**Abstract** |
Consider applying the Conjugate Residual (CR) method toˇˇsystems 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. |