 
Summary: Performance analysis of Krylov iterative solvers using Ritz values
Irina Craciun
Advisor: Dr. Burak Aksoylu
Center for Computation and Technology
Louisiana State University  Baton Rouge
(irina@cct.lsu.edu)
1. Introduction
The problem of solving linear systems has become a crucial issue for scientific computing during
the past 50 years. A majority of problems in physics, astrophysics, computational fluid dynamics,
chemistry, engineering etc. reduce to solving a differential or integral equation, which in turn,
after discretisation and linearisation, reduce to a linear system of the form Ax = b. Coming from
reallife problems, the matrices are usually nonnormal and very large  often reaching orders of
millions. For such large matrices, any linear solver fails because of computational errors.
On the positive side, they are also often sparse matrices (contain very few nonzero entries), which
makes the computation of matrixvector products a very easy, noncosting task. This is why we
use iterative methods to solve large sparse linear systems: iterative methods generate successive
approximations of the solution, such that each approximation is computed from the previous one
through a series of matrixvector products.
Currently there is no "perfect" iterative method; each known iterative method has been proved
efficient for some particular types of matrices and inefficient for others, each having many circum
