, length=1000)
nodes = chebyshev_nodes(N, -1, 1)

yy = [poly_Lagrange(runge, nodes, xx[i]) for i in 1:length(xx)]
yL = [poly_Lagrange(runge, x, xx[i]) for i in 1:length(xx)]
scatter(x, y, label="titik data",
    xlabel="x",
    ylabel="y"
)

plot!(xx, yy, label="interpolasi Chebyshev", linewidth=2, ylimits=(-0.2,1.5))
plot!(xx, yL, label="interpolasi Lagrange", linewidth=2, ylimits=(-0.2,1.5), linestyle=:dash)