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 method of steepest ortho-descent*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**9**(1961), 575–580. MR**0137297**

M. Altman,*A general method of steepest ortho-descent*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**9**(1961), 645–651. MR**0137298**

M. Altman,*A generalized method of steepest descent*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**9**(1961), 733–737. MR**0137299**

M. Altman,*A general method with minimum ortho-residua*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**9**(1961), 739–744. MR**0137300****[2]**Noël Gastinel,*Procédé itératif pour la résolution numérique d’un système d’équations linéaires*, C. R. Acad. Sci. Paris**246**(1958), 2571–2574 (French). MR**0094895****[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, National Bureau of Standards Applied Mathematics Series No. 39, U. S. Government Printing Office, Washington, D. C., 1954, pp. 71–103. MR**0066563****[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 and F. L. Bauer,*On certain iterative methods for solving linear systems*, Numer. Math.**2**(1960), 55–59. MR**0116464**, https://doi.org/10.1007/BF01386209**[7]**L. V. Kantorovič and G. P. Akilov,*\cyr Funktsional′nyĭ analiz v normirovannykh prostranstvakh*, Gosudarstv. Izdat. Fis.-Mat. Lit., Moscow, 1959 (Russian). MR**0119071****[8]**M. A. Krasnosel′skiĭ and S. G. Kreĭn,*An iteration process with minimal residuals*, Mat. Sbornik N.S.**31(73)**(1952), 315–334 (Russian). MR**0052885****[9]**M. Kerner, "Die Differentiale in der allgemeinen Analysis,"*Ann. of Math.*, v. 34, 1933, p. 546-572.**[10]**Cornelius Lanczos,*Solution of systems of linear equations by minimized-iterations*, J. Research Nat. Bur. Standards**49**(1952), 33–53. MR**0051583****[11]**Yu. Lumiste,*The method of steepest descent for nonlinear equations*, Tartu. Gos. Univ. Trudy Estest.-Mat. Fak.**37**(1955), 106–113 (Russian, with Estonian summary). MR**0076444****[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.**13**(1964), 765–794. MR**0166638****[14]**W. V. Petryshyn,*Direct and iterative methods for the solution of linear operator equations in Hilbert space*, Trans. Amer. Math. Soc.**105**(1962), 136–175. MR**0145651**, https://doi.org/10.1090/S0002-9947-1962-0145651-8**[15]**E. H. Rothe,*Gradient mappings*, Bull. Amer. Math. Soc.**59**(1953), 5–19. MR**0052681**, https://doi.org/10.1090/S0002-9904-1953-09649-5**[16]***Survey of numerical analysis*, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1962. MR**0135221****[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.**2**(1961), 201–220 (Russian). MR**0126732**

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

Article copyright:
© Copyright 1965
American Mathematical Society