Model curves from known points with reliable interpolation tools. Inspect coefficients, estimates, and numeric trends. Built for students, analysts, engineers, and precise problem solving.
| x | y | Meaning |
|---|---|---|
| 0 | 1 | Starting sample point |
| 1 | 3 | Second measured value |
| 2 | 2 | Third known coordinate |
| 3 | 5 | Fourth measured value |
Use these four points to estimate the value at x = 1.5 and compare interpolation methods.
P(x) = a0 + a1(x - x0) + a2(x - x0)(x - x1) + ... + an(x - x0)...(x - x[n-1])
The coefficients come from recursive divided differences. This form is efficient when you add one more point later.
P(x) = Σ yi Li(x), where Li(x) = Π[(x - xj) / (xi - xj)] for j ≠ i
Each basis polynomial equals 1 at its own node and 0 at the others, so the curve passes through every data point.
V · c = y
Here V contains powers of x, c contains standard coefficients, and solving the linear system produces the unique interpolating polynomial.
If P(x) = c0 + c1x + c2x² + ... + cnxⁿ, then P'(x) = c1 + 2c2x + ... + ncnxⁿ⁻¹
The definite integral is found from the antiderivative and evaluated between the chosen start and end limits.
1. Enter matching x and y lists. Each x value must be unique.
2. Pick a method. Newton is usually convenient for table-based work.
3. Add a target x to estimate the interpolated y value.
4. Optionally set graph limits and integral bounds for extra analysis.
5. Press Submit. The result appears above the form and below the header.
6. Review the polynomial, derivative, divided differences, residuals, chart, and export files.
It creates one polynomial that passes through all given data points exactly. You can then estimate values between points or study the curve’s derivative and integral.
Repeated x values create division by zero in interpolation formulas and make the polynomial undefined for distinct y values at the same x location.
Newton works well for stepwise tabular work. Lagrange is easy to understand analytically. Vandermonde is useful when you want a direct matrix solution.
Large-degree polynomials may oscillate strongly, especially near interval edges. This is a classic numerical issue called Runge-like behavior.
Residuals compare actual input values with polynomial predictions at the same x points. Exact interpolation should give residuals very close to zero.
Yes, but use caution. Interpolation is most reliable inside the known interval. Extrapolation can grow inaccurate very quickly.
Use interpolation when every point must be matched exactly. Use regression when data contains noise and you want a best-fit trend instead.
The CSV includes coefficients and core result values. The PDF summarizes method, equation, estimates, integral, and dataset values for sharing.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.