Enter up to 6 known (x, y) coordinate pairs and a target x value to interpolate. The Polynomial Interpolation Calculator fits a polynomial through your data points using Newton's divided differences method and returns the interpolated y value, the polynomial degree, and a visual curve of the fitted polynomial.
Interpolated Y Value
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Polynomial Degree
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Data Points Used
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Polynomial Expression (approx.)
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Polynomial interpolation is a method of finding a polynomial function that passes exactly through a given set of data points. Given n+1 points, a unique polynomial of degree at most n exists that fits all of them — this polynomial can then be used to estimate values between (or beyond) the known data.
Both Lagrange and Newton methods find the same unique interpolating polynomial but express it differently. Lagrange uses basis polynomials constructed directly from the data points. Newton's divided differences method builds the polynomial incrementally, making it computationally efficient when you need to add more points. This calculator uses Newton's method.
Linear interpolation connects two known points with a straight line and estimates values along that line. Polynomial interpolation uses three or more points to fit a curved polynomial, which can capture nonlinear relationships in the data much more accurately.
Enter at least three (x, y) coordinate pairs using the X0/Y0, X1/Y1, and X2/Y2 fields. You can add up to six points using the optional fields X3–X5. All x-values must be distinct (no two x-values can be the same). Then enter the x-value you want to interpolate in the 'X Value to Interpolate' field.
High-degree polynomials can suffer from Runge's phenomenon — wild oscillations between data points, especially near the edges of the interval. Using more than 6–7 data points with this method can produce unreliable results. For large datasets, spline interpolation is often preferred.
Yes, you can enter an x-value outside the range of your data points, but extrapolation is inherently less reliable than interpolation. Polynomial functions can behave unpredictably beyond the known data range, so extrapolated results should be treated with caution.
Newton's divided differences is an algorithm that computes the coefficients of the interpolating polynomial in Newton's forward form. It works by building a triangular table of differences from the (x, y) data points. The diagonal entries become the polynomial coefficients, and the result can be evaluated at any x using nested multiplication (Horner's method).
Polynomial interpolation is widely used in numerical analysis, engineering, computer graphics (curve design), economics (trend estimation), and scientific computing. It's useful any time you have discrete data points and need to estimate values at unobserved locations.