On the numerical solution of \((\lambda^2 A + \lambda B + C)x = b\) and application to structural dynamics (Q2780608)

The authors discuss the numerical solution of large linear systems \(L(\lambda)x = b\) with \(L(\lambda) = \lambda^2 A + \lambda B + C\), where \(\lambda\) varies in a wide range. It is assumed that \(A\), \(B\), and \(C\) are \(n \times n\) complex symmetric matrices, and that either \(A\) or \(C\) is nonsingular. In a first step the system is linearized in \(\lambda\) such that one gets a system \(({\mathcal A} + \lambda {\mathcal B})z = d\) with \((2n \times 2n)\) complex symmetric matrices \({\mathcal A}\) and \({\mathcal B}\). If \({\mathcal B}\) is nonsingular one obtains the system \(({\mathcal A}{\mathcal B}^{-1} + \lambda I_{2n}){\hat z} = d\), where \(I_{2n}\) is the identity matrix of the order \(2n\) and \({\hat z} = {\mathcal B}z\). For solving this problem iteratively a simplified shifted Lanczos method with QMR procedure is proposed.NEWLINENEWLINENEWLINEThe main advantage of this algorithm is that it allows to approximate the solution of all systems with values \(\lambda_j\), \(j=1,2,\ldots , s\), by generating only the Krylov subspace associated with one of them. In each iteration step one has to solve a system of equations with the matrix \({\mathcal B}\). The influence of inexact solution procedures on the convergence behaviour of the outer iteration is analyzed. Numerical experiments with problems arising from structural dynamics illustrate the convergence behaviour of the proposed solution method. Additionally, comparisons with other known solution strategies are presented.