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We present an error analysis of the symmetric Lanczos algorithm in finite precision arithmetic. The loss of orthogonality among the computed Lanczos vectors is explained with the help of a recurrence formula. A backward error analysis shows that semiorthogonality among the Lanczos vectors is enough to guarantee the accuracy of the computed quantities up to machine precision. The results of this analysis are then extended to the more general case of the Lanczos algorithm with a semiorthogonalization strategy. Based on the recurrence formula, a new reorthogonalization method called partial reorthogonalization is introduced. We show that both partial reorthogonalization and selective orthogonalization as introduced by Parlett and Scott [15] are semiorthogonalization strategies. Finally we discuss the application of our results to the solution of linear systems of equations and to the eigenvalue problem.