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


Author: A. Melman
Journal: Math. Comp. 66 (1997), 333-344
MSC (1991): Primary 65F15, 65H05
DOI: https://doi.org/10.1090/S0025-5718-97-00787-4
MathSciNet review: 1370854
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Abstract: 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.


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Additional Information

A. Melman
Affiliation: Department of Industrial Engineering and Management, Ben-Gurion University, Beer-Sheva 84105, Israel
Email: melman@bgumail.bgu.ac.il

DOI: https://doi.org/10.1090/S0025-5718-97-00787-4
Keywords: Symmetric eigenvalues, secular equation, nonlinear approximation, global convergence
Received by editor(s): February 12, 1995
Received by editor(s) in revised form: November 13, 1995
Article copyright: © Copyright 1997 American Mathematical Society

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