Numerical Methods

  1. Foundational Concepts in Numerical Computation
    1. Computer Number Representation
      1. Integer Representation
        1. Signed Integers
          1. Unsigned Integers
            1. Integer Overflow
              1. Integer Underflow
              2. Floating-Point Representation
                1. IEEE 754 Standard
                  1. Sign Bit
                    1. Exponent Field
                      1. Mantissa
                        1. Normalization
                          1. Special Values
                            1. Not-a-Number
                              1. Positive Infinity
                                1. Negative Infinity
                                  1. Denormalized Numbers
                                  2. Range and Precision
                                  3. Machine Epsilon
                                    1. Calculation Methods
                                      1. Significance in Computation
                                    2. Sources of Error
                                      1. Modeling Errors
                                        1. Incomplete Models
                                          1. Oversimplified Models
                                            1. Incorrect Assumptions
                                            2. Human Errors
                                              1. Programming Mistakes
                                                1. Data Entry Errors
                                                  1. Conceptual Errors
                                                  2. Data Uncertainty
                                                    1. Measurement Errors
                                                      1. Experimental Uncertainty
                                                        1. Uncertainty Propagation
                                                        2. Truncation Error
                                                          1. Series Truncation
                                                            1. Discretization Error
                                                              1. Approximation Error
                                                              2. Round-off Error
                                                                1. Floating-Point Limitations
                                                                  1. Arithmetic Operations
                                                                    1. Error Accumulation
                                                                  2. Error Analysis
                                                                    1. Error Definitions
                                                                      1. Absolute Error
                                                                        1. Relative Error
                                                                          1. Percentage Error
                                                                          2. Error Propagation
                                                                            1. Addition and Subtraction
                                                                              1. Multiplication and Division
                                                                                1. Function Evaluation
                                                                                  1. Chain Rule Applications
                                                                                  2. Condition Numbers
                                                                                    1. Problem Conditioning
                                                                                      1. Well-Conditioned Problems
                                                                                        1. Ill-Conditioned Problems
                                                                                          1. Condition Number Calculation
                                                                                          2. Algorithm Stability
                                                                                            1. Forward Stability
                                                                                              1. Backward Stability
                                                                                                1. Numerical Stability
                                                                                                  1. Stability Analysis
                                                                                                2. Convergence Theory
                                                                                                  1. Convergence Definitions
                                                                                                    1. Convergence Rates
                                                                                                      1. Linear Convergence
                                                                                                        1. Quadratic Convergence
                                                                                                          1. Superlinear Convergence
                                                                                                            1. Order of Convergence
                                                                                                            2. Stopping Criteria
                                                                                                              1. Absolute Tolerance
                                                                                                                1. Relative Tolerance
                                                                                                                  1. Maximum Iterations
                                                                                                                    1. Residual-Based Criteria

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                                                                                                                1. Introduction to Numerical Methods

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                                                                                                                3. Root Finding Methods