Enlargement procedure for resolution of singularities at simple singular solutions of nonlinear equations
Summary
A solution of a nonlinear equation in Hilbert spaces is said to be a simple singular solution if the Fréchet derivative at the solution has one-dimensional kernel and cokernel. In this paper we present the enlargement procedure for resolution of singularities at simple singular solutions of nonlinear equations. Once singularities are resolved, we can compute accurately the singular solution by Newton's method. Conditions for which the procedure terminates in finite steps are given. In particular, if the equation defined in ℝn is analytic and the simple singular solution is geometrically isolated, the procedure stops in finite steps, and we obtain the enlarged problem with an isolated solution. Numerical examples are given.
Subject Classifications
AMS(MOS): 65J15 CR: G1.5Preview
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References
- 1.Decker, D.W., Keller, H.B., Kelley, C.T.: Convergence rates for Newton's method at singular points. SIAM J. Numer. Anal.20, 296–314 (1983)Google Scholar
- 2.Dieudonné, J.: Foundations of modern analysis. New York: Academic Press 1969Google Scholar
- 3.Keller, H.B.: Geometrically isolated nonisolated solutions and their approximation. SIAM J. Numer. Anal.18, 822–838 (1981)Google Scholar
- 4.Rall, L.B.: Convergence of the Newton process to multiple solutions. Numer. Math.9, 23–37 (1966)Google Scholar
- 5.Rall, L.B.: Computational solution of nonlinear operator equations. New York: John Wiley 1969Google Scholar
- 6.Seydel, R.: Numerical computation of branch points in nonlinear equations. Numer. Math.33, 339–352 (1979)Google Scholar
- 7.Weber, H., Werner, W.: On the accurate determination of nonisolated solutions of nonlinear equations. Computing26, 315–326 (1981)Google Scholar
- 8.Yamamoto, N.: Newton's method for singular problems and its application to boundary value problem. J. Math. Tokushima Univ.17, 27–88 (1983)Google Scholar
- 9.Yamamoto, N.: Regularization of solutions of nonlinear equations with singular Jacobian matrices. J. Inf. Process.7, 16–21 (1984)Google Scholar