Majorizing sequences and error bounds for iterative methods

Author:
George J. Miel

Journal:
Math. Comp. **34** (1980), 185-202

MSC:
Primary 65J05; Secondary 47H10

DOI:
https://doi.org/10.1090/S0025-5718-1980-0551297-4

MathSciNet review:
551297

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Given a sequence in a Banach space, it is well known that if there is a sequence such that and , then converges to some and the error bounds hold. It is shown that certain stronger hypotheses imply sharper error bounds,

**[1]**J. E. Dennis Jr.,*Toward a unified convergence theory for Newton-like methods*, Nonlinear Functional Anal. and Appl. (Proc. Advanced Sem., Math. Res. Center, Univ. of Wisconsin, Madison, Wis., 1970) Academic Press, New York, 1971, pp. 425–472. MR**0278556****[2]**J. E. Dennis Jr.,*A brief introduction to quasi-Newton methods*, Numerical analysis (Proc. Sympos. Appl. Math., Atlanta, Ga., 1978), Proc. Sympos. Appl. Math., XXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 19–52. MR**533049****[3]**J. E. Dennis Jr. and Jorge J. Moré,*A characterization of superlinear convergence and its application to quasi-Newton methods*, Math. Comp.**28**(1974), 549–560. MR**0343581**, https://doi.org/10.1090/S0025-5718-1974-0343581-1**[4]**J. E. Dennis Jr. and Jorge J. Moré,*Quasi-Newton methods, motivation and theory*, SIAM Rev.**19**(1977), no. 1, 46–89. MR**0445812**, https://doi.org/10.1137/1019005**[5]**W. B. Gragg and R. A. Tapia,*Optimal error bounds for the Newton-Kantorovich theorem*, SIAM J. Numer. Anal.**11**(1974), 10–13. MR**0343594**, https://doi.org/10.1137/0711002**[6]**Harris Hancock,*Elliptic integrals*, Dover Publications, Inc., New York, 1958. MR**0099454****[7]**L. V. Kantorovič,*Functional analysis and applied mathematics*, Uspehi Matem. Nauk (N.S.)**3**(1948), no. 6(28), 89–185 (Russian). MR**0027947****[8]**K. KNOPP,*Theory and Application of Infinite Series*, Blackie & Son Ltd., London and Glasgow, 1928.**[9]**N. S. Kurpel′,*Projection-iterative methods for solution of operator equations*, American Mathematical Society, Providence, R. I., 1976. Translated from the Russian; Translations of Mathematical Monographs, Vol. 46. MR**0405140****[10]**P. Lancaster,*Error analysis for the Newton-Raphson method*, Numer. Math.**9**(1966), 55–68. MR**0210315**, https://doi.org/10.1007/BF02165230**[11]**George Miel,*On a posteriori error estimates*, Math. Comp.**31**(1977), no. 137, 204–213. MR**0426418**, https://doi.org/10.1090/S0025-5718-1977-0426418-4**[12]**George J. Miel,*Cones and error bounds for linear iterations*, Aequationes Math.**20**(1980), no. 2-3, 170–183. MR**577486**, https://doi.org/10.1007/BF02190512**[13]**G. J. Miel,*The Kantorovich theorem with optimal error bounds*, Amer. Math. Monthly**86**(1979), no. 3, 212–215. MR**522348**, https://doi.org/10.2307/2321528**[14]**G. J. MIEL,*Exit Criteria for Newton-Type Iterations*, Research Paper No. 363, Dept. of Math. and Stat., Univ. of Calgary, 1977.**[15]**George J. Miel,*Unified error analysis for Newton-type methods*, Numer. Math.**33**(1979), no. 4, 391–396. MR**553349**, https://doi.org/10.1007/BF01399322**[16]**James M. Ortega,*The Newton-Kantorovich theorem*, Amer. Math. Monthly**75**(1968), 658–660. MR**0231218**, https://doi.org/10.2307/2313800**[17]**J. M. Ortega and W. C. Rheinboldt,*Iterative solution of nonlinear equations in several variables*, Academic Press, New York-London, 1970. MR**0273810****[18]**Alexandre Ostrowski,*La méthode de Newton dans les espaces de Banach*, C. R. Acad. Sci. Paris Sér. A-B**272**(1971), A1251–A1253. MR**0285110****[19]**L. B. Rall,*Quadratic equations in Banach spaces*, Rend. Circ. Mat. Palermo (2)**10**(1961), 314–332. MR**0144184**, https://doi.org/10.1007/BF02843677**[20]**Louis B. Rall,*Computational solution of nonlinear operator equations*, With an appendix by Ramon E. Moore, John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR**0240944****[21]**L. B. Rall,*A note on the convergence of Newton’s method*, SIAM J. Numer. Anal.**11**(1974), 34–36. MR**0343599**, https://doi.org/10.1137/0711004**[22]**Werner C. Rheinboldt,*A unified convergence theory for a class of iterative processes*, SIAM J. Numer. Anal.**5**(1968), 42–63. MR**0225468**, https://doi.org/10.1137/0705003**[23]**J. ROCKNE, "Newton's method under mild differentiability conditions with error analysis,"*Numer. Math.*, v. 18, 1972, pp. 401-412.**[24]**Andrew H. Sherman,*On Newton-iterative methods for the solution of systems of nonlinear equations*, SIAM J. Numer. Anal.**15**(1978), no. 4, 755–771. MR**0483382**, https://doi.org/10.1137/0715050**[25]**A. I. ZINČENKO, "A class of approximate methods for solving operator equations with nondifferentiable operators,"*Dopovīdī Akad. Nauk Ukrdïn. RSR*,**1963**, pp. 852-855. (Ukrainian)

Retrieve articles in *Mathematics of Computation*
with MSC:
65J05,
47H10

Retrieve articles in all journals with MSC: 65J05, 47H10

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1980-0551297-4

Keywords:
Majorizing sequences,
iterative methods,
exit criteria

Article copyright:
© Copyright 1980
American Mathematical Society