Approximate zeros of quadratically convergent algorithms
Author:
Pengyuan Chen
Journal:
Math. Comp. 63 (1994), 247270
MSC:
Primary 65H05; Secondary 65E05, 65Y20
MathSciNet review:
1240655
Fulltext PDF Free Access
Abstract 
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Abstract: Smale's condition for a point to be an approximate zero of a function for Newton's method is extended to the general quadratically convergent iterative algorithm. It is shown in which way the bound in the condition is affected by the characteristics of the algorithm. This puts the original condition of Smale for Newton's method in a more general perspective. The results are also discussed in the light of numerical evidence.
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 J. H. Curry, On zero finding methods of higher order from data at one point, J. Complexity 5 (1989), 219237. MR 1006107 (91a:65134)
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 J. H. Curry and E.S. Van Vleck, On the theory and computation of approximate zeros, preprint.
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 M. Kim, On approximate zeros and rootfinding algorithms for a complex polynomial, Math. Comp. 51 (1988), 707719. MR 958638 (90f:65073)
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 , Computational complexity: on the geometry of polynomials and a theory of costs. II, SIAM J. Comput. 15 (1986), 145161. MR 822199 (87m:68044)
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 , On the efficiency of algorithms of analysis, Bull. Amer. Math. Soc. (N.S.) 13 (1985), 87121. MR 799791 (86m:65061)
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 W.C. Rheinboldt, On a theorem of S. Smale about Newton's method for analytic mappings, Appl. Math. Lett. 1 (1988), 6972. MR 947170 (89h:65079)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199412406550
PII:
S 00255718(1994)12406550
Article copyright:
© Copyright 1994 American Mathematical Society
