Linear equations:
Most every real problem may start out non-linear and difficult,
but is approximated and solved as systems of linear equations.
Example:
a/ Analyzing forces in an airplane:
1 Using the rules of physics we model external forces
2 External forces relate to internal stresses through
partial differential equations (PDE)
3 PDE is discretized and solved by e.g. FEM
4 Result of FEM analysis is a *system of linear equations*
(This sustem can be HUGE, but in principle no different from
ones you solve in 120 and 340)
b/ Solving a non-linear optimization problems
Define a search method using a series of linear problems
converging to the solution of the non-linear problem
Lecture 2: Intro to matrices,
1/2 of geometry slides.
Lecture 3: 2nd half of geometry slides
Review of IEEE floating point numbers (black board)
Lecture 4: Forward-Backward analysis (black board)
First part of solving lin syst.
Review of vector and matrix calculations
Singularity example
Applications in geometry
Solving linear equation systems
Norms and Condition numbers
Linear Least Squares
Measurement errors inevitable in observational and experimental sciences
Errors smoothed out by averaging over many cases, i.e., taking more
measurements than strictly necessary to determine parameters of system
Resulting system overdetermined, so usually no exact solution
Project higher dimensional data into lower dimensional space to suppress
irrelevant detail
Projection most conveniently accomplished by method of least squares
Singular Value Decomposition solves both over and underdetermined systems
Terry's SVD slides
(Note Heath also covers this in both Chapter 3 and 4)
Eigenvalue problems
Eigenvalue problems occur in many areas of science and engineering,
such as structural analysis
Eigenvalues also important in analyzing numerical methods
