NA - 90 - 06 * . . The Nonsymmetric Lanczos Algorithm May 1990 and Controllability

We give a brief description of a non-symmetric Lanczos algorithm that does not require strict bi-orthogonality among the generated vectors. We show how the vectors generated are algebraically related to “Controllable Space” and “Observable Space” for a related linear dynamical system. The algorithm described is particularly appropriate for large sparse systems. 1. Intr~uction The Lanczos Algorithm was originally proposed by Lanczos [15] as a method for the computation of eigenvalues of symmetric and nonsymmetric matrices. The idea was to reduce a general matrix to tridiagonal form, from which the eigenvalues could be easily determined, For symmetric matrices, the Lanczos Algorithm has been studied extensively [5] [17]. In that case, the convergence of the algorithm, when used to compute eigenvalues, has been extensively analyzed in [ 141 [ 161 [20] [21] [22, p27OffJ. This algorithm is particularly suited for large sparse matrix problems. A block Lanczos analog has been studied and analyzed by Underwood (cf. Golub and Underwood [lo], Cullum and Willoughby [S] and Parlett [17]). However, until recently, the nonsymmetric Lanczos Algorithm has received much less attention. Some recent computational exprience with this algorithm can be found in [4]. Besides some numerical stability problems, the method suffered from the possibility of an incurable breakdown from which the only way to “recover” was to restart the whole process from the beginning with different starting vectors [22, p388ffJ. More recently, several modifications allowing the Lanczos process . ‘Computer Science Dept., University of Minnesota, Mimeapolis, MN 55455. E-mail: [email protected]. The research reported by this author was supported in part by NSF grant CCR-8813493. 2Computer Science Dept., Stanford University, Stanford, CA 94305. E-mail: [email protected]. The research reported by this author was supporkd in part by AR0 grant DAAU)3-90-G-0105 and in part by NSF grant DCR-8412314.”