Computational advantages of the recursion and lanczos methods

Abstract

In exact arithmetic, the recursion, Lanczos, Givens and Householder methods of tridiagonalizing symmetric real matrices give the same result if the first column of the orthogonal transformation is the same for each method. The recursion and Lanczos methods require store for only two vectors while the Givens and Householder methods require store for an entire matrix. The recursion and Lanczos methods are significantly faster for sparse and other special matrices. Thus, along with their numerical stability, the recursion and Lanczos methods make possible calculations which would be impossible using the Givens or Householder methods.