Summary
Nonlinear stationary fixed point iterations inR n are considered. The Perron-Ostrowski theorem [23] guarantees convergence if the iteration functionG possesses an isolated fixed pointu. In this paper a sufficient condition for convergence is given ifG possesses a manifold of fixed points.
As an application, convergence of a nonlinear extension of the method of Kaczmarz is proved. This method is applicable to underdetermined equations; it is appropriate for the numerical treatment of large and possibly ill-conditioned problems with a sparse, nonsquare Jacobian matrix. A practical example of this type (nonlinear image reconstruction in ultrasound tomography) is included.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Altman, M.: Connection between gradient methods and Newton's method for functionals. Bull. de l'Acad. Polon. Sciences, Série math., astr., phys. Vol.IX, 12, 877–880 (1961)
Ansorge, R.: Über die Konvergenz der Iterationsverfahren zur Auflösung linearer Gleichungssysteme im Falle einer singulären Koeffizientenmatrix. Z. Angew. Math. Mech.40, 427 (1960)
Censor, Y.: Row-action methods for huge and sparse systems and their applications. SIAM Rev.23, 4, 444–466 (1982)
Censor, Y., Gustafson, D.E., Lent, A., Tuy, H.: A new approach to the emission computerized tomography problem: simultaneous calculation of attenuation and activity coefficients. IEEE Trans. Nucl. Soc. NS26, 2, 2775–2779 (1979)
Censor, Y., Lent, A.: A cyclic subgradient projection method for the convex feasibility problem. Preprint, University of Haifa 1980
Elfving, T.: Block-iterative methods for consistent and inconsistent linear equations. Numer. Math.35, 1–12 (1980)
Eremin, I.I.: Certain iteration methods in convex programming. Ekonom. i Mat. Metody2, 870–886 (1966) (in Russian)
Friedrich, V.: Zur iterativen Behandlung unterbestimmter und nichtkorrekter linearer Aufgaben. Beiträge zur Numer. Math.3, 11–20 (1975)
Gordon, R., Bender, R., Herman, G.T.: Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography. J. Theor. Biol.29, 471–481 (1970)
Hart, W.L., Motzkin, T.: A composite Newton-Raphson gradient iteration method for the solution of systems of equations. Pacific J. Math.6, 691–707 (1956)
Herman, G.T., Lent, A., Lutz, P.H.: Iterative relaxation methods for image reconstruction. Proc. ACM '75, Minneapolis Conf. pp. 169–174 (1975)
Huang, T.S.: Introduction. In: Huang, T.S. (ed.). Picture processing and digital filtering. Berlin, Heidelberg, New York: Springer 1979
Johnson, L.E., Gilbert, F.E.: Inversion and inference for teleseismic ray data. In: Bolt, B.A. (ed.). Methods of computational physics 12 (Seismology): Body waves and sources. New York, London: Academic Press 1972
Kaczmarz, S.: Angenäherte Auflösung von Systemen linearer Gleichungen. Bull. Akad. Pol. Soc. Lett. a35, 355–357 (1937)
Kak, A.C.: Computerized tomography with X-ray, emission and ultrasound sources. Proc. IEEE67, 1245–1272 (1979)
Keller, H.B.: On the solution of singular and semidefinite linear systems by iteration. SIAM J. Numer. Anal.2, 281–290 (1965)
McCormick, S.F.: An iterative procedure for the solution of constrained nonlinear equations with application to optimization problems. Numer. Math.23, 371–385 (1975)
McCormick, S.F.: The methods of Kaczmarz and iterative row orthogonalization for solving linear equations and least squares problems in Hilbert space. Indiana University. Math. J.26, (1977)
McKinnon, G.C., Bates, R.H.T.: A limitation on ultrasonic transmission tomography. Ultrasonic Imaging2, 48–54 (1980)
Muhometov, R.G.: The problem of recovery of a two-dimensional Riemannian metric and integral geometry. Soviet. Math. Dokl.18, 27–31 (1977)
Oldenburger, R.: Infinite powers of matrices and characteristic roots. Duke Math. J.6, 357–361 (1940)
Ortega, J., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. New York: Academic Press 1970
Ostrowski, A.: Solution of equations and systems equations. New York: Academic Press 1966
Ostrowski, A.: Vorlesungen über Differential- und Integralrechnung, Bd. II. Basel: Birkhäuser 1968
Raik, E.: Fejér type methods in Hilbert space. Eesti NSV Tead. Akad. Toimetised Füüs.-Mat.16, 286–293 (1967) (in Russian)
Schomberg, H.: An improved approach to reconstructive ultrasound tomography. J. Phys., D: Appl. Phys.11, L181–L185 (1978)
Schomberg, H.: Nonlinear image reconstruction from projections of ultrasonic travel times and electric current densities. In: Herman, G.T., Natterer, F. (eds.). Mathematical aspects of computerized tomography, Lect. Notes in Med. and Biol. 8. Berlin, Heidelberg, New York: Springer 1981
Tanabe, K.: Projection method for solving a singular system of linear equations, Numer. Math.17, 203–214 (1971)
Tanabe, K.: Characterization of linear stationary iterative processes for solving a singular system of linear equations. Numer. Math.22, 349–359 (1974)
Tompkins, C.: Projection methods in calculation. Proc. Sec. Symp. Lin. Progr., Washington D.C., pp. 425–448, 1955
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Meyn, KH. Solution of underdetermined nonlinear equations by stationary iteration methods. Numer. Math. 42, 161–172 (1983). https://doi.org/10.1007/BF01395309
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01395309