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Convergence of the Conjugate Gradient Method on Singular Systems

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arxiv 1809.00793 v2 pith:3HY4LZCA submitted 2018-09-04 math.NA cs.NA

Ken Hayami This is my paper · ORCID
classification math.NAcs.NA
keywords conjugateconvergencegradientmethodalgorithmanalysisanalyzeapply
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abstract

We analyze the convergence of the Conjugate Gradient (CG) method in exact arithmetic, when the coefficient matrix $A$ is symmetric positive semidefinite and the system is consistent. To do so, we diagonalize $A$ and decompose the algorithm into the range and the null space components of $A$. Further, we apply the analysis to the CGLS and CGNE (CG Normal Error) methods for rank-deficient least squares problems.

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  1. Auxiliary space theory for the analysis of iterative methods for semidefinite linear systems

    math.NA 2025-09 conditional novelty 6.0 of 10

    A single auxiliary space framework yields exact error-propagation and condition-number identities for iterative methods on semidefinite systems, unifying and generalizing prior SPD and semidefinite analyses.

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