Analysis reveals convergence behaviors in singular systems using the Conjugate Gradient method, suggesting improvements for least squares problems.
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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Ken Hayami (2018) studied this question.
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