powermethod.txt

this is version 1.0

Initial study of power method for a symmetrix matrix that returns dominant eigenvalue/eigenvector
current model:   		symmetric matrix.  ( tested on some symmetric matrices )
						checks for convergence
						see below for todo list.

Instructions:
In a q session. Go to k.  You can a free version at kx.com
(1) copy/paste variables.
(2) copy/paste while expression.
(3) Returns eigenvector and eigenvalue for dominant eigenvalue.


Choose a matrix 3X3 or 6x6. Dominant eigenvalue and eigenvector for the next two matrices were checked using excel.
A:((-1 2 -2);(-2 -6 3);(-2 -4 1))
A:((7  1  1  7  4  1);(4  4  4  4  4  4);(1  7  7  1  4  7);(7  1  1  7  4  1);(4  4  4  4  4  4);(1  7  7  1  4  7));
/ variables:
v:0.1,1_(#A)#0   / vector parameter used to compute eigenvalue.  See that initial value is 1/10
vN:(#A)#10     / vector parameter used to compute eigenvalue. Add code to prevent these values and 'v' from becoming too large.
el: 2          / current eigenvalue
el2: 1         / previous eigenvalue
e:0.0000001    / error limit to test if eigenvalues have converged. epsilon.
ev: (#A)#1     / eigenvectors for this eigenvalue
iter:1 
a:1            / boolean eigenvalues converges
b:1            / boolean iter max reached.
iterM:10000	   / iter max

/ 3 lines of code.
while[1<a+b; vN:A(+/*)\:v;el2:el;el:(+/vN*v) % (+/v*v);ev:vN % el xexp iter;v:vN;a:e < abs(el2-el);iter+:1;b:iter < iterM];
$[iter=iterM;"Did not converge. Results are incorrect";"Converged in #iterations: ", $iter]
ev:ev % sqrt(+/ev*ev);R:(ev;el);::R

///To do
1. Test powermethod on various large symmetric matrix sizes. Ex: 1000x1000
2. Add code to prevent v and vN from having very large values. i.e. at some stage, v:%vN

///Notes
1. Power method isn't suitable to calculate all eigenvalues eigenvectors.
    due to rounding off errors.  Have not found any documentation identifying a deflation method that corrects
	rounding off errors.