Testing the Efficiency of Iterative Methods for Sparse Matrices. BRIAN LANGSTRAAT (Central College Pella, IA 50219) MASHA SOSONKINA (Ames Laboratory, Ames, IA, 50011)

Matrices are used to store mathematical and scientific information. If a large matrix is made up of mostly zeros, then it is called sparse. Sparse matrices are stored such that each nonzero is recorded by its location within a matrix and zeros are generally omitted. Thus the matrix storage is minimized. Sparse matrices arise in many real-world applications from energy networks and biological systems to weather prediction and engineering. Iterative methods are used to find the value of vector x in the linear system of equations: Ax=b, where A is a matrix (often sparse) and b is a vector. Physicists and chemists utilize iterative methods on parallel computers at the Scalable Computing Laboratory (SCL) to solve large-scale problems involving sparse matrices. MATLAB 7.0 was used to test a variety of iterative methods that solve sparse matrices. Several programs were created to test the iterative methods of generalized minimum residual (GMRES), minimum residual (MINRES), preconditioned conjugate gradient (PCG), and quasi-minimal residual (QMR) on a variety of sparse matrices. A preconditioner is a matrix whose inverse is multiplied to both sides of the linear system of equations which changes how the system is solved to obtain the same solution. Preconditioners were used in several tests which decreased the number of iterations needed. Also, an accurate prediction of final results lessened the quantity of iterations greatly. The residual norms can be found more quickly, if the iterative method, preconditioner, and estimation are selected well. This work will supplement the SCL's ability to use MATLAB for solving sparse matrices.