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Method for verifying solutions of sparse linear systems with general coefficients

Author

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  • Terao, Takeshi
  • Ozaki, Katsuhisa

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 ATAx=ATb by exploiting symmetric and positive definiteness; however, the condition number of ATA 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.

Suggested Citation

  • Terao, Takeshi & Ozaki, Katsuhisa, 2025. "Method for verifying solutions of sparse linear systems with general coefficients," Applied Mathematics and Computation, Elsevier, vol. 490(C).
    • Handle: RePEc:eee:apmaco:v:490:y:2025:i:c:s0096300324006659
    DOI: 10.1016/j.amc.2024.129204
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