Spline Interpolation Calculator
Interpolate between data points using natural cubic splines and evaluate at a given x.
Inputs
At least 3 distinct x values in ascending order.
Corresponding y values.
S(x)
6.23214286
Segment used
[2, 3] (segment 3 of 4)
Step by step
Values used
x values (comma-separated) = 0, 1, 2, 3, 4; y values (comma-separated) = 0, 1, 4, 9, 16; Evaluate at x = 2.50
Cubic spline on segment i
Sᵢ(x) = aᵢ + bᵢ(x-xᵢ) + cᵢ(x-xᵢ)² + dᵢ(x-xᵢ)³
S(x)
= 6.23214286
Segment used
= [2, 3] (segment 3 of 4)
How it works
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.
Formula
Cubic spline on segment i
Sᵢ(x) = aᵢ + bᵢ(x-xᵢ) + cᵢ(x-xᵢ)² + dᵢ(x-xᵢ)³
- aᵢ
- yᵢ (interpolation condition)
- bᵢ
- First derivative coefficient
- cᵢ
- Second derivative / 2
- dᵢ
- Third derivative / 6
Frequently Asked Questions
Why use splines instead of a single polynomial?
A single high-degree polynomial can oscillate wildly (Runge's phenomenon). Cubic splines use low-degree pieces that join smoothly, avoiding this problem.
What are natural boundary conditions?
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.
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