Calculator
Example Data Table
| x | y | Use Case |
|---|---|---|
| 0 | 1 | Starting known point |
| 1 | 3 | Second observation |
| 2 | 2 | Middle control point |
| 4 | 5 | Final known point |
Formula Used
Lagrange form:
P(x) = Σ yᵢ Lᵢ(x)
Lᵢ(x) = Π (x - xⱼ) / (xᵢ - xⱼ), where j ≠ i
Newton divided difference form:
P(x) = a₀ + a₁(x - x₀) + a₂(x - x₀)(x - x₁) + ...
a₀ = f[x₀], a₁ = f[x₀,x₁], a₂ = f[x₀,x₁,x₂]
Divided difference:
f[xᵢ,...,xᵢ₊ₖ] = (f[xᵢ₊₁,...,xᵢ₊ₖ] - f[xᵢ,...,xᵢ₊ₖ₋₁]) / (xᵢ₊ₖ - xᵢ)
How to Use This Calculator
Enter your known data points in the text box. Use one point per line. Separate each x and y value with a comma, space, or tab. Choose the interpolation method. Enter the x value where you want an estimated y value. Select the precision level. Submit the form. The result appears above the form and below the header section. Use the coefficient table to inspect the generated polynomial. Use the divided difference table to audit the Newton method. Download the result as CSV for spreadsheets or PDF for sharing.
Interpolation Polynomial Calculator Guide
Purpose of Polynomial Interpolation
Polynomial interpolation builds a curve through known data points. The curve passes exactly through every supplied point when the x values are unique. This makes it useful for estimation, missing value analysis, numerical methods, calibration, and classroom exercises. The calculator converts scattered points into a readable polynomial.
Why This Tool Is Useful
Manual interpolation can be slow. It also becomes error prone when four or more points are used. This calculator handles the arithmetic and displays each important result. You can view the final polynomial, estimated value, coefficients, residual checks, and divided differences. The layout keeps the process clear.
Supported Methods
The Newton divided difference method is efficient for building a polynomial step by step. It is helpful when new points may be added later. The Lagrange method builds basis polynomials around every point. It is direct and easy to compare with textbook formulas. Both methods create the same polynomial when the same valid data is used.
Understanding the Output
The polynomial is shown in standard power form. Coefficients are listed from constant term upward. The point check table evaluates the polynomial at every original x value. Residuals should be near zero. Small residuals may appear because decimal arithmetic has limited precision. The derivative and integral add extra insight for advanced study.
Best Practices
Use unique x values. Keep points ordered when possible. Avoid using too many points unless the data is smooth. High degree interpolation can oscillate between points. This is common near the ends of the interval. For noisy measurements, regression may be better than exact interpolation. For trusted samples, interpolation is a strong estimating method.
Exporting Results
The CSV file is useful for spreadsheets, reports, and further analysis. The PDF file gives a compact summary of the method, polynomial, estimate, and points. These downloads help preserve calculations for teaching, homework, technical notes, and review records.
FAQs
What is an interpolation polynomial?
It is a polynomial that passes through a given set of data points. Each unique x value has one matching y value on the curve.
How many points are required?
At least two points are required. Two points create a line. Three points can create a quadratic polynomial.
Can x values repeat?
No. Repeated x values create division by zero in interpolation formulas. Each x value must be unique.
Which method should I choose?
Use Newton divided differences for stepwise tables. Use Lagrange basis when you want a direct textbook style formula.
Why are residuals not always zero?
Very tiny residuals can appear because computers round decimal values. They are usually harmless when close to zero.
Can this estimate values outside the point range?
Yes, but outside estimates are extrapolation. They can be unstable, especially with high degree polynomials.
What does the coefficient table mean?
It lists the polynomial coefficients by power. The constant term is x^0, followed by x^1, x^2, and higher powers.
Is interpolation the same as regression?
No. Interpolation passes exactly through points. Regression fits a trend and usually does not pass through every point.