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Linear Algebra, Optimisation and Numerical Methods / Spectral and factor methods

Numerical Conditioning and Stability

A theoretically unique hedge Ax=bA\mathbf{x}=\mathbf{b} can still be unusable: a 0.01%0.01\% relative error in b\mathbf{b} can become a 100%100\% relative error in x\mathbf{x} when AA is ill-conditioned. The same pathology hits OLS when factors are nearly collinear, and hits Σ−1μ\Sigma^{-1}\boldsymbol\mu when a covariance has a tiny eigenvalue. This note keeps the load-bearing numerical linear algebra — floating-point error, the condition number, perturbation bounds — and reads each through hedges, regressions, and covariances.
Learning objectives. By the end of this note you should be able to:
  • distinguish problem conditioning from algorithmic stability;
  • define κ(A)=σmax⁡/σmin⁡\kappa(A)=\sigma_{\max}/\sigma_{\min} and apply the perturbation bound for Ax=bA\mathbf{x}=\mathbf{b};
  • explain why κ\kappa, not det⁡\det, diagnoses near-singularity;
  • diagnose an ill-conditioned covariance or design matrix on a desk.

In this lesson

  1. Floating-point error (the ambient noise)
  2. Condition numbers
  3. Desk failure modes

Before you start

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