I have collected information regarding latest progress made in solving of system of Linear Equations. Interestingly this works uses spanning tree (approximate) in solving Linear Equations. Its was amazing that tree has to do anything with system of linear equations . See the attached beamer. Apparently they claim that they have found the fastest algorithm. In nutshell, they combined support theory with conjugate gradient methods. A lot of mathematicians work has been an integral part of this progress. Dan Spielman to ex EE-IITB Praveen Vaidya who chose not to publish his work but to produce commercial software instead.
There are some good insight also about interior point methods and Laplacian. May be in the next version of this document, I’ll try to review it.
Please report back any mistakes.