Numerical Methods

1. Root-Finding Algorithms
   1. Basic Concepts
      1. Function Zeros
         1. Definition of a root of a function
         2. Types of roots: simple roots, multiple roots
         3. Importance of finding roots in mathematical and real-world contexts
      2. Convergence Criteria
         1. Definition of convergence in root-finding
         2. Stopping criteria: absolute error, relative error, maximum iterations
         3. Role of tolerance levels in convergence
         4. Trade-offs between accuracy and computational effort
   2. Methods
      1. Bisection Method
         1. Principle of the method: intermediate value theorem
         2. Algorithm steps
         3. Advantages and limitations
         4. Convergence properties: monotonic and guaranteed convergence
      2. Newton-Raphson Method
         1. Derivation from tangent line approximation
         2. Algorithm steps
         3. Requirements: derivative calculation
         4. Advantages: quadratic convergence
         5. Limitations: failure in cases of zero derivative, poor initial guesses
         6. Modifications: use of derivative-free variants
      3. Secant Method
         1. Iterative process: using secant line instead of tangent
         2. Algorithm steps
         3. Comparison with Newton-Raphson: no need for derivatives
         4. Convergence properties: superlinear convergence
         5. Limitations: choice of initial guesses, possible divergence
      4. Fixed Point Iteration
         1. Concept: rewriting equation in the form \(x = g(x)\)
         2. Algorithm steps
         3. Convergence: Banach fixed-point theorem
         4. Analysis of rate of convergence
         5. Example transformations for common functions
      5. Regula Falsi Method (False Position Method)
         1. Similarity with bisection and secant methods
         2. Algorithm steps
         3. Characteristics: bracketing method, conserves convergence guarantee
         4. Comparison with other bracketing methods
      6. Brent's Method
         1. Combination of bisection, secant, and inverse quadratic interpolation
         2. Algorithm steps
         3. Merits: robust and efficient, does not require derivatives
         4. Application in complex situations: multiple roots, ill-behaved functions
   3. Algorithm Comparison
      1. Convergence Rate
         1. Defining convergence rate: linear, superlinear, quadratic
         2. Comparative analysis of methods based on speed of convergence
      2. Computational Cost
         1. Metric for measuring efficiency: function evaluations, time complexity
         2. Balancing cost against accuracy
         3. Examples: overhead in calculating derivatives, additional iterations
      3. Stability
         1. Sensitivity to initial conditions and function behavior
         2. Comparison of methods in terms of stability and robustness
         3. Identifying stable vs. unstable methods for specific scenarios
   4. Practical Considerations
      1. Selection of Appropriate Method
         1. Factors influencing choice: type of problem, resources, desired accuracy
         2. Guidelines for selecting an algorithm based on problem characteristics
      2. Handling Multiple Roots
         1. Challenges posed by multiple roots: slow convergence, divergence
         2. Adaptations and modifications to methods
      3. Implementation Issues
         1. Common pitfalls: rounding errors, overflow/underflow
         2. Strategies for robust implementation: rescaling, adaptive algorithms
      4. Case Studies
         1. Application of algorithms to real-world problems
         2. Analysis of performance and outcomes in practical settings