[binomial]

Please elaborate. Solving linear equations with software is nothing new, this algorithm just does it faster. So I'd like to see prior art for what you're describing here, sounds interesting. How exactly would one apply linear algebra to solving the first and third questions for example?

[jpren]

Back in college a few years ago, we mostly used Simplex and other tableau-pivot based algorithms for solving large linear programming problems. Link to wikipedia article on the Simplex algorithm http://en.wikipedia.org/wiki/Simplex_algorithm

For most problems that individuals face each day, linear programs are fairly simple to model and solve. However, there are lots of complex problems that are solved each day.

A few examples of large-scale linear programming problems: - For Amazon.com: what quantity of each item should be stocked at each warehouse each day to minimize inventory while also optimizing for shipping time and cost to demand nodes (customers). - For an airline: how to plan and schedule flights to all domestic and international airports to maximize profit - In shipping logistics: how to allocate trucks and set routes to minimize fuel cost while satisfying delivery time.

For complex systems, you can easily run up a linear program with millions of independent variables (producing millions of rows in the linear system).

[lmkg]

Solving a linear system of equations is not the same as solving a linear program. The article is about the former, not the latter. Linear programming is an optimization problem, subject to constraints that are described by inequalities. Linear systems of equations are finding values of relationships defined by equalities. There are lots of matrix multiplications involved in both, but other than that the solutions don't really resemble each other. For example, Linear Programming is not even known to be solvable in strongly polynomial time (simplex is exponential in the worst case, and the best method is weakly polynomial), while the algorithm in the paper improves from one strongly polynomial result ot another.