Lagrange Interpolation Calculator

Reminder: what Lagrange interpolation is

Suppose you have n + 1 distinct data points . There exists a unique polynomial P(x) of degree at most n such that:

P(x_i) = y_i for every index i = 0, 1, …, n.

Lagrange interpolation expresses this polynomial as a sum of Lagrange basis polynomials that are constructed directly from the x-values of your data. Each basis polynomial is equal to 1 at one data point and 0 at all the others, so their weighted sum reproduces all the y-values exactly.

Lagrange interpolation formulas

The interpolating polynomial is written as

P(x) = ∑i=0n yi Li(x),

where the Lagrange basis polynomials are

Li(x) = ∏j=0, j ≠ in (x − xj)/(xi − xj).

Each Li(x) satisfies Li(xi) = 1 and Li(xk) = 0 for all k ≠ i, so the sum P(x) = ∑ yiLi(x) passes exactly through all points.

The same idea in MathML form is:

The calculator implements these products numerically. For a moderate number of points this is efficient and avoids solving a separate linear system for the coefficients.

How to enter data into the calculator

To use the Lagrange interpolation calculator effectively, follow these steps:

1. List your data points.
   • Enter one point per line in the format x,y.
   • You may use integers or decimals (for example 0,1, 1.5,2.75).
   • The x-values must all be distinct (no duplicates).
2. Choose an evaluation x-value.
   • Enter the value of x where you want to evaluate the interpolating polynomial.
   • This can be inside the range of your data (interpolation) or outside (extrapolation), but see the limitations below.
3. Click the Interpolate button.
   • The script parses your points and constructs the Lagrange basis polynomials.
   • It forms P(x) = ∑ yiLi(x) and evaluates it at the chosen x.
   • If any point is invalid or there are duplicate x-values, an error message is shown instead of a result.

For numerical stability and interpretability, many practical problems work well with between 3 and 10 points. Very high-degree interpolating polynomials can behave unpredictably.

Interpreting the results

After running the calculator, you will typically see:

• An expression for the interpolating polynomial. This may be shown in a Lagrange-like factored form or as a simplified polynomial. You can copy this expression into other software or into a symbolic algebra system to plot or differentiate it.
• The interpolated value at your chosen x. This numeric value is P(xeval), where xeval is the x-value you entered. If this x is equal to one of your original x-values, the result should match the corresponding y-value (up to rounding error).

You can check consistency in several ways:

• Verify that plugging each original x-value into the polynomial reproduces your input y-values.
• Plot the polynomial and the original points in an external tool to confirm that the curve passes through all points.
• Compare interpolated values obtained from this calculator with known or theoretical values when available.

Worked example

Consider three data points:

• (0, 1)
• (1, 2)
• (2, 0)

We want the polynomial P(x) of degree at most 2 that passes through all three points, and we will evaluate it at x = 0.5.

Step 1: Enter the data

In the data points field, type:

0,1
1,2
2,0
  

Then set the evaluation x-value to 0.5 and click Interpolate.

Step 2: Form the Lagrange basis polynomials

For three points we have indices i = 0, 1, 2:

• L0(x) = (x - 1)(x - 2) / ((0 - 1)(0 - 2)) = (x - 1)(x - 2) / 2
• L1(x) = (x - 0)(x - 2) / ((1 - 0)(1 - 2)) = -x(x - 2)
• L2(x) = (x - 0)(x - 1) / ((2 - 0)(2 - 1)) = x(x - 1) / 2

The polynomial is then

P(x) = 1 × L0(x) + 2 × L1(x) + 0 × L2(x).

Substituting the expressions for L0 and L1 and simplifying gives:

P(x) = -½x2 + &frac32;x + 1.

Step 3: Evaluate at x = 0.5

Now compute:

P(0.5) = -½(0.5)2 + &frac32;(0.5) + 1 = -0.125 + 0.75 + 1 = 1.625.

The calculator will display this value as the interpolated result at x = 0.5. It will also show the symbolic polynomial, which you can compare with the analytic expression above.

When to use Lagrange interpolation vs. other approaches

Lagrange interpolation is one of several ways to construct a curve through data points. The table below contrasts it with a few common alternatives at a high level.

This calculator focuses specifically on the Lagrange form, which is especially suitable for education, small sets of experimental data, and situations where you want a clear analytical expression for the interpolating polynomial.

Assumptions, limitations, and good practices

The Lagrange interpolation implemented here rests on several important assumptions:

• Distinct x-values. All input data points must have different x-coordinates. If two points share the same x but different y-values, no single-valued function can pass through both.
• Exact polynomial fit. The method assumes that your data are samples of (or can be reasonably approximated by) a single polynomial of degree at most n.
• Finite, moderate number of points. The implementation is designed for a moderate number of points (for example, up to a few dozen). Very large sets can be slow and numerically unstable.

Keep these practical limitations in mind:

• Runge’s phenomenon and oscillations. Using a high-degree polynomial through many points, especially evenly spaced ones, can cause large oscillations between points. A visually smooth curve is not guaranteed.
• Extrapolation risk. Evaluating the polynomial far outside the range of your data can give unreliable results. Interpolation (inside the range of x-values) is usually much safer than extrapolation.
• Sensitivity to noise. If your data are noisy measurements, a polynomial that passes exactly through every point may amplify measurement error instead of smoothing it. Curve fitting or regression may be more appropriate in that case.
• Floating-point arithmetic. Results are computed with standard double-precision floating-point arithmetic. For extreme scales or very ill-conditioned point sets, round-off error can appear in the coefficients or evaluated values.

Some guidelines for using this calculator responsibly:

• Prefer a small, representative set of points rather than many closely spaced samples.
• Use interpolation primarily inside the convex hull of your x-values.
• Cross-check important results with analytical formulas, experimental data, or alternative numerical methods when possible.

Typical application scenarios

This Lagrange interpolation calculator can support a range of tasks:

• Physics and engineering labs. Interpolate between a few measured operating points to estimate a value that was not measured directly, and compare with theoretical predictions.
• Numerical analysis coursework. Demonstrate how polynomial interpolation works, verify manual computations, and explore the impact of adding or removing data points on the interpolating polynomial.
• Computer graphics and animation. Fit a simple curve through control points to prototype trajectories, then compare with more advanced spline-based methods.
• General data exploration. Obtain a smooth analytical curve through a small set of key samples when you need symbolic expressions for further calculus operations.

By understanding both the capabilities and the limitations described above, you can use the Lagrange Interpolation Calculator as a precise and insightful tool for fitting polynomials to data points.