Numerical Methods

  1. Linear Systems of Equations
    1. Linear Algebra Review
      1. Vector Spaces
        1. Matrix Operations
          1. Addition
            1. Multiplication
              1. Transpose
                1. Inverse
                2. Determinants
                  1. Calculation Methods
                  2. Matrix Norms
                    1. Vector Norms
                      1. Matrix Norms
                        1. Condition Numbers
                      2. Direct Solution Methods
                        1. Gaussian Elimination
                          1. Forward Elimination
                            1. Back Substitution
                              1. Pivoting Strategies
                                1. Partial Pivoting
                                  1. Complete Pivoting
                                    1. Scaled Partial Pivoting
                                    2. Operation Count
                                    3. Gauss-Jordan Elimination
                                      1. Reduced Row Echelon Form
                                        1. Matrix Inversion
                                          1. Computational Efficiency
                                          2. LU Decomposition
                                            1. Doolittle Method
                                              1. Crout Method
                                                1. Pivoting in LU
                                                  1. Forward Substitution
                                                    1. Backward Substitution
                                                    2. Cholesky Decomposition
                                                      1. Symmetric Positive Definite Matrices
                                                        1. Algorithm Implementation
                                                          1. Computational Advantages
                                                          2. Special Matrix Systems
                                                            1. Tridiagonal Systems
                                                              1. Thomas Algorithm
                                                                1. Storage Requirements
                                                                2. Banded Systems
                                                                  1. Storage Schemes
                                                                    1. Solution Algorithms
                                                                3. Iterative Solution Methods
                                                                  1. Jacobi Method
                                                                    1. Algorithm Description
                                                                      1. Matrix Formulation
                                                                        1. Convergence Criteria
                                                                        2. Gauss-Seidel Method
                                                                          1. Algorithm Description
                                                                            1. Convergence Analysis
                                                                              1. Comparison with Jacobi
                                                                              2. Successive Over-Relaxation
                                                                                1. Relaxation Parameter
                                                                                  1. Optimal Parameter Selection
                                                                                    1. Convergence Properties
                                                                                    2. Convergence Theory
                                                                                      1. Diagonal Dominance
                                                                                        1. Spectral Radius
                                                                                          1. Convergence Conditions
                                                                                            1. Stopping Criteria

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                                                                                        3. Root Finding Methods

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                                                                                        5. Interpolation and Approximation