Eigenvalues without eigenproblems or characteristic equations

This chapter shows how the eigenvalues can be obtained without actually solving the eigenproblem or the characteristic equation by using the Shanks transform and the Rutishauser quotient difference (QD) algorithm. It illustrates that the Shanks transform, the Wynn algorithm and Rutishauser QD algorithm can all give the eigenvalues without resorting to the more conventional procedures, e.g. the eigenproblem or the characteristic equation. The chapter also shows how the need for the introduction of two vectors from the QD algorithm can be naturally motivated within the analysis of the Shanks transform.