Approximate zeros of quadratically convergent algorithms

Chen, Pengyuan

doi:10.1090/S0025-5718-1994-1240655-0

Approximate zeros of quadratically convergent algorithms
HTML articles powered by AMS MathViewer

by Pengyuan Chen PDF

Math. Comp. 63 (1994), 247-270 Request permission

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.

References

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, DOI 10.1016/0885-064X(89)90005-8

On the theory and computation of approximate zeros

Myong-Hi Kim, On approximate zeros and rootfinding algorithms for a complex polynomial, Math. Comp. 51 (1988), no. 184, 707–719. MR 958638, DOI 10.1090/S0025-5718-1988-0958638-1
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
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, DOI 10.1137/0215011
Steve Smale, The fundamental theorem of algebra and complexity theory, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 1, 1–36. MR 590817, DOI 10.1090/S0273-0979-1981-14858-8
Steve Smale, On the efficiency of algorithms of analysis, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 2, 87–121. MR 799791, DOI 10.1090/S0273-0979-1985-15391-1

Newton’s method estimates from data at one point

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
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, DOI 10.1016/0893-9659(88)90179-6