Spline Interpolation Calculator

Calculator Inputs

Data Points (x, y)

Enter one pair per line. Example: 0, 1
x values must be strictly increasing.

Query x Values

Separate values with commas, spaces, or new lines. Leave blank to auto-test interval midpoints.

Spline Method

Start Slope f'(x₀)

End Slope f'(xₙ)

Graph Sample Points

Decimal Places

Allow end-segment extrapolation for out-of-range queries

Example Data Table

x	y
0	1.0
1	2.2
2	1.8
3	3.6
4	3.2

This sample set is preloaded in the form and works well for testing natural, clamped, and linear spline behavior.

Formula Used

1) Cubic spline on each interval

For each interval [x_i, x_i+1], the calculator uses S_i(x) = a_i + b_i(x - x_i) + c_i(x - x_i)² + d_i(x - x_i)³.

2) Interval width

h_i = x_i+1 - x_i

3) Interior spline equations

h_i-1M_i-1 + 2(h_i-1 + h_i)M_i + h_iM_i+1 = 6[(y_i+1 - y_i)/h_i - (y_i - y_i-1)/h_i-1]

4) Natural boundary conditions

M₀ = 0 and M_n = 0

5) Clamped boundary conditions

2h₀M₀ + h₀M₁ = 6[(y₁ - y₀)/h₀ - f'(x₀)]
h_n-1M_n-1 + 2h_n-1M_n = 6[f'(x_n) - (y_n - y_n-1)/h_n-1]

6) Coefficient formulas

a_i = y_i
b_i = (y_i+1 - y_i)/h_i - h_i(2M_i + M_i+1)/6
c_i = M_i/2
d_i = (M_i+1 - M_i)/(6h_i)

7) Derivatives

S'_i(x) = b_i + 2c_i(x - x_i) + 3d_i(x - x_i)²
S''_i(x) = 2c_i + 6d_i(x - x_i)

8) Linear spline option

On each interval, the linear method uses S_i(x) = y_i + [(y_i+1 - y_i)/(x_i+1 - x_i)](x - x_i).

How to Use This Calculator

Enter your tabulated data points as one x,y pair per line.
Choose Natural Cubic, Clamped Cubic, or Linear spline.
If you select clamped spline, enter the starting and ending slopes.
Add one or more query x values for interpolation checks.
Set graph sample points and decimal precision as needed.
Press Calculate Spline to show results above the form.
Review interpolated values, coefficients, equations, and second derivatives.
Use the CSV or PDF buttons to download your results.

Frequently Asked Questions

1) What does spline interpolation do?

Spline interpolation builds piecewise curves between known data points. It estimates values between points while keeping the curve smoother than ordinary straight-line interpolation.

2) When should I use natural cubic spline?

Use natural cubic spline when endpoint curvature is unknown or assumed to be minimal. It sets the second derivative to zero at both ends.

3) When is clamped cubic spline better?

Clamped spline is better when you know or can estimate endpoint slopes. It forces the curve to match those boundary gradients, often improving physical realism.

4) What is the linear spline option for?

Linear spline connects each data pair with a straight segment. It is simpler, faster, and useful when smooth curvature is not required.

5) Why must x values be strictly increasing?

Spline formulas depend on interval widths. Repeated or decreasing x values create zero or negative widths, which break the interpolation system.

6) What do the derivatives in the results mean?

The first derivative shows slope at a query point. The second derivative shows curvature. Together they help analyze trend direction and local shape.

7) Can this calculator extrapolate outside the data range?

Yes. Enable extrapolation to use the nearest end segment for outside queries. Keep in mind that extrapolated values are usually less reliable than interpolated ones.

8) What is included in the downloads?

The CSV export includes result tables. The PDF export captures the visible result section, including summary metrics, tables, equations, and the graph.