Linear Interpolation

Linear interpolation is the simplest method for estimating values between two known data points. It assumes a straight line relationship between the points.

Applications:

• Engineering calculations
• Financial modeling
• Scientific data analysis
• Computer graphics

Polynomial Interpolation

Polynomial interpolation finds a polynomial that passes exactly through all given data points. It provides more accurate results for curved relationships.

Advantages:

• Exact fit through all points
• Handles non-linear relationships
• Smooth curve generation
• Higher accuracy for complex data

When to Use Each Method

Linear Interpolation is best for:

• Simple, approximately linear relationships
• Quick approximations
• Two data points only
• Real-time calculations

Polynomial Interpolation is best for:

• Complex, curved relationships
• Multiple data points available
• High accuracy requirements
• Smooth curve fitting

Mathematical Concepts

Extrapolation vs Interpolation:

Interpolation estimates values within the range of known data points, while extrapolation estimates values outside this range. Interpolation is generally more reliable.

Degree of Polynomial:

For n data points, the interpolating polynomial has degree at most n-1. Higher degrees can lead to oscillations (Runge’s phenomenon).

Accuracy Considerations:

Linear interpolation provides first-order accuracy, while polynomial interpolation can achieve higher-order accuracy but may be sensitive to data spacing.