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Interpolation Calculator

Estimate values between known data points using linear or polynomial interpolation.

1.x =y =
2.x =y =
3.x =y =

About this tool

Interpolation is a mathematical technique for estimating unknown values that fall between known data points. Whether you're analyzing experimental data, predicting trends, or filling gaps in datasets, this calculator helps you find intermediate values using linear interpolation (for straight-line estimates) or polynomial interpolation (for curved relationships). It's invaluable in science, engineering, finance, and data analysis where precise intermediate values are needed without running expensive additional experiments or measurements.

To use the calculator, enter your known data points as X and Y pairs, then specify the X value where you want to estimate Y. For linear interpolation, the tool finds the straight line between two adjacent points and calculates the Y value directly. For polynomial interpolation, it fits a curve through multiple points to capture more complex relationships. Simply input your data, choose your method, and the calculator instantly provides the interpolated result with step-by-step reasoning.

Linear interpolation works best when your data has a consistent, roughly straight trend; use polynomial interpolation when your data follows a curved or non-linear pattern. Keep your data points sorted by X value for most accurate results, and remember that estimates become less reliable far outside your data range (extrapolation beyond boundaries carries higher uncertainty). This tool is especially useful for researchers, engineers, students, and analysts who work with empirical data.

Frequently Asked Questions

Code Implementation

def linear_interpolate(x1, y1, x2, y2, x):
    """Linear interpolation between two points."""
    if x2 == x1:
        raise ValueError("x1 and x2 must be different")
    return y1 + (x - x1) * (y2 - y1) / (x2 - x1)

def lagrange_interpolate(points, x):
    """Lagrange polynomial interpolation.
    points: list of (x, y) tuples.
    Returns the estimated y value at x.
    """
    n = len(points)
    result = 0.0
    for i in range(n):
        xi, yi = points[i]
        basis = 1.0
        for j in range(n):
            if i != j:
                xj, _ = points[j]
                basis *= (x - xj) / (xi - xj)
        result += yi * basis
    return result

def newton_divided_diff(points):
    """Newton's divided differences table."""
    n = len(points)
    coef = [y for _, y in points]
    xs = [x for x, _ in points]
    for j in range(1, n):
        for i in range(n - 1, j - 1, -1):
            coef[i] = (coef[i] - coef[i - 1]) / (xs[i] - xs[i - j])
    return coef

# Examples
print(linear_interpolate(0, 0, 10, 100, 5))  # 50.0

pts = [(1, 1), (2, 4), (3, 9), (4, 16)]  # y = x²
print(f"Lagrange at x=2.5: {lagrange_interpolate(pts, 2.5):.4f}")  # 6.25

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