# NII Technical Report (NII-2001-003J)

 Title On the Convergence of the Conjugate Residual Method for Singular Systems Authors Ken HAYAMI Abstract Consider applying the Conjugate Residual (CR) method to¡¡systems of linear equations $A \bx = \bb$ or least squares problems ${\displaystyle \min_{\bx \in \rn} \| \bb - A \bx \|_2 }$, where $A \in \rnn$ is singular and nonsymmetric. First, we prove the following. When $R(A)^\perp = \ker A$, the CR method can be decomposed into the $R(A)$ and $\ker A$ components, and the necessary and sufficient condition for the method to converge to the least squares solution without breaking down for arbitrary $\bb$ and initial approximate solution $\bx_0$ is that the symmetric part $M(A)$ of $A$ is semi-definite and $\rank \, M(A) = \rank A$. Furthermore, when $\bx_0 \in R(A),$ the approximate solution converges to the pseudo inverse solution. Next, for the case when $R(A) \oplus \ker A = \rn$ and $\bb \in R(A),$ the necessary and sufficient condition for the CR method to converge to the least squares solution without breaking down for arbitrary $\bx_0$, is also derived. Finally, we will give examples corresponding to the above two cases arising in the finite difference discretization of two-point boundary value problems of an ordinary differential equation. Language Japanese Published Aug 03, 2001 Pages 33p PDF File 01-003J.pdf

ISSN:1346-5597
NII Technical Reports
National Institute of Informatics