or ###The Rise and Fall of Matrix Stardust and the Vectors from Mars
#####A project aiming to implement the Gauss-Newton method, find the convergence of the power method, and produce a rudimentary 2D animation (full explanations below). This is a group project, with members (in alphabetical order) Jayden Gardiner, Brenda Lin, and Mitchell Manguno. So, here is a description of all three parts of the project, in their full glory.
##Methods
###1. The Gauss-Newton Method
Read up on the Gauss-Newton method here.
For this portion of the project, four executable programs will be written (one for each of the curves listed below) that, when run, do the following:
- prompt the user for a text file containing a list of points,
- prompt the user for initial guesses for the parameters a, b, and c,
- prompt the user for a number of iterations to run the Gauss-Newton algorithm, and
- output the parameters giving the best approximation for the appropriate curve matching the given points.
The curves to be approximated will be the following.
- Quadratic
- Exponential
- Logarithmic
- Rational
These programs will be named gn_qua, gn_exp, gn_log, and gn_rat.
In order to do this, the programs will use Householder Reflections and Givens Rotations to QR-factorize the Jacobian of the input points, and then use a modified version of the Gauss-Newton algorithm to minimize the residual).
###2. Convergence of the Power Method
For this part of the project, over 1000 2 x 2 matrices with its values in the boundaries [-2, 2] will be randomly generated.
The power method will then be run on all these matrices, which is used to find the doiminant eigenvalue for a matrix. The method will also be run for the matrices' inverses, and also the matrices' determinants will be calculated.
This information is graphed using Java FX and written to a txt file, for study and analysis of trends between eigenvalues and determinants.
##External Libraries Used
###JAMA, A Java Matrix Package
JAMA is a basic linear algebra package for Java. It provides user-level classes for constructing and manipulating real, dense matrices. It is meant to provide sufficient functionality for routine problems, packaged in a way that is natural and understandable to non-experts. It is intended to serve as the standard matrix class for Java, and will be proposed as such to the Java Grande Forum and then to Sun. A straightforward public-domain reference implementation has been developed by the MathWorks and NIST as a strawman for such a class.
JAMA website: Background section.
JAMA is a software package in the public domain, and thus may be used freely in this project. Visit their website for more information on the JAMA software package, or their licensing section for more information on public use.