Interpolation Calculator
Find values between data points using linear, Lagrange, and Newton interpolation methods. Our calculator provides step-by-step solutions for numerical analysis, data fitting, and polynomial interpolation with detailed mathematical explanations.
Last updated: December 15, 2024
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Interpolation Result
Interpolated Value:
Using lagrange interpolation with 4 data points
Method Details:
Interpolating Polynomial:
Data Points Used:
Quick Example Result
For data points (0,1), (1,4), (2,9), (3,16) interpolating at x = 1.5:
f(1.5) = 6.25
How This Calculator Works
Our interpolation calculator uses proven numerical methods to estimate values between known data points. The process involves constructing polynomials that pass through your data points and evaluating them at the target location. Each method offers different advantages for various types of data and accuracy requirements.
Interpolation Methods
Linear Interpolation:
f(x) = y₀ + (y₁ - y₀) × (x - x₀) / (x₁ - x₀)Lagrange Interpolation:
P(x) = Σ yᵢ × Πⱼ≠ᵢ (x - xⱼ) / (xᵢ - xⱼ)Newton's Method:
P(x) = f[x₀] + f[x₀,x₁](x-x₀) + f[x₀,x₁,x₂](x-x₀)(x-x₁) + ...Shows how different methods fit curves through data points
Mathematical Foundation
Interpolation is based on the Weierstrass approximation theorem, which states that any continuous function can be uniformly approximated by polynomials. Each method constructs a unique polynomial of degree n-1 that passes through n data points, providing different computational advantages and numerical properties.
- Linear interpolation provides first-order accuracy with minimal computation
- Lagrange method offers explicit polynomial form but can be numerically unstable
- Newton's divided differences allow efficient addition of new data points
- Higher-degree polynomials can suffer from Runge's phenomenon with many points
Sources & References
- Numerical Recipes - Press, Teukolsky, Vetterling, and FlanneryComprehensive reference for numerical interpolation methods
- Burden & Faires - Numerical Analysis TextbookStandard academic reference for interpolation theory
- NIST Handbook - Mathematical Functions ReferenceGovernment standard for numerical computation methods
Need help with other numerical calculations? Check out our derivative calculator and quadratic formula calculator.
Get Custom Calculator for Your BusinessExample Calculation
Given Data Points:
- Point 1: (0, 1)
- Point 2: (1, 4)
- Point 3: (2, 9)
- Point 4: (3, 16)
- Target X: 1.5
Calculation Steps:
- Identify data points and target value
- Apply Lagrange interpolation formula
- Calculate each Lagrange basis polynomial
- Sum weighted contributions: f(1.5) = 6.25
Result: f(1.5) = 6.25
This interpolated value estimates the function at x = 1.5 based on the given data points.
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