3次の自然スプライン

cf. スプライン関数入門, 桜井明編著, 東京電機大学出版局

　

(t₀, x₀), (t₁, x₁), ..., (t_n, x_n) : データ点

S_i(t) : [t_i, t_i+1] における3次の自然スプライン. S_i(t_i) = x_i, S_i(t_i+1) = x_i+1

a_i = S"_i(t_i) : 2階微分

h_i = t_i+1 - t_i

u = (t - t_i) / h_i : 正規化 dt = h_i du

S"_i(u) = a_i (1-u) + a_i+1 u

S_i(u) = h_i²a_i/6(1-u)³ + h_i²a_i+1/6 u³ + (x_i - h_i²a_i/6) (1-u) + (x_i+1 - h_i²a_i+1/6) u

S'_i(u) = - h_ia_i/2(1-u)² + h_ia_i+1/2 u² + (x_i+1 - x_i) / h_i - h_i/6 (a_i+1 - a_i)

S'_i(0) = S'_i-1(1)

h_i-1a_i-1 + 2 (h_i-1+h_i)a_i + h_ia_i+1 = d_i

(i = 1, ..., n-1)

d_i = 6 ((x_i+1 - x_i) / h_i - (x_i - x_i-1) / h_i-1)

a₀ = a_n = 0

a_i-1 = ρ_ia_i + τ_i

ρ_i+1 = - h_i / (h_i-1ρ_i + 2h_i + 2h_i-1)

τ_i+1 = (d_i - h_i-1τ_i) / (h_i-1ρ_i + 2h_i + 2h_i-1)

ρ₁ = τ₁ = 0