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AbstractAbstract. Third-order numerical methods are analyzed for secular equations. These equations arise in several matrix problems and numerical linear algebra applications. A closer look at an existing method shows that it can be considered as a classical method for an equivalent problem. This not only leads to other third-order methods, it also provides the means for a unifying convergence analysis of these methods and for their comparisons. Finally, we consider approximated versions of the aforementioned methods. 1.