A novel nonsymmetric \(K\)-Lanczos algorithm for the generalized nonsymmetric \(K\)-eigenvalue problems (Q676019)

The generalized eigenvalue problem \(Mx=\lambda Nx\) is studied, where the matrices \(M\), \(N\) have certain symmetries, namely the \(K_{\pm}\)-structure. A new algorithm, the nonsymmetric \(K\)-Lanczos algorithm, is developed to calculate extreme eigenvalues. For this a tridiagonalization procedure preserving the symmetries is given. It generalizes an algorithm by \textit{U. Flaschka} [Eine Variante des Lanczos-Algorithmus für große, dünn besetzte symmetrische Matrizen mit Blockstruktur, Diss. Univ. Bielefeld (1992; Zbl 0830.65023)] in the case that \(N^{-1}M\) is in addition symmetric. This algorithm can be viewed as a \(2\times 2\)-block nonsymmetric Lanczos algorithm. Numerical experiments are reported.