Applications to dynamics: eigenvalues of K, solution of Mu'' + Ku = F(t)

Lecture Description

Course Description

This course provides a review of linear algebra, including applications to networks, structures, and estimation, Lagrange multipliers. Also covered are: differential equations of equilibrium; Laplace's equation and potential flow; boundary-value problems; minimum principles and calculus of variations; Fourier series; discrete Fourier transform; convolution; and applications.

Course Index

1. Positive definite matrices K = A'CA
2. One-dimensional applications: A = difference matrix
3. Network applications: A = incidence matrix
4. Applications to linear estimation: least squares
5. Applications to dynamics: eigenvalues of K, solution of Mu'' + Ku = F(t)
6. Underlying theory: applied linear algebra
7. Discrete vs. continuous: differences and derivatives
8. Applications to boundary value problems: Laplace equation
9. Solutions of Laplace equation: complex variables
10. Delta function and Green's function
11. Initial value problems: wave equation and heat equation
12. Solutions of initial value problems: eigenfunctions
13. Numerical linear algebra: orthogonalization and A = QR
14. Numerical linear algebra: SVD and applications
15. Numerical methods in estimation: recursive least squares and covariance matrix
16. Dynamic estimation: Kalman filter and square root filter
17. Finite difference methods: equilibrium problems
18. Finite difference methods: stability and convergence
19. Optimization and minimum principles: Euler equation
20. Finite element method: equilibrium equations
21. Spectral method: dynamic equations
22. Fourier expansions and convolution
23. Fast fourier transform and circulant matrices
24. Discrete filters: lowpass and highpass
25. Filters in the time and frequency domain
26. Filter banks and perfect reconstruction
27. Multiresolution, wavelet transform and scaling function
28. Splines and orthogonal wavelets: Daubechies construction
29. Applications in signal and image processing: compression
30. Network flows and combinatorics: max flow = min cut
31. Simplex method in linear programming
32. Nonlinear optimization: algorithms and theory