A unifying convergence analysis of second-order methods for secular equations

Existing numerical methods of second-order are considered for a so-called secular equation. We give a brief description of the most important of these methods and show that all of them can be interpreted as improvements of Newton’s method for an equivalent problem for which Newton’s method exhibits convergence from any point in a given interval. This interpretation unifies the convergence analysis of these methods, provides convergence proofs where they were lacking and furnishes ways to construct improved methods. In addition, we show that some of these methods are, in fact, equivalent. A second secular equation is also briefly considered.