1. What is Polynomial Interpolation?

Definition: Polynomial interpolation is a method of estimating values between known data points using a polynomial function that passes through all the given points.

Purpose: It's used in numerical analysis, engineering, and scientific computing to estimate unknown values from discrete data.

2. How Does the Calculator Work?

The calculator uses the Lagrange interpolation formula:

Where:

• \( y \) — Interpolated value at target x
• \( y_i \) — Known y-values from data points
• \( L_i(x) \) — Lagrange basis polynomials
• \( n \) — Number of data points minus one

Explanation: For each data point, the calculator computes a basis polynomial that is 1 at that point and 0 at all other points, then combines them weighted by the y-values.

3. Importance of Polynomial Interpolation

Details: Provides smooth curve fitting through data points, useful for function approximation, numerical integration, and computer graphics.

4. Using the Calculator

Tips: Enter data points as space-separated x,y pairs (e.g., "1,3 2,5 3,7"). The target x should be within the range of your data points for best results.

5. Frequently Asked Questions (FAQ)

Q1: What's the maximum number of points I can use?
A: While technically unlimited, high-degree polynomials (n > 10) can become numerically unstable (Runge's phenomenon).

Q2: What if my target x is outside my data range?
A: This is called extrapolation and is generally less accurate than interpolation.

Q3: Are there other interpolation methods?
A: Yes, including Newton's divided differences, spline interpolation, and piecewise linear interpolation.

Q4: Why does the polynomial degree matter?
A: Higher degrees can fit more points exactly but may produce unrealistic oscillations between points.

Q5: Can I use this for non-numeric data?
A: No, this method requires numeric x and y values.