Numerical Methods

1. Interpolation
   1. Basic Concepts
      1. Interpolation vs. Extrapolation
         1. Definition of Interpolation: Estimating values within the range of a discrete set of known data points.
         2. Definition of Extrapolation: Extending a function beyond the range of given data points.
         3. Situations and implications for choosing interpolation over extrapolation due to increased uncertainty in extrapolation.
      2. Polynomial Interpolation
         1. Definition and introduction to polynomial curves passing through a set of points.
         2. Degree of the polynomial related to the number of points.
         3. Runge’s Phenomenon: The problem of oscillation at the edges for higher-degree polynomials.
         4. Piecewise Polynomial Interpolation: Handling of Runge’s Phenomenon by using piecewise lower-degree polynomials.
   2. Methods
      1. Lagrange Interpolation
         1. Introduction to the Lagrange polynomial formula.
         2. Construction of Lagrange basis polynomials.
         3. Computational considerations and complexities.
         4. Advantages and limitations in terms of simplicity vs. computational expense.
      2. Newton's Divided Differences
         1. Explanation of constructing Newton’s interpolation polynomial.
         2. Use of divided differences to recursively calculate polynomial coefficients.
         3. Benefits of Newton's method such as ease of adding more points.
         4. Comparison with Lagrange method in terms of efficiency and flexibility.
      3. Hermite Interpolation
         1. Overview of Hermite interpolation when derivative information is available.
         2. Construction of Hermite polynomials.
         3. Application scenarios requiring continuity in the function’s derivatives.
         4. Advantages over simple polynomial interpolation regarding derivative preservation.
      4. Spline Interpolation
         1. General description of splines as piecewise polynomial functions.
         2. Linear Spline
            1. Characteristics and mathematical formulation of piecewise linear functions between data points.
            2. Situations where linear splines are adequate due to simplicity.
            3. Comparative benefits and scenarios vs. higher-order splines.
         3. Cubic Spline
            1. Definition and benefits of using cubic splines for smoothness.
            2. Natural Cubic Spline: Boundary condition setting second derivatives to zero at endpoints.
            3. Clamped Cubic Spline: Controlling the slope at the endpoints.
            4. Application areas where cubic splines excel due to smoother transitions.
         4. Differences and benefits over global polynomial interpolation regarding robustness and flexibility.
   3. Practical Considerations
      1. Error Analysis
         1. Introduction to truncation error and stability in interpolation.
         2. Error bounds in polynomial interpolation.
         3. Sensitivity to noise and implications for data with high variability.
      2. Application in Data Fitting
         1. Using interpolation for curve fitting when a mathematical model is not easily obtainable.
         2. Comparing to regression, interpolating vs. approximating data in terms of preserving the original data points.
         3. Scenarios emphasizing interpolation like modeling sensor data.
      3. Handling Large Datasets
         1. Challenges in interpolating large numbers of data points.
         2. Dealing with computational limitations and stability issues.
         3. Optimization techniques to improve efficiency, like reducing dimensionality or selectively interpolating.