Polynomial Interpolation Tool
Enter your data points as x,y pairs (one per line). The calculator builds the unique interpolating polynomial that passes through every point.
Example format: 1.5, 2.75 (at least 2 points, all x-values must be distinct).
What is Polynomial Interpolation?
Polynomial interpolation is a method of constructing a polynomial that exactly passes through a set of known data points. If you provide n points with distinct x-values, there is one and only one polynomial of degree at most n - 1 that fits all points perfectly.
This is useful when you need a smooth algebraic model of measured data, want to estimate values between known samples, or need an explicit equation for analysis, simulation, or further calculus operations.
How to Use This Calculator
- Enter each point on its own line using the format
x,y. - Choose whether you want to view results in standard (Lagrange-derived) form or Newton form.
- Optionally enter a target x-value to compute
P(x). - Click Calculate Polynomial to generate the polynomial and evaluation output.
Example Input
0,1 1,3 2,2
For this dataset, the calculator finds the exact quadratic polynomial that intersects all three points.
Lagrange vs. Newton Interpolation
Lagrange Form
Lagrange interpolation builds the polynomial as a sum of weighted basis polynomials. It is conceptually straightforward and good for direct formula construction.
Newton Form
Newton interpolation uses divided differences and expresses the polynomial in nested factors like (x - x0)(x - x1). It is often more efficient when adding new points incrementally.
Practical Notes and Limitations
- Distinct x-values are required. Duplicate x-values make interpolation undefined in the standard sense.
- High degree can oscillate. With many points, especially unevenly spaced points, you may see Runge-type oscillations.
- Interpolation vs. regression: interpolation passes through every point exactly, while regression finds a best-fit curve that may not hit every sample.
- Numeric precision: floating-point arithmetic may introduce tiny rounding artifacts for large datasets.
Where This Helps
A polynomial interpolation calculator is commonly used in numerical analysis, engineering data calibration, physics experiments, graphics curve fitting, and educational contexts for understanding approximation methods.