论文信息 - Convergence of the Conjugate Gradient Method on Singular Systems

Convergence of the Conjugate Gradient Method on Singular Systems

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.

Ken Hayami

[1] J. Navarro-Pedreño. Numerical Methods for Least Squares Problems , 1996 .

[2] E. F. Kaasschieter,et al. Preconditioned conjugate gradients for solving singular systems , 1988 .

[3] M. Z. Nashed,et al. On the Convergence of the Conjugate Gradient Method for Singular Linear Operator Equations , 1972 .

[4] M. Hestenes,et al. Methods of conjugate gradients for solving linear systems , 1952 .

[5] Masaaki Sugihara,et al. A geometric view of Krylov subspace methods on singular systems , 2011, Numer. Linear Algebra Appl..

[6] Yousef Saad,et al. Iterative methods for sparse linear systems , 2003 .

[7] Nobuhiro Yoshikawa,et al. A new approach for solving singular systems in topology optimization using Krylov subspace methods , 2013, ArXiv.