Method for verifying solutions of sparse linear systems with general coefficients

Abstract

This paper proposes a verification method for sparse linear systems $Ax=b$ with general and nonsingular coefficient matrices. A verification method produces the error bound for a given approximate solution. Practical methods use one of two approaches. One approach is to verify the computed solution of the normal equation $A^{T}Ax=A^{T}b$ by exploiting symmetric and positive definiteness; however, the condition number of $A^{T}A$ is the square of that for A. The other approach applies an approximate inverse of A; however, the approximate inverse of A may be dense even if A is sparse. Additionally, several other methods have been proposed; however, they are considered impractical due to various issues. Here, this paper provides a computing method for verified error bounds using the previous verification method and the latest equilibration. The proposed method can reduce the fill-in and is applicable to many problems. Moreover, we will show the efficiency of an iterative refinement method to obtain accurate solutions.