42.4 Error Analysis and Stability

Finite‑precision arithmetic accumulates error. Understand where it arises and how to mitigate it.

42.4.1 Sources of Error

• Rounding in floating‑point operations
• Catastrophic cancellation (subtracting nearly equal numbers)
• Accumulation across long sums/products

42.4.2 Mitigation Techniques

• Kahan summation for dot products
• Reorder operations to improve conditioning (e.g., sum small to large)
• Scale/normalize inputs to avoid extreme magnitudes

42.4.3 Validation Metrics

• Relative/absolute error
• Unit in last place (ULP) differences
• Residual checks for linear systems

float ulpDiff(float a, float b) { return Math.ulp(Math.max(Math.abs(a), Math.abs(b))); }