Lagrange Polynomial Calculator: Interpolation Tool

This calculator builds the Lagrange interpolating polynomial through a set of given points and evaluates it at any x-value you choose. You enter the x- and y-coordinates of your data points, plus the target x, and you get back the interpolated y-value, the full polynomial expression, and a step-by-step breakdown of the calculation.

Formula

The Lagrange interpolation formula constructs a polynomial that passes exactly through every given point \((x_i, y_i)\). For \(n+1\) points, the polynomial is:

\(P(x) = \sum_{i=0}^{n} y_i \cdot L_i(x)\)

where each Lagrange basis polynomial \(L_i(x)\) is:

\(L_i(x) = \prod_{j \neq i} \frac{x – x_j}{x_i – x_j}\)

Here, \(x_i\) and \(y_i\) are the coordinates of the \(i\)-th data point, and the product runs over all other points \(j\). The basis polynomial \(L_i(x)\) equals 1 at \(x = x_i\) and 0 at every other data point, so the weighted sum passes through all points exactly.

Worked Example

Using the default points (0,1), (1,3), and (2,2), we want the interpolated value at \(x = 1.5\).

1. Compute \(L_0(1.5)\): \(\frac{(1.5-1)(1.5-2)}{(0-1)(0-2)} = \frac{0.5 \times (-0.5)}{(-1) \times (-2)} = \frac{-0.25}{2} = -0.125\)
2. Compute \(L_1(1.5)\): \(\frac{(1.5-0)(1.5-2)}{(1-0)(1-2)} = \frac{1.5 \times (-0.5)}{1 \times (-1)} = \frac{-0.75}{-1} = 0.75\)
3. Compute \(L_2(1.5)\): \(\frac{(1.5-0)(1.5-1)}{(2-0)(2-1)} = \frac{1.5 \times 0.5}{2 \times 1} = \frac{0.75}{2} = 0.375\)
4. Multiply each by the corresponding \(y\): \(1 \times (-0.125) + 3 \times 0.75 + 2 \times 0.375 = -0.125 + 2.25 + 0.75 = 2.875\)

So the interpolated value at \(x = 1.5\) is \(2.875\), matching the calculator output.

When to Use Lagrange Interpolation

Lagrange interpolation is ideal when you have a handful of data points from an experiment, sensor readings, or a table and need to estimate values between them. It is especially useful when you want a polynomial that exactly fits the data, without solving a system of equations. Common applications include:

• Estimating intermediate values in a data table (e.g., temperature at a time between measurements)
• Constructing a smooth curve through points in computer graphics
• Approximating a complicated function by a polynomial when you only know its values at certain points
• Numerical integration and differentiation in scientific computing

However, for many data points (more than about 10), other methods like spline interpolation often give more stable results.

How to Choose the Right Interpolation Method

Lagrange interpolation is not always the best choice. Here is a quick guide:

• Lagrange polynomial: Use when you need an exact polynomial fit and have few points (say, up to 10). It is simple to implement but can oscillate wildly at the edges if points are unevenly spaced.
• Newton divided differences: Equivalent to Lagrange but easier to extend when adding new points. Use if you plan to add points incrementally.
• Linear or spline interpolation: Use for many points or when you need smoothness without oscillations. Splines are piecewise polynomials that avoid the Runge phenomenon.
• Least-squares fitting: Use when your data has noise and you want a trend line, not an exact fit.

Common Mistakes and Pitfalls

Several errors are common when using Lagrange interpolation:

• Duplicate x-values: If two points share the same x, the denominator \(x_i – x_j\) becomes zero, and the interpolation fails. Your data must have distinct x-values.
• Extrapolation: Evaluating outside the range of your data points is risky. The polynomial may give wildly inaccurate values, especially with higher-degree polynomials.
• Too many points: A polynomial of degree \(n-1\) through \(n\) points can oscillate severely between points (Runge’s phenomenon). For more than about 10 points, consider splines.
• Unit mistakes: Ensure all x-values are in the same units and all y-values are in the same units. Mixing units (e.g., meters and centimeters) will produce nonsense.
• Rounding too early: Rounding intermediate basis values can shift the final result. Keep full precision until the end.

Limitations of the Lagrange Formula

The Lagrange polynomial is exact at the given points, but it has limitations:

• Numerical instability: For large \(n\), the basis polynomials can become very large, leading to cancellation errors in floating-point arithmetic.
• No error control: The formula gives no built-in estimate of interpolation error. The error depends on the unknown higher derivatives of the underlying function.
• Not suitable for extrapolation: Values outside the data range are often meaningless, as the polynomial behavior is dominated by the highest-degree term.
• Computational cost: Evaluating the polynomial at many points requires \(O(n^2)\) operations, which is slower than Newton’s form for repeated evaluations.

For most practical purposes, if you have more than a handful of points, consider using piecewise interpolation (linear, cubic spline) or a least-squares fit instead.

Frequently Asked Questions

What is the difference between Lagrange and Newton interpolation?

Both produce the same unique polynomial through the given points, but they are constructed differently. Newton’s divided differences allow you to add a new point and update the polynomial without recomputing everything, while Lagrange requires a full recomputation. Lagrange is easier to write in code, but Newton is more efficient when points are added incrementally.

Can I use Lagrange interpolation for extrapolation?

Technically yes, but the results are often unreliable. The polynomial may shoot up or down dramatically outside the data range, especially with higher-degree polynomials. It is safer to use a linear extrapolation or a model based on the underlying physics of your data.

Why does my interpolated value oscillate so much?

This is likely due to Runge’s phenomenon, which occurs with high-degree polynomials on evenly spaced points. The polynomial overshoots near the edges of the interval. Using Chebyshev nodes (unevenly spaced points clustered near the ends) or switching to spline interpolation reduces this effect.

What if I have more than 10 points?

For more than about 10 points, a single high-degree polynomial becomes numerically unstable and oscillatory. Consider using a piecewise method like cubic splines, which fit a separate low-degree polynomial between each pair of points, ensuring smoothness and stability.

Does the order of the points matter?

No, the Lagrange polynomial is independent of the order in which you list the points. The same set of points always yields the same polynomial, regardless of the sequence.

Can Lagrange interpolation handle non-equally spaced points?

Yes, this is one of its strengths. Unlike finite difference methods, Lagrange interpolation works with any spacing of x-values, which makes it useful for irregularly sampled data.

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in xs | x-values (comma-separated) | - | 0, 1, 2 | 0
in ys | y-values (comma-separated) | - | 1, 3, 2 | 0
in x0 | Evaluate at x = | - | 1.5 | 0
out y | Interpolated y at x0 | - | lagrange(xs, ys, x0) | 4
out poly | Polynomial (expanded) | - | lagrange_poly(xs, ys) | 0