Inexact inverse iteration for generalized eigenvalue problems
To solve the generalized eigenvalue problem \(Ax=\lambda Bx\), one can use the inverse iteration method where in each iteration a linear system of equation \(Az_{k+1}=Bx_k\), has to be solved. In recent years, it has been proposed to solve that system by iterative schemes leading to an inexact inverse iteration method. In this paper, convergence results in function of two stopping criteria are given. In particular, the stopping criteria used are \(\|q_k\|<\varepsilon \|Bx_k-Ay_k\|\) or \(\|q_k\|<\varepsilon_k \|y_{k+1}\|\), where \(y_{k+1}\) is an approximate to \(z_{k+1}\) and \(\varepsilon\) is a threshold. In the convergence analysis, under the hypotheses that the matrix \(B^{-1}A\) is diagonalizable and the two minimum eigenvalues have different absolute value, the authors prove that the convergence is linear under the natural assumption that \(\varepsilon_k\) decreases linearly with respect to a parameter \(\gamma\leq 1\). In particular, they prove that the convergence is linear at the rate of the maximum of \(\rho\) and the parameter \(\gamma\leq 1\), where \(\rho\) is the quotient of the absolute value of the two minimum eigenvalues of \(B^{-1}A\). When using the other stopping criterion with fixed \(\varepsilon\) the convergence is linear but not to zero. Further, the computational work is analyzed. More precisely, the effect of \(\gamma\) on the total amount of work is studied proving that the optimal choice of this parameter is greater than \(\rho\) and as close to it as possible. Finally, there are some numerical results for solving the eigenvalue problem of the convection diffusion equation. These numerical problems illustrate the theoretical results of.
scientific article; zbMATH DE number 1556488