p_{AB}

p_{BC}

p_{CD}

p_{DE}

Polynom

Spline_{ggb}

A = (1, 1)
B = (3, 4)
C = (4.74, 1.32)
D = (7.68, 2.52)
A1: x(A)
A2: x(B)
A3: x(C)
A4: x(D)
B1: y(A)
B2: y(B)
B3: y(C)
B4: y(D)
E = (11.56, 4.04)
A5: x(E)
B5: y(E)
h0: x(B) - x(A)
h1: x(C) - x(B)
h2: x(D) - x(C)
h3: x(E) - x(D)
A9: (2 * (h0 + h1))
B9: h1
C9: 0
A10: h1
B10: (2 * (h1 + h2))
C10: h2
A11: 0
B11: h2
C11: (2 * (h2 + h3))
E9: y(C) / h1 - ((1 / h0 + 1 / h1) * y(B)) + y(A) / h0
E10: y(D) / h2 - ((1 / h2 + 1 / h1) * y(C)) + y(B) / h1
E11: y(E) / h3 - ((1 / h3 + 1 / h2) * y(D)) + y(C) / h2
M: {{A9, B9, C9}, {A10, B10, C10}, {A11, B11, C11}}
delta: {{E9}, {E10}, {E11}}
m1: M^(-1)
m2: ((6 * m1) * delta)
b0(x) = 1 - x
b1(x) = (((1 - x) * (x - 2)) * x / 6)
b2(x) = (((x - 1) * (x + 1)) * x / 6)
b3(x) = x
b0AB(x) = b0((x - x(A)) / (x(B) - x(A)))
b1AB(x) = (b1((x - x(A)) / (x(B) - x(A))) * h0^(2))
b3AB(x) = b3((x - x(A)) / (x(B) - x(A)))
b0BC(x) = b0((x - x(B)) / (x(C) - x(B)))
b3BC(x) = b3((x - x(B)) / (x(C) - x(B)))
b0CD(x) = b0((x - x(C)) / (x(D) - x(C)))
b3CD(x) = b3((x - x(C)) / (x(D) - x(C)))
b0DE(x) = b0((x - x(D)) / (x(E) - x(D)))
b3DE(x) = b3((x - x(D)) / (x(E) - x(D)))
b2AB(x) = (b2((x - x(A)) / (x(B) - x(A))) * h0^(2))
b1BC(x) = (b1((x - x(B)) / (x(C) - x(B))) * h1^(2))
b1CD(x) = (b1((x - x(C)) / (x(D) - x(C))) * h2^(2))
b1DE(x) = (b1((x - x(D)) / (x(E) - x(D))) * h3^(2))
b2BC(x) = (b2((x - x(B)) / (x(C) - x(B))) * h1^(2))
b2CD(x) = (b2((x - x(C)) / (x(D) - x(C))) * h2^(2))
b2DE(x) = (b2((x - x(D)) / (x(E) - x(D))) * h3^(2))
c1: m2(1, 1)
c2: m2(2, 1)
c3: m2(3, 1)
pBC(x) = (y(B) * b0BC(x)) + (c1 * b1BC(x)) + (c2 * b2BC(x)) + (y(C) * b3BC(x))
pCD(x) = (y(C) * b0CD(x)) + (c2 * b1CD(x)) + (c3 * b2CD(x)) + (y(D) * b3CD(x))
pDE(x) = (y(D) * b0DE(x)) + (c3 * b1DE(x)) + (y(E) * b3DE(x))
pAB(x) = (y(A) * b0AB(x)) + (c1 * b2AB(x)) + (y(B) * b3AB(x))
m3: (M * m1)
f = Polynomial({A, B, C, D, E})
a = Spline({A, B, C, D, E}, 3)
p2(x) = If[x < x(D) ∧ x > x(C) ∨ x < x(C) ∧ x > x(D), pCD]
p1(x) = If[x < x(C) ∧ x > x(B) ∨ x < x(B) ∧ x > x(C), pBC(x)]
p3(x) = If[x < x(E) ∧ x > x(D) ∨ x < x(D) ∧ x > x(E), pDE]
p0(x) = If[x < x(B) ∧ x > x(A) ∨ x < x(A) ∧ x > x(B), pAB]