1. What is Lagrange Interpolation?

Lagrange interpolation is a method of polynomial interpolation that finds the polynomial of least degree that passes through a given set of points. It's particularly useful for estimating values between known data points.

2. How Does the Calculator Work?

The calculator uses the Lagrange interpolation formula:

Where:

• \( y_i \) — y-coordinate of the i-th data point
• \( l_i(x) \) — Lagrange basis polynomial for the i-th point
• \( l_i(x) = \prod_{j=0, j\neq i}^{n} \frac{x - x_j}{x_i - x_j} \)

Explanation: The method constructs a polynomial that exactly passes through all given data points and uses it to estimate values at other x-coordinates.

3. Importance of Lagrange Interpolation

Details: Lagrange interpolation is widely used in numerical analysis, computer graphics, and scientific computing for curve fitting, data approximation, and function estimation when analytical solutions are not available.

4. Using the Calculator

Tips: Enter data points as x,y pairs separated by semicolons (e.g., "1,2; 3,4; 5,6"). Provide the x-value where you want to estimate the function value. Ensure x-values are distinct to avoid division by zero.

5. Frequently Asked Questions (FAQ)

Q1: What are the limitations of Lagrange interpolation?
A: The method can suffer from Runge's phenomenon (oscillations at edges) with many equally spaced points and becomes computationally expensive for large datasets.

Q2: How many data points can I use?
A: While theoretically unlimited, practical limitations exist due to computational complexity and numerical instability with high-degree polynomials.

Q3: When should I use Lagrange interpolation?
A: Ideal for small datasets and when you need an exact fit through all points. For larger datasets, consider spline interpolation or least squares fitting.

Q4: Can Lagrange interpolation extrapolate beyond data range?
A: While possible, extrapolation with polynomial interpolation is generally unreliable and can produce large errors outside the data range.

Q5: What's the difference between interpolation and extrapolation?
A: Interpolation estimates values within the range of known data points, while extrapolation estimates values outside this range.