Solving Linear Systems

Direct Methods

  • Gaussian Elimination: Row reduction and backsubstitution
  • LU Decomposition: Efficient for multiple right-hand sides
  • Cholesky Decomposition: For symmetric positive-definite matrices
  • QR Decomposition: Numerically stable

Iterative Methods

  • Jacobi Iteration: Simple but slow
  • Gauss-Seidel: Improved convergence
  • SOR (Successive Over-Relaxation): Acceleration technique
  • Conjugate Gradient: For symmetric positive-definite systems

Matrix Properties

  • Condition number
  • Matrix norms
  • Ill-conditioned systems

Applications

  • Finite element analysis
  • Image processing
  • Machine learning optimization

Numerical Stability

  • Pivoting strategies
  • Error propagation
  • Algorithm selection