A Comparison of highly efficient iterative linear solvers

A number of iterative techniques have recently been developed which are extremely efficient at solving systems of linear equations. Of these methods probably the most recognized is the Conjugate Gradient Method (CG). This is an extremely efficient solver and has been used successfully for a number of years now. A newer method proposed initially by Davidson [1] is studied in this paper. This method has proven itself in terms of efficiency by solving the same system (of order 2000) that was solved by the CG method. It converged in approximately 40 iterations, taking less than five minutes to do so[5], compared to the CG method which took nearly 100 iterations, converging after about 1 5 minutes. Very little documentation about the derivation or development of Davidson\u27s method exists, and his paper was written in terms of an eigenvalue problem. A portion of a program developed by NASA Ames Research Center uses a variation of Davidson\u27s method as a linear solver. Davidson\u27s method was explored and derived using his paper and the FORTRAN code from NASA. The purpose of this thesis is to provide some insight into the analytical aspect of Davidson\u27s method, using the CG method for comparison