Calculation Excel — Spline

[ (x_i - x_i-1) z_i-1 + 2(x_i+1 - x_i-1) z_i + (x_i+1 - x_i) z_i+1 = 6 \left( \fracy_i+1 - y_ix_i+1 - x_i - \fracy_i - y_i-1x_i - x_i-1 \right) ]

Manual solution: From first: z2 = (-9 - 2*z3)/6 Sub into second: 2*[(-9 - 2*z3)/6] + 10*z3 = 9 → (-18 - 4*z3)/6 + 10*z3 = 9 → -3 - (2/3)z3 + 10*z3 = 9 → (28/3)z3 = 12 → z3 = 9/7 ≈ 1.285714 Then z2 = (-9 - 2*(9/7))/6 = (-9 - 18/7)/6 = (-81/7)/6 = -81/42 = -27/14 ≈ -1.92857 spline calculation excel

Equation for i=3: h2*z2 + 2*(h2+h3)*z3 + h3*z4 = 6*(slope3 - slope2) → 2*z2 + 2*(2+3)*z3 + 3*0 = 6*(1 - (-0.5)) → 2*z2 + 10*z3 = 9 [ (x_i - x_i-1) z_i-1 + 2(x_i+1 -

Solve in Excel: Use and MMULT or manual algebra. z_2 = -1.92857

So: (z_1 = 0, z_2 = -1.92857, z_3 = 1.285714, z_4 = 0) For each interval ([x_i, x_i+1]):