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CONSOLE APPLICATION : Matrix Project Overview
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Running a couple of tests on Systematic solution of linear equations.

Completed Procedures

1. Gaussian Elimination
   
2. Gauss-Jordan Elimination
   
3. Determinant of a triangular matrix
   
4. Solve lower triangular system (back substitution)
   
5. Solve upper triangular system (back substitution)
   
6. LU Decomposition Doolittle
   
7. LU Decomposition Crout
   
8. Solve LU system (gemeric)
   
9. LU Decomposition Cholesky
   
10. Gauss-Siedel iteration
    
11. Householder tridiagonalization
    

had problems with the process of constructing a Hessenburg matrix for the general case
of an n*n square matrix because I thought the Householder algorithm only worked for
symmetric matrices.

I implemented a method that performs a similarity transform and returns
H and S from http://www.mymathlib.com/matrices/eigen/,
(these are removed, I used the Krezig method, see much older versions)

I found a paper
http://www.ams.org/journals/mcom/1969-23-108/S0025-5718-1969-0258255-3/S0025-5718-1969-0258255-3.pdf that proves numerical instability of the row reduction methods.(this showed up again in the complex matrix).

Anyway I discovered that the Householder method can be used to transform a general N*N matrix
into the Hessenburg form ... The matrix in the example is from this page
http://mathfaculty.fullerton.edu/mathews/n2003/hessenberg/HessenbergMod/Links/HessenbergMod_lnk_9.html
and for more information
http://mathfaculty.fullerton.edu/mathews/n2003/HessenbergMod.html


12) implemented QR algorithm for finding real eigen-values.
13) the QR procedure is working well for real valued solutions and especially symmetric matrices
14) the QR procedure solved for the complex number pairs
15) There are now constructors for various static initializer types
(see examples at top for information, the max is a 6x6 matrix)
16) memory optimized for real matrices
17) Implemented a Complex matrix type - still experimental
18) Set up matrix inversion using LU decomposition thanks to
http://mathworld.wolfram.com/MatrixInverse.html
and
http://www.gamedev.net/page/resources/_/technical/math-and-physics/matrix-inversion-using-lu-decomposition-r3637
for completely explaining it
19) Added some functions for finding a determinant up to 5x5 however the input function isn't implemented.
20) Added method for newton iteration to improve bad inverses.
21) first complex matrix version of the householder algorithm is implemented
22) implemented QR factorisation and tested here - the output produces the same R matrix
23) Reduction to Hessenburg is *exactly* the same as the python version, the solution was to use CSGN instead of SGN
24) Eigenvalues for the Hermitian matrix are *exactly* the same as the python version
25) Eigenvalues for general complex matrix are working (recognizing the fact that the Hermitian is its own conjugate transpose)