Competitive Algorithms for Large Scale Positive Definite Linear Systems of Equations Using an Error in Variables Optimization Model

Abstract

We seek positive definite solutions of large scale overdetermined linear systems of equations with multiple right hand sides, where the coefficient and the right hand side matrices are respectively called data and target matrices. We recently proposed a new approach for solving such problems in small scales considering error in both the measured data and the target matrices and using a novel error formulation. We proposed four algorithms for solving small scale positive definite linear systems of equations with both full rank and rank deficient data matrices. Here, we extend our algorithms to solve large scale problems. The algorithms are implemented in Matlab and Fortran (using the ScaLAPACK library) respectively for small and large scale problems . ScaLAPACK is a parallel library written in Fortran to solve linear algebra problems such as computing matrix decompositions for large scale matrices. Block cyclic data distribution is used in ScaLAPACK to provide proper scalability, reliability, portability, flexibility and ease of use. Comparative numerical experiments with two available methods, the interior point method and a method based on quadratic programming, show that our approach produce a smaller standard deviation of the error entries and a smaller effective rank, as being desirable for control problems. Using the Dolan-More performance profiles we show that our proposed algorithms are more efficient and more accurate in computing proper solutions.