An analysis of the Rayleigh-Ritz method for approximating eigenspaces (Q2701555)

This paper concerns the computation of approximations to an eigenspace \({\mathcal X}\) of a general matrix \(A\) without the assumptions that the eigenvalues of \(A\) are distinct or that \(A\) is diagonalizable. Using a subspace \({\mathcal W}\) containing an approximation to \({\mathcal X}\), the method produces Ritz pairs \((N,\widetilde X)\) approximating \((L,X)\), \(X\) a basis of \({\mathcal X}\). Under a ``uniform separation condition'' and with uniform adjustment it converges linearly as the sine and the angle between \({\mathcal X}\) and \({\mathcal W}\) approaches zero (and without that condition when \(\dim{\mathcal X}=1\)). Alternatively, convergence without that condition is obtained for ``refined Ritz vectors''.