AkCalculators

Interpolation — from simple linear guesses to full polynomial fits

Interpolation is the process of estimating values of a function between known data points. Unlike extrapolation (estimating outside the range of known data), interpolation is generally safer and often yields accurate estimates when the function behaves smoothly. Interpolation is ubiquitous: filling missing samples in digital signals, estimating temperature between recorded measurements, generating animation frames, numerical analysis, computational physics and many engineering tasks.

Linear interpolation — the simplest case

Linear interpolation estimates the value at a point x between two known points (x₀, y₀) and (x₁, y₁) by assuming the function is linear on the interval. The formula for the interpolated value y at x is:

y = y₀ + ( (y₁ − y₀) / (x₁ − x₀) ) × (x − x₀)

Geometrically, you draw the straight line through the two known points and read the value at x. Linear interpolation is fast, robust and often sufficient when sampling is dense or the function is near-linear in small intervals.

Polynomial interpolation — exact fit through points

Given n distinct data points (x₀, y₀), ..., (x_{n−1}, y_{n−1}), there exists a unique polynomial P(x) of degree at most n−1 that passes through all points. Two common constructive approaches are Lagrange interpolation and Newton's divided differences. This calculator uses the Lagrange basis to compute coefficients and then expands the result so you can see the polynomial in standard power form:

P(x) = a₀ + a₁ x + a₂ x² + ... + a_{n−1} x^{n−1}

The coefficients aₖ can be computed by summing the contributions of each Lagrange basis polynomial multiplied by the corresponding y-value. Lagrange basis polynomials L_i(x) are defined as:

L_i(x) = ∏_{j ≠ i} (x − x_j) / (x_i − x_j)
P(x) = Σ_i y_i * L_i(x)
        

Each L_i(x) is 1 at x = x_i and 0 at all other x_j, ensuring P(x_i)=y_i. The algebraic expansion produces the coefficients shown in the calculator's "expanded polynomial" output.

Advantages and caveats of polynomial interpolation

Polynomial interpolation is powerful: for smooth functions with few points it can approximate the underlying function very well. However, there are practical caveats:

• Runge's phenomenon: using high-degree polynomials on equally spaced points can lead to oscillations between points — errors grow near the interval ends.
• Overfitting: fitting a very high-degree polynomial to noisy data reproduces noise rather than the true underlying trend.
• Numerical conditioning: Vandermonde systems can be ill-conditioned, causing coefficient inaccuracies. Lagrange or Newton forms are often preferred for stability and incremental computation.

When to use which method

Linear interpolation — use when you have two nearby samples or need a quick, monotone estimate that will not overshoot. It's commonly used in rendering, simple sensor smoothing and quick lookups.

Polynomial interpolation — use when you need an exact polynomial through a small number of data points and the function is expected to be smooth. For larger datasets consider piecewise approaches (splines), least-squares regression for noisy data, or basis functions (Chebyshev polynomials) to reduce oscillation.

Practical examples

Example 1 — Linear: Known temperatures at 10:00 (14°C) and 11:00 (18°C); estimate at 10:30: y = 14 + (18−14)/(11−10) × 0.5 = 16°C.

Example 2 — Polynomial: Given three points from a physics experiment you need the quadratic that fits them to compute acceleration behavior — polynomial interpolation gives an exact quadratic passing through those points, and the expanded coefficients relate directly to kinematic parameters.

Implementation notes (how this calculator works)

This calculator supports fractions as inputs (e.g., 3/4) which are parsed to decimals. In polynomial mode the calculator computes Lagrange basis polynomials as coefficient arrays (multiplies linear factors to produce coefficient vectors), scales them by y_i/(denominator) and sums to get the final coefficient vector. The expanded polynomial is shown with coefficients rounded to your chosen precision. The plot samples P(x) densely across the x-range of the data points to draw a smooth curve.

When interpolation is not appropriate

Avoid high-degree polynomial interpolation for large n with noisy data — prefer spline interpolation (piecewise polynomials) or least-squares regression which reduces sensitivity to noise. If you need monotonic or shape-preserving interpolation, specialized methods like monotone cubic interpolation are better.

Best practices

• Keep data points distinct in x (no repeated x values for polynomial interpolation).
• Use polynomial interpolation for small n (typically n ≤ 6–8). This tool limits n to 6 for stability and clarity.
• Check plots visually — spurious oscillations often reveal overfitting.
• Prefer piecewise linear or spline methods for large datasets.

Interpolation is a flexible toolbox: linear interpolation gives quick estimates, while polynomial interpolation enables exact fits and analytic access to the fitted function. Use the interactive controls here to experiment, inspect the expanded polynomial, and plot results to build intuition.