Approximate zeros of quadratically convergent algorithms

Author:
Pengyuan Chen

Journal:
Math. Comp. **63** (1994), 247-270

MSC:
Primary 65H05; Secondary 65E05, 65Y20

MathSciNet review:
1240655

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Abstract | References | Similar Articles | Additional Information

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.

**[1]**James H. Curry,*On zero finding methods of higher order from data at one point*, J. Complexity**5**(1989), no. 2, 219–237. MR**1006107**, 10.1016/0885-064X(89)90005-8**[2]**J. H. Curry and E.S. Van Vleck,*On the theory and computation of approximate zeros*, preprint.**[3]**Myong-Hi Kim,*On approximate zeros and rootfinding algorithms for a complex polynomial*, Math. Comp.**51**(1988), no. 184, 707–719. MR**958638**, 10.1090/S0025-5718-1988-0958638-1**[4]**Mike Shub and Steven Smale,*Computational complexity. On the geometry of polynomials and a theory of cost. I*, Ann. Sci. École Norm. Sup. (4)**18**(1985), no. 1, 107–142. MR**803197****[5]**M. Shub and S. Smale,*Computational complexity: on the geometry of polynomials and a theory of cost. II*, SIAM J. Comput.**15**(1986), no. 1, 145–161. MR**822199**, 10.1137/0215011**[6]**Steve Smale,*The fundamental theorem of algebra and complexity theory*, Bull. Amer. Math. Soc. (N.S.)**4**(1981), no. 1, 1–36. MR**590817**, 10.1090/S0273-0979-1981-14858-8**[7]**Steve Smale,*On the efficiency of algorithms of analysis*, Bull. Amer. Math. Soc. (N.S.)**13**(1985), no. 2, 87–121. MR**799791**, 10.1090/S0273-0979-1985-15391-1**[8]**-,*Newton's method estimates from data at one point*, The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics (Richard E. Ewing, Kenneth I. Gross, and Clyde F. Martin, eds.), Springer-Verlag, 1986, pp. 185-196.**[9]**Steve Smale,*Algorithms for solving equations*, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 172–195. MR**934223****[10]**Werner C. Rheinboldt,*On a theorem of S. Smale about Newton’s method for analytic mappings*, Appl. Math. Lett.**1**(1988), no. 1, 69–72. MR**947170**, 10.1016/0893-9659(88)90179-6

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DOI:
https://doi.org/10.1090/S0025-5718-1994-1240655-0

Article copyright:
© Copyright 1994
American Mathematical Society