Numerical Methods

Numerical methods are a collection of algorithms used to find approximate numerical solutions to mathematical problems that are difficult or impossible to solve analytically. Instead of deriving an exact symbolic formula, these techniques employ iterative arithmetic operations, typically performed by a computer, to progressively refine an estimate until it reaches a desired level of accuracy. This practical approach is fundamental to modern science and engineering, enabling the solution of complex real-world problems involving differential equations, optimization, large systems of linear equations, and integration.

  1. Introduction to Numerical Methods
    1. Definition and Scope of Numerical Methods
      1. Role of Numerical Methods in Science and Engineering
        1. Applications in Physical Sciences
          1. Applications in Engineering
            1. Applications in Data Science and Economics
            2. Analytical vs. Numerical Solutions
              1. Exact Solutions
                1. Approximate Solutions
                  1. Advantages of Analytical Methods
                    1. Limitations of Analytical Methods
                      1. Advantages of Numerical Methods
                        1. Limitations of Numerical Methods
                        2. Major Problem Classes
                          1. Root Finding
                            1. Systems of Linear Equations
                              1. Optimization
                                1. Curve Fitting and Interpolation
                                  1. Numerical Differentiation
                                    1. Numerical Integration
                                      1. Ordinary Differential Equations
                                        1. Partial Differential Equations
                                          1. Eigenvalue Problems
                                          2. Numerical Computing Process
                                            1. Mathematical Modeling
                                              1. Problem Formulation
                                                1. Model Simplification
                                                  1. Assumptions and Constraints
                                                  2. Algorithm Selection and Development
                                                    1. Selection Criteria
                                                      1. Design Principles
                                                        1. Efficiency Considerations
                                                        2. Implementation and Programming
                                                          1. Programming Languages
                                                            1. Software Libraries
                                                              1. Code Verification
                                                                1. Code Validation
                                                                2. Results Analysis
                                                                  1. Output Interpretation
                                                                    1. Verification Methods
                                                                      1. Sensitivity Analysis