<p>For the generalized eigenvalue problem, a quotient function is devised for estimating eigenvalues in terms of an approximate eigenvector. This gives rise to an infinite family of quotients, a disc centered at the Rayleigh quotient, all entirely arguable to be used in estimation. Although the Rayleigh quotient is among them, one can suggest using it only in an auxiliary manner to choose the quotient in a more optimal way. In normal eigenvalue problems, for any approximate eigenvector, there always exists a ’perfect’ quotient exactly giving an eigenvalue. For practical estimates in the self-adjoint case, an approximate midpoint of the spectrum is a good choice for reformulating the eigenvalue problem, yielding apparently the fastest known quotient iteration. No distinction is made between estimating extreme or interior eigenvalues. Preconditioning from the left results in changing the inner-product and affects the estimates accordingly. Preconditioning from the right preserves self-adjointness and can hence be performed without any restrictions. Right preconditioning is combined with variational methods for generating starting vectors for quotient iterations.</p>