Interpolation Methods

Estimate values between known data points using polynomial interpolation. These methods help approximate functions from discrete data sets.

Newton Forward Difference

Forward difference interpolation for equally spaced data points

Complexity:O(n²)
Accuracy:High for beginning intervals
Features:
  • Forward differences
  • Equally spaced data
  • Good for extrapolation
Formula:
P(x) = f(x₀) + uΔf(x₀) + u(u-1)/2!Δ²f(x₀) + ...
Best For:

Interpolating near the beginning of data

Use Newton Forward Difference
Newton Backward Difference

Backward difference interpolation for equally spaced data points

Complexity:O(n²)
Accuracy:High for ending intervals
Features:
  • Backward differences
  • Equally spaced data
  • Good for extrapolation
Formula:
P(x) = f(xₙ) + u∇f(xₙ) + u(u+1)/2!∇²f(xₙ) + ...
Best For:

Interpolating near the end of data

Use Newton Backward Difference
Lagrange Interpolation

Polynomial interpolation for unequally spaced data points

Complexity:O(n²)
Accuracy:Exact for polynomials
Features:
  • Unequal spacing
  • Direct formula
  • No difference table
Formula:
P(x) = Σ yᵢ ∏(x-xⱼ)/(xᵢ-xⱼ) for j≠i
Best For:

General interpolation problems

Use Lagrange Interpolation
When to Use Each Method

Newton Forward

  • • Equally spaced data points
  • • Interpolation near beginning
  • • Forward extrapolation

Newton Backward

  • • Equally spaced data points
  • • Interpolation near end
  • • Backward extrapolation

Lagrange

  • • Unequally spaced data
  • • General interpolation
  • • Direct calculation
About Interpolation

Interpolation constructs a polynomial \( P(x) \) that passes through all given data points \( (x_0, y_0), (x_1, y_1), ..., (x_n, y_n) \). Applications include:

  • Data analysis: Filling gaps in experimental data
  • Computer graphics: Smooth curves and surfaces
  • Signal processing: Upsampling and reconstruction
  • Scientific computing: Function approximation