• Overview
• Details
• Exercises

Conditioning describes sensitivity of a mathematical problem to input perturbations; stability describes how an algorithm propagates rounding errors. Well-conditioned problems with unstable algorithms still yield inaccurate results.

• Floating-point model: fl(x ⊕ y) = (x ⊕ y)(1 + δ), |δ| ≤ u (unit roundoff).
• Condition number κ = |x f'(x)/f(x)| or matrix κ(A) = ||A||·||A^{-1}||; κ measures relative error amplification.
• Stability: forward, backward, and mixed error analyses. Backward stable algorithm solves a nearby problem.
• Catastrophic cancellation: subtracting close numbers; remedy via algebraic reformulation.
• Examples: solving Ax=b via Gaussian elimination with partial pivoting (GEPP) is usually backward stable; normal equations can be unstable vs QR.
• Practical tips: scaling, pivoting, use orthogonal transformations, avoid forming A^T A explicitly.

1. Estimate κ(A) from singular values and predict relative solution error for Ax=b.
2. Rewrite a quadratic formula evaluation to avoid cancellation when b^2 ≈ 4ac.
3. Compare least-squares via normal equations vs QR on an ill-conditioned example.