Spline Interpolation Calculator

Interpolate between data points using natural cubic splines and evaluate at a given x.

About Spline Interpolation Calculator

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

Worked example

With the default values below, Spline Interpolation Calculator returns S(x) of 6.23214286. Change any input above and every figure here is recalculated in your browser.

Inputs used

x values (comma-separated): 0, 1, 2, 3, 4
y values (comma-separated): 0, 1, 4, 9, 16
Evaluate at x: 2.5

Results

S(x): 6.23214286
Segment used: [2, 3] (segment 3 of 4)

Step by step

1Values used
2Cubic spline on segment i
3S(x)= 6.23214286
4Segment used= [2, 3] (segment 3 of 4)

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

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.