Lagrange Polynomial Interpolation Calculator

Estimate missing values from known data points with confidence. Review basis terms and polynomial details. Download clean reports for homework and research tasks today.

Calculator Inputs

Data Points

Formula Used

The calculator uses the Lagrange interpolation polynomial:

P(x) = Σ yiLi(x)

Li(x) = Π (x - xj) / (xi - xj), where j ≠ i

Each basis term equals one at its own data point. It equals zero at every other data point. The weighted sum passes through all entered points.

If a derivative bound is entered, the calculator estimates this error bound:

|R(x)| ≤ M / n! × Π |x - xi|

How to Use This Calculator

  1. Enter the x and y values for each known point.
  2. Make sure every x value is unique.
  3. Enter the target x value where you need the estimate.
  4. Select decimal places for the final output.
  5. Add an optional derivative bound if you know it.
  6. Press Calculate to view the polynomial and basis terms.
  7. Use CSV or PDF export for saving your work.

Example Data Table

x y Purpose
0 1 First known point
1 3 Second known point
2 2 Third known point
3 5 Fourth known point

Using x = 1.5, this data gives an interpolation estimate between the known table values.

About the Lagrange Polynomial Interpolation Calculator

This calculator finds a polynomial through known data points. It then estimates the y value at a chosen x. The method is useful when a table is known, but the function is not. It works without solving a linear system. Each point builds one basis term. Those terms are combined into one interpolation formula.

Why This Method Matters

Lagrange interpolation is common in numerical analysis. It helps engineers, students, and analysts fill gaps in measured data. The method is exact at every supplied point. It is also transparent, because every contribution can be inspected. This tool shows the basis value, the weighted term, and the expanded polynomial coefficients.

Choosing Good Data Points

Good results depend on sensible input points. Points should be close to the target x value. Very distant points may create large swings. Repeated x values are not allowed, because a single x cannot hold two different table positions in one function. Sort order does not matter. The calculator checks duplicates before it computes.

Understanding the Output

The main result is P(x), the interpolated estimate. The degree is one less than the number of points. The expanded polynomial shows coefficients from the constant term upward. The basis table shows how each original point affects the final value. A large positive or negative contribution can warn you about unstable data spacing.

Practical Uses

Use this calculator for tabulated physics data, calibration charts, finance curves, engineering records, and math homework. It is best for interpolation inside the point range. Extrapolation outside the range is possible, but it may be unreliable. Add an optional derivative bound to estimate the theoretical error term.

Best Practice

Start with three or four nearby points. Compare the result after adding another point. If the estimate changes sharply, inspect the data spacing. Dense points near the target usually work better. Keep decimal precision high while calculating. Round only the final answer for reporting.

This page also supports record keeping. The export buttons save the final value, each basis term, and polynomial coefficients. That makes checking easier after a class, lab, or report session. Use the example table first, then replace it with your own data carefully. For better review.

FAQs

What is Lagrange interpolation?

It is a method that builds one polynomial through known data points. The polynomial can estimate a missing value at another x position.

How many points do I need?

You need at least two points. More points can build a higher degree polynomial, but too many wide points may reduce stability.

Can x values repeat?

No. Each x value must be unique. Repeated x values cause division by zero inside the basis formula.

Is this best for interpolation or extrapolation?

It is best for interpolation inside the known x range. Extrapolation outside the range may produce unreliable values.

What does the basis table show?

It shows each Lagrange basis value and its weighted contribution. This helps explain how every point affects the final estimate.

What is the polynomial degree?

The degree is usually one less than the number of valid points. Four points normally create a cubic polynomial.

What is the derivative bound field?

It is optional. If you know a bound for the next derivative, the calculator estimates a theoretical error limit.

Can I export the result?

Yes. Use the CSV or PDF buttons after calculation. They save the final result, basis rows, and coefficients.

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