Numerical Methods

  1. Interpolation and Approximation
    1. Interpolation Problem
      1. Problem Definition
        1. Existence and Uniqueness
        2. Polynomial Interpolation
          1. Vandermonde Matrix Approach
            1. Matrix Construction
              1. Numerical Stability
                1. Computational Complexity
                2. Lagrange Interpolation
                  1. Lagrange Basis Functions
                    1. Formula Derivation
                      1. Computational Considerations
                      2. Newton Interpolation
                        1. Divided Differences
                          1. Newton's Formula
                            1. Recursive Construction
                              1. Computational Advantages
                              2. Polynomial Evaluation
                                1. Horner's Method
                                  1. Nested Multiplication
                                    1. Efficiency Analysis
                                  2. Interpolation Error Analysis
                                    1. Error Formula
                                      1. Error Bounds
                                        1. Runge's Phenomenon
                                          1. High-Degree Polynomials
                                            1. Oscillatory Behavior
                                              1. Remedies
                                            2. Piecewise Interpolation
                                              1. Piecewise Linear Interpolation
                                                1. Construction
                                                  1. Continuity Properties
                                                    1. Error Analysis
                                                    2. Spline Interpolation
                                                      1. Cubic Splines
                                                        1. Continuity Conditions
                                                          1. Smoothness Requirements
                                                            1. Tridiagonal System
                                                            2. Boundary Conditions
                                                              1. Natural Splines
                                                                1. Clamped Splines
                                                                  1. Not-a-Knot Splines
                                                                  2. B-Splines
                                                                    1. Basis Functions
                                                                      1. Local Support
                                                                        1. Computational Advantages

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                                                                  4. Linear Systems of Equations

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                                                                  6. Curve Fitting and Least Squares