On the approximate computation of extreme eigenvalues and the condition number of nonsingular matrices (Q1192728)

The author shows that the preconditioned conjugate gradient method applied to the solution of the system \(Ax=b\in\mathbb{R}^ n\), \(A=A^ T\) positive definite, produces the similarity relation \((AM^{-1})R=RB\), which can be used to determine the extreme eigenvalues of the generalized eigenvalue problem (1) \(Ax=\lambda Mx\), where \(M=M^ T\) positive definite, denotes the preconditioner. The matrices \(R\) and \(B\) are generated by the residual vectors and the \(cg\)-iteration parameters, respectively. \(B\) is tridiagonal. So, the determination of the eigenvalues is easy. In practice, the process is stopped after \(s\) iterations, with \(s\ll N\) for good preconditioners. On the basis of a corresponding modification of (1), one obtains approximations to the extreme eigenvalues. Numerical examples are given.