"A conjecture in numerical linear algebra"

Abstract

In [1] and [2] we proposed a conjecture in numerical linear algebra and here we would like to clarify an implicit assumption in our note. Throughout we meant by a matrix always a <u>dense</u> matrix. It was clear to us that for real symmetric band matrices the "chasing" strategy proposed by Rutishauser [3] is superior to the unmodified form of tridiagonalization. It has also been pointed out [4] that for very sparse matrices the Givens' method can take more advantage of the sparsity than Householder's method. At present intensive research is being done in the field of very large matrix problems [5] and here new approach is necessary. In spite of the development of a fast Givens transformation [6], [7], we still feel that our conjecture may hold true for dense matrices, because we request minimum of operations and optimal numerical properties.