The Lanczos algorithm with partial reorthogonalization

Abstract

The Lanczos algorithm is becoming accepted as a powerful tool for finding the eigenvalues and for solving linear systems of equations. Any practical implementation of the algorithm suffers however from roundoff errors, which usually cause the Lanczos vectors to lose their mutual orthogonality. In order to maintain some level of orthogonality, full reorthogonalization (FRO) and selective orthogonalization (SO) have been used in the past as a remedy. Here partial reorthogonalization (PRO) is proposed as a new method for maintaining semiorthogonality among the Lanczos vectors. PRO is based on a simple recurrence, which allows us to monitor the loss of orthogonality among the Lanczos vectors directly without computing the inner products. Based on the information from the recurrence, reorthogonalizations occur only when necessary. Thus substantial savings are made as compared to FRO. In some numerical examples we apply the Lanczos algorithm with PRO to the solution of large symmetric systems of linear equations and show that it is a robust and efficient algorithm for maintaining semiorthogonality among the Lanczos vectors. The results obtained compare favorably with the conjugate gradient method.