Numerical Methods

  1. Root Finding Methods
    1. Problem Formulation
      1. Definition of Roots
        1. Existence Theorems
          1. Uniqueness Conditions
            1. Multiple Roots
            2. Bracketing Methods
              1. Bisection Method
                1. Algorithm Description
                  1. Convergence Analysis
                    1. Error Bounds
                      1. Implementation Details
                      2. False Position Method
                        1. Algorithm Description
                          1. Linear Interpolation
                            1. Convergence Properties
                              1. Comparison with Bisection
                            2. Open Methods
                              1. Fixed-Point Iteration
                                1. Fixed-Point Theorem
                                  1. Convergence Conditions
                                    1. Contraction Mapping
                                      1. Implementation
                                      2. Newton's Method
                                        1. Derivation
                                          1. Geometric Interpretation
                                            1. Convergence Analysis
                                              1. Derivative Requirements
                                                1. Implementation Issues
                                                2. Secant Method
                                                  1. Finite Difference Approximation
                                                    1. Algorithm Steps
                                                      1. Convergence Rate
                                                        1. Comparison with Newton's Method
                                                        2. Müller's Method
                                                          1. Quadratic Interpolation
                                                            1. Complex Roots
                                                              1. Algorithm Implementation
                                                            2. Method Comparison
                                                              1. Convergence Properties
                                                                1. Computational Efficiency
                                                                  1. Robustness
                                                                    1. Implementation Complexity
                                                                    2. Polynomial Roots
                                                                      1. Horner's Method
                                                                        1. Polynomial Evaluation
                                                                          1. Derivative Calculation
                                                                            1. Efficiency Analysis
                                                                            2. Bairstow's Method
                                                                              1. Quadratic Factorization
                                                                                1. Real Coefficients
                                                                                  1. Complex Root Pairs
                                                                                  2. Deflation Techniques
                                                                                    1. Synthetic Division
                                                                                      1. Error Propagation

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                                                                                  2. Foundational Concepts in Numerical Computation

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                                                                                  4. Linear Systems of Equations