A new efficient solution method for a system of linear equations: Partially Solving Method (PSM)

Abstract

We present a new method of solving a system of linear equations, called Partially Solving Method (PSM). The PSM can essentially deal with only a subsystem at each processing stage without complete knowledge of the entire system. For dense systems, it reduces the necessary memory space e¡ by a factor of four as compared with the conventional LU-decomposition method. For sparse systems, the method operates up to twice as fast as Gaussian elimination method, and the efficiency in both space and time is further enhanced by a proper ordering of selections of the equations. It may be feasible to apply the PSM in a parallel processing environment, when the entire system is divided into subsystems.

Very fast chip-level thermal analysis

“…We decompose Eq. (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) into four subsystems of equations that correspond to the four materials. We denote the inverse matrix of each coefficient matrix of the above equations by the matrix of 16 submatrices of the form u B as shown below.…”

“…Note also that the last two lines of Eq. (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) describe the correspondence between the variable names of each of the two materials (For the reasoning to derive such equations, see [7]. ).…”

“…Recently, however, a new efficient direct method, called Symbolic Partial Solution Method (S-PSM) was introduced in the area of computational fluid dynamics [2] [3]. This method was derived from the general linear equation solver, called Partial Solution Method (PSM) [8], and is applicable to the case of tridiagonal coefficient matrices. S-PSM was later applied in a Laplace equation solution process for thermal conduction analysis of two adjacent materials [6].…”

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

“…At present, it is widely considered that the fundamental methods of solving linear systems have been established, among which Gaussian elimination method (GEM) is one of the most prominent formulations. However, we introduced a new efficient method called the Partially Solving Method (PSM) in our previous paper [2].…”

“…Various methods have been studied so far to efficiently treat sparse systems by proper clustering or ordering of variables [3,4,5,6,7], and we also have discussed the problem of ordering in PSM in our previous paper [2]. In the present paper, we evaluate the efficiency of PSM and its dependence on sparsity and size of linear systems, by comparing quantitatively the number of operations required to solve linear systems by PSM and the traditional GEM.…”