Hence one may use the same storage for all three. Likewise, if only the tridiagonal matrix formula_52 is sought, then the raw iteration does not need formula_53 after having computed formula_49, although some schemes for improving the numerical stability would need it later on. Sometimes the subsequent Lanczos vectors are recomputed from formula_39 when needed. The Lanczos algorithm is most often brought up in the context of finding the eigenvalues and eigenvectors of a matrix, but whereas an ordinary diagonalization of a matrix would make eigenvectors and eigenvalues apparent from inspection, the same is not true for the tridiagonalization performed by the Lanczos algorithm; nontrivial additional steps are needed to compute even a single eigenvalue or eigenvector. Nonetheless, applying the Lanczos algorithm is often a significant step forward in computing the eigendecomposition. If formula_56 is an eigenvalue of formula_6, and if formula_58 (formula_59 is an eigenvector of formula_52) then formula_61 is the corresponding eigenvector of formula_6 (since formula_63). Thus the Lanczos algorithm transforms the eigendecomposition problem for formula_6 into the eigendecomposition problem for formula_52. The combination of good performance for sparse matrices and the ability to compute several (without computing all) eigenvalues are the main reasons for choosing to use the Lanczos algorithm. Though the eigenproblem is often the motivation for applying the Lanczos algorithm, the operation the algorithm primarily performs is tridiagonalization of a matrix, for which numerically stable Householder transformations have been favoured since the 1950s. During the 1960s the Lanczos algorithm was disregarded. Interest in it was rejuvenated by the Kaniel–Paige convergence theory and the development of methods to prevent numerical instability, but the Lanczos algorithm remains the alternative algorithm that one tries only if Householder is not satisfactory. Aspects in which the two algorithms differ include:  There are several lines of reasoning which lead to the Lanczos algorithm. The power method for finding the eigenvalue of largest magnitude and a corresponding eigenvector of a matrix formula_6 is roughly A critique that can be raised against this method is that it is wasteful: it spends a lot of work (the matrix–vector products in step 2.1) extracting information from the matrix formula_6, but pays attention only to the very last result; implementations typically use the same variable for all the vectors formula_100, having each new iteration overwrite the results from the previous one.