Evaluation of the efficiency of the Partially Solving Method (PSM) in comparison with the Gaussian elimination method

Abstract

We have proposed a new efficient solution method for linear systems called the Partially Solving Method (PSM), which essentially deals with a subsystem at each processing stage without complete knowledge of the entire system, resulting in significant reduction in necessary memory space and computation time. In the present paper, we estimate the complexity of PSM and compare it with that of the prominent Gaussian elimination method. We derive an analytic expression of the total number of operations required to get a final solution of a system based on each of these schemes. Dependence of the total number of operations on the size and sparsity of a coefficient matrix of a linear system is examined. It is demonstrated that PSM is approximately twice as efficient as the Gaussian method when linear systems are sufficiently large and moderately sparse. We clarify the origin of these essential features of PSM by analyzing operations in different levels of procedures. In addition, as a benchmark test we have confirmed the effectiveness of PSM by estimating numerically the complexity to solve linear systems with coefficient matrices given statistically. To our best knowledge, we are the first to quantitatively evaluate the complexity of the solving methods, including the Gaussian elimination method.