What question did this study set out to answer?

This paper aims to analyze the convergence of the Conjugate Gradient method for singular systems.

April 1, 2026Open Access

NII Technical Report (NII-2018-001E)：Convergence of the Conjugate Gradient Method on Singular Systems

Key Points

This paper aims to analyze the convergence of the Conjugate Gradient method for singular systems.
Analyzed the Conjugate Gradient (CG) method under exact arithmetic conditions.
Diagonalized the coefficient matrix A, assuming it is symmetric positive semidefinite.
Decomposed the algorithm into range and null space components of A.
Applied the analysis to CGLS and CGNE methods for rank-deficient least squares problems.
Identified specific convergence behaviors of the CG method in singular systems.
Demonstrated how diagonalization affects the algorithm's behavior in the range and null space.
Provided insights into the effectiveness of CGLS and CGNE methods for solving rank-deficient problems.

Abstract

In this short paper, 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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