Interpolation Error Calculator

Calculator inputs

Enter nodes as one x,y pair per line. The calculator can compare interpolation output with an exact value when available.

Interpolation method

Target x value

Exact y at target x

Polynomial degree

Polynomial methods use degree + 1 nearest nodes.

Max |f⁽ⁿ⁺¹⁾(ξ)| bound

Decimal places

Interpolation nodes

Example format: 0, 1

Example data table

This sample uses values from e^x. Test x = 0.8 with exact value 2.225541.

x	y
0.0	1.000000
0.5	1.648721
1.0	2.718282
1.5	4.481689

Formula used

Lagrange polynomial: P_n(x) = Σ y_iL_i(x), where L_i(x) = ∏[(x − x_j) / (x_i − x_j)] for j ≠ i.

Newton polynomial: P_n(x) = a₀ + a₁(x − x₀) + a₂(x − x₀)(x − x₁) + ...

Linear interpolation: P(x) = y₀ + (x − x₀)[(y₁ − y₀) / (x₁ − x₀)]

Absolute error: E_abs = |f(x) − P_n(x)|

Relative error: E_rel = |f(x) − P_n(x)| / |f(x)|

Percentage error: E_% = E_rel × 100

Error bound: |R_n(x)| ≤ [M / (n + 1)!] × |∏(x − x_i)|, where M bounds |f⁽ⁿ⁺¹⁾(ξ)| on the interval.

The calculator selects the nearest nodes for polynomial methods, or the best local pair for linear interpolation. It then computes the approximation, compares it with the exact value when given, and estimates a theoretical bound when a derivative bound is supplied.

How to use this calculator

Choose an interpolation method.
Enter one x,y node pair per line.
Set the target x where you want the estimate.
Optionally enter the exact y value for true error analysis.
Set the degree for polynomial methods.
Optionally enter a derivative bound for an error estimate.
Click Calculate error to show the result above the form.
Review the graph, diagnostics, and export the results as CSV or PDF.

Frequently asked questions

1. What does interpolation error mean?

Interpolation error is the difference between the exact function value and the value predicted from sampled nodes. Smaller error means the approximation is closer to the true value at the chosen target x.

2. Why is the exact value optional?

Sometimes you only know the nodes and need an estimated value. In that case, the calculator still computes the interpolation result. When an exact value is available, it adds absolute, relative, and percentage error metrics.

3. Which method should I choose?

Use linear for quick local estimates between two nearby points. Use Newton or Lagrange for polynomial interpolation across several nodes. Newton is often easier for repeated updates, while Lagrange is useful for direct formulation.

4. Does a higher degree always improve accuracy?

Not always. A higher degree may reduce local error when data is smooth, but it can also amplify oscillation and sensitivity, especially with unevenly spaced nodes or targets near the ends of the interval.

5. What happens outside the node range?

That becomes extrapolation, not interpolation. Error often grows faster outside the known range because the fitted polynomial or line is no longer anchored by surrounding data on both sides.

6. What is the derivative bound used for?

It estimates a theoretical upper limit for the interpolation remainder. This helps you judge whether the observed or expected error is consistent with the smoothness of the original function.

7. Why do node spacing and selection matter?

Interpolation is usually more stable when nodes are well distributed near the target x. Large gaps or irregular spacing can increase sensitivity and produce larger approximation errors.

8. Can I export both results and sample data?

Yes. The calculator includes CSV and PDF export buttons for the computed results and for the example table, making it easier to document test cases or share outputs.