On general iterative methods for the solutions of a class of nonlinear operator equations

Author:
M. Z. Nashed

Journal:
Math. Comp. **19** (1965), 14-24

MSC:
Primary 65.10

DOI:
https://doi.org/10.1090/S0025-5718-1965-0179906-4

MathSciNet review:
0179906

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References | Similar Articles | Additional Information

**[1]**M. Altman, "A general method of steepest ortho-descent,"*Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys.*, v. 9, 1961, p. 645-651. MR**25**#7526. MR**0137298 (25:752b)****[2]**N. Gastinel, "Procédé itératif pour la résolution numérique d'un système d'équations linéaires,"*C. R. Acad. Sci. Paris*, v. 246, 1958, p. 2571-2574. MR**20**#1404. MR**0094895 (20:1404)****[3]**L. M. Graves, "Riemann integration and Taylor's theorem in general analysis,"*Trans. Amer. Math. Soc.*, v. 29, 1927, p. 163-177.**[4]**R. M. Hayes, "Iterative methods of solving linear problems on Hilbert space,"*Contributions to the Solution of Systems of Linear Equations and the Determination of Eigenvalues*, Nat. Bur. Standards Appl. Math. Ser., No. 39, U. S. Government Printing Office, Washington, D. C, 1954, p. 71-103. MR**16**, 597. MR**0066563 (16:597b)****[5]**M. R. Hestenes, "Hilbert space methods in variation theory and numerical analysis," Proc. Internat. Congress of Mathematicians, v. 3, 1954, p. 229-236.**[6]**A. S. Householder & F. L. Bauer, "On certain iterative methods for solving linear systems,"*Numer. Math.*, v. 2, 1960, p. 55-59. MR**22**#7251. MR**0116464 (22:7251)****[7]**L. V. Kantorovich & G. P. Akilov,*Functional Analysis in Normed Spaces*, Fizmatgiz, Moscow, 1959. (Russian) MR**22**#9837. MR**0119071 (22:9837)****[8]**M. A. Krasnosel'skiĭ & S. G. Kreĭn, "An iteration process with minimal residuals,"*Mat. Sb. (N. S.)*, v. 31 (73), 1952, p. 315-334. MR**14**, 692. MR**0052885 (14:692d)****[9]**M. Kerner, "Die Differentiale in der allgemeinen Analysis,"*Ann. of Math.*, v. 34, 1933, p. 546-572.**[10]**C. Lanczos, "Solution of systems of linear equations by minimized iterations,"*J. Res. Nat. Bur. Standards*, v. 49, 1952, p. 33-53. MR**14**, 501. MR**0051583 (14:501g)****[11]**Ju. Lumiste, "The method of steepest descent for nonlinear equations,"*Tartu. Gos. Univ. Trudy Estest.-Mat. Fak.*, v. 37, 1955, p. 106-113. (Russian. Estonian Summary) MR**17**, 900. MR**0076444 (17:900b)****[12]**M. Z. Nashed, "Iterative methods for the solutions of nonlinear operator equations in Hilbert space," Ph. D. Dissertation, The University of Michigan, Ann Arbor, Mich., 1963.**[13]**M. Z. Nashed, "The convergence of the method of steepest descents for nonlinear equations with variational or quasi-variational operators,"*J. Math. Mech.*, v. 13, 1964, p. 765-794. MR**0166638 (29:3911)****[14]**W. V. Petryshyn, "Direct and iterative methods for the solution of linear operator equations in Hilbert space,"*Trans. Amer. Math. Soc.*, v. 105, 1962, p. 136-175. MR**26**#3180. MR**0145651 (26:3180)****[15]**E. H. Rothe, "Gradient mappings,"*Bull. Amer. Math. Soc.*, v. 59, 1953, p. 5-19. MR**14**, 657. MR**0052681 (14:657d)****[16]**J. Todd, Ed.,*Survey of Numerical Analysis*, McGraw-Hill, New York, 1962. MR**0135221 (24:B1271)****[17]**M. M. Vaĭnberg,*Variational Methods for Investigation of Non-Linear Operators*, GITTL, Moscow, 1956. (Russian) MR**19**, 567.**[18]**M. M. Vaĭnberg, "On the convergence of the method of steepest descents for nonlinear equations,"*Dokl. Akad. Nauk SSSR*, v. 130, 1960, p. 9-12.*Soviet Math Dokl.*, v. 1, 1960, p. 1-4. MR**25**#751.**[19]**M. M. Vaĭnberg, "On the convergence of the process of steepest descent for nonlinear equations,"*Sibirsk. Mat. Ž*, v. 2, 1961, p. 201-220. (Russian) MR**23**#A4026. MR**0126732 (23:A4026)**

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DOI:
https://doi.org/10.1090/S0025-5718-1965-0179906-4

Article copyright:
© Copyright 1965
American Mathematical Society