645:572 Computational Mathematics II / 640:497 Advanced Computational Mathematics - Spring 2021
Course Outline (Tentative):
  1. Matrices and Systems of Linear Equations
    1. Introduction
    2. Gaussian Elimination
      1. Pivoting Strategy
      2. Triangular Factorization
      3. Inverses and Determinants
    3. Cholesky Factorization
    4. Error Analysis, Norms and Condition Numbers
    5. Iterative Improvement
    6. Iterative Methods
      1. Newton's Method
      2. Fixed-Point Iteration
      3. Iteration/Relaxation Methods
        1. Jacobi
        2. Gauss-Seidel
        3. Successive Overelaxation (SOR)
      4. Gradient and Conjugate Gradient Methods
  2. Matrix Eigenvalue Problem
    1. Introduction
    2. Power Method
    3. Deflation
    4. Inverse Power Method
    5. QR Method
  3. Ordinary Differential Equations (ODEs)
    1. Initial Value Problems (IVPs)
      1. One-step and Multi-step Methods
      2. Stability
    2. Boundary Value Problems (BVPs)
      1. Shooting Methods
      2. Finite Difference Methods
      3. Finite Element Methods