Interpolate between data points using natural cubic splines and evaluate at a given x.
Natural cubic spline interpolation fits a piecewise cubic polynomial through the data points with continuous first and second derivatives (C² smoothness). The 'natural' boundary condition sets the second derivative to zero at both endpoints. This avoids the oscillation problems of high-degree polynomial interpolation while giving a smooth curve.
Cubic spline on segment i
Sᵢ(x) = aᵢ + bᵢ(x-xᵢ) + cᵢ(x-xᵢ)² + dᵢ(x-xᵢ)³
A single high-degree polynomial can oscillate wildly (Runge's phenomenon). Cubic splines use low-degree pieces that join smoothly, avoiding this problem.
Natural splines set S''(x₀) = S''(xₙ) = 0, meaning the curve is straight (zero curvature) at the endpoints. Other options include clamped (specified slopes) or not-a-knot conditions.