MAT 581 Introduction to Numerical Methods

Fall 2020, Spring 2021, Fall 2021

Lecture Notes

Fall 2016

Exams

Lecture Notes

Homework Set

Unit 1: MATLAB
  1. Matrix/vector manipulation
  2. Matrix/vector creation; elementwise operations
  3. Built-in functions, input/output, 2D graphics
  4. Scripting
  5. Functions
  6. Linear systems
Unit 2: Root Finding
  1. Root finding: bisection method
  2. Fixed point method
  3. Newton and secant methods
  4. Multivariable Newton's method
  5. Broyden's method
Unit 3: Numerical Calculus
  1. Numerical differentiation
  2. Numerical integration: Trapezoidal and midpoint rules
  3. Simpson's rule, Newton-Cotes
  4. Orthogonal polynomials: Legendre, Laguerre
  5. Gaussian integration, Gauss-Laguerre integration
  6. Adaptive integration
  7. Multiple integrals
  8. ODE: Euler's method, RK2 methods
  9. ODE systems, predator-prey model
Unit 4: Data Fitting
  1. Polynomial interpolation
  2. Spline interpolation
  3. Spline approximation
  4. Discrete Fourier transform
  5. Linear least squares
  6. Model comparison
  7. Nonlinear least squares
  8. Transforming data
Unit 5: Optimization
  1. Linear programming
  2. Single-variable minimization
  3. Gradient method and conjugate gradient method
  4. Nelder-Mead method
  5. Applications of optimization: data classification
  6. Applications of optimization: model selection

Fall 2015

Lecture Notes

Homework (Matlab)

  1. Vectors and plotting
  2. Matrices
  3. Experiments with probability
  4. Floating point arithmetics
  5. Matlab functions
  6. Polynomial interpolation
  7. Newton polynomial
  8. Error of interpolating polynomial
  9. Flop Count
  10. Piecewise linear interpolation
  11. Adaptive piecewise linear interpolation
  12. Natural cubic splines, theory
  13. Construction of cubic splines
  14. B-splines
  15. Curve fitting by least squares
  16. More on least squares
  17. Training and testing data sets
  18. Root finding: bisection
  19. Root finding: Newton's method
  20. Root finding: secant method
  21. Root finding: Multivariable Newton's method
  22. Minimization: golden section search
  23. Minimization: gradient descent
  24. Multivariable minimization: gradient descent
  25. Multivariable minimization: Nelder-Mead
  26. Multivariable minimization: application to classification of data
  27. Numerical differentiation
  28. Numerical integration: midpoint and trapezoid
  29. Numerical integration: adaptive Simpson's rule
  30. Numerical integration: Newton-Cotes formulas
  31. Discrete Fourier transform
  32. Numerical integration: Clenshaw-Curtis method
  33. Discrete cosine transform, signal compression
  34. ODE: Euler's and Runge-Kutta methods
  35. ODE: boundary value problems