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Numerical Methods - Overview

Introduction​

Overview of computational aspects of linear algebra: norms, conditioning, and iterative methods.

Key Concepts​

Matrix Norms​

  • Vector norms: ||x||₁, ||x||₂, ||x||∞
  • Matrix norms
  • Induced norms
  • Frobenius norm

Condition Numbers​

  • κ(A) = ||A|| ||A⁻¹||
  • Ill-conditioned vs well-conditioned matrices
  • Impact on numerical accuracy
  • Relationship to eigenvalues

Eigenvalue Computation​

  • Power method
  • QR algorithm
  • Jacobi method
  • Applications to large-scale problems

Iterative Methods​

  • Jacobi iteration
  • Gauss-Seidel
  • Conjugate gradient
  • When to use iterative vs direct methods

Applications​

  • Large-scale linear systems
  • Sparse matrices
  • Machine learning optimization
  • Scientific computing

References​