On Location and Approximation of Clusters of Zeros: Case of Embedding Dimension One

AbstractIsolated multiple zeros or clusters of zeros of analytic maps with several variables are known to be difficult to locate and approximate. This paper is in the vein of the α-theory, initiated by M. Shub and S. Smale in the beginning of the 1980s. This theory restricts to simple zeros, i.e., where the map has corank zero. In this paper we deal with situations where the analytic map has corank one at the multiple isolated zero, which has embedding dimension one in the frame of deformation theory. These situations are the least degenerate ones and therefore most likely to be of practical significance. More generally, we define clusters of embedding dimension one. We provide a criterion for locating such clusters of zeros and a fast algorithm for approximating them, with quadratic convergence. In the case of a cluster with positive diameter our algorithm stops at a distance of the cluster which is about its diameter.

[1]  A. Griewank.  On Solving Nonlinear Equations with Simple Singularities or Nearly Singular Solutions , 1985 .

[2]  Anton Leykin,et al.  Newton's method with deflation for isolated singularities of polynomial systems , 2006, Theor. Comput. Sci..

[3]  S. Smale,et al.  Computational complexity: on the geometry of polynomials and a theory of cost. I , 1985 .

[4]  A. Griewank,et al.  Characterization and Computation of Generalized Turning Points , 1984 .

[5]  Marc Giusti,et al.  On Location and Approximation of Clusters of Zeros of Analytic Functions , 2005, Found. Comput. Math..

[6]  Annegret Hoy.  A relation between Newton and Gauss-Newton steps for singular nonlinear equations , 2005, Computing.

[7]  Andreas Griewank,et al.  The Approximate Solution of Defining Equations for Generalized Turning Points , 1996 .

[8]  A.-E. Pellet.  Sur un mode de séparation des racines des équations et la formule de Lagrange , 1881 .

[9]  J. M. Boardman.  Singularties of differentiable maps , 1967 .

[10]  A. Morgan,et al.  Computing singular solutions to polynomial systems , 1992 .

[11]  W. Govaerts.  Computation of Singularities in Large Nonlinear Systems , 1997 .

[12]  Andrew J. Sommese,et al.  Numerical Homotopies to Compute Generic Points on Positive Dimensional Algebraic Sets , 2000, J. Complex..

[13]  J. Faugère.  A new efficient algorithm for computing Gröbner bases (F4) , 1999 .

[14]  Mauro C. Beltrametti,et al.  A method for tracking singular paths with application to the numerical irreducible decomposition , 2002 .

[15]  L. Aĭzenberg,et al.  Integral Representations and Residues in Multidimensional Complex Analysis , 1983 .

[16]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[17]  A. Ostrowski.  Solution of equations and systems of equations , 1967 .

[18]  P. Kunkel.  Quadratically Convergent Methods for the Computation of Unfolded Singularities , 1988 .

[19]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

[20]  S. Smale.  Newton’s Method Estimates from Data at One Point , 1986 .

[21]  Takeo Ojika,et al.  Modified deflation algorithm for the solution of singular problems. I. A system of nonlinear algebraic equations , 1987 .

[22]  Alexander Varchenko,et al.  The classification of critical points, caustics and wave fronts , 1985 .

[23]  H. Vel.  A method for computing a root of a single nonlinear equation, including its multiplicity , 2005, Computing.

[24]  C. Kelley,et al.  Newton’s Method at Singular Points. I , 1980 .

[25]  Marc Van Barel,et al.  Computing the Zeros of Analytic Functions , 2000 .

[26]  X. Wang.  ON DOMINATING SEQUENCE METHOD IN THE POINT ESTIMATE AND SMALE THEOREM , 1990 .

[27]  Klaus Böhmer,et al.  Direct Methods for Solving Singular Nonlinear Equations , 1999 .

[28]  Frank Stenger,et al.  Computing the topological degree of a mapping inRn , 1975 .

[29]  G. Reddien.  Newton's method and high order singularities , 1979 .

[30]  B. Mourrain,et al.  Computation of a specified root of a polynomial system of equations using eigenvectors , 2000 .

[31]  A. Griewank.  Starlike domains of convergence for Newton's method at singularities , 1980 .

[32]  Michael Shub,et al.  On simple zeros of analytic functions of n variables , 2001 .

[33]  H. Van de Vel,et al.  Multiple root-finding methods , 1992 .

[34]  Michael Shub,et al.  On simple double zeros and badly conditioned zeros of analytic functions of n variables , 2001, Math. Comput..

[35]  Peter Kunkel.  A Tree-based analysis of a family of augmented systems for the computation of singular points , 1996 .

[36]  L. B. Rall.  Convergence of the newton process to multiple solutions , 1966 .

[37]  Lenore Blum,et al.  Complexity and Real Computation , 1997, Springer New York.

[38]  M. R. Osborne,et al.  Analysis of Newton’s Method at Irregular Singularities , 1983 .

[39]  Grégoire Lecerf.  Quadratic Newton Iteration for Systems with Multiplicity , 2002, Found. Comput. Math..

[40]  Peter Kunkel.  Efficient Computation of Singular Points , 1989 .

[41]  Myong-Hi Kim,et al.  Implicit Gamma Theorems (I): Pseudoroots and Pseudospectra , 2003, Found. Comput. Math..

[42]  G. Reddien.  On Newton’s Method for Singular Problems , 1978 .

[43]  Jan Verschelde,et al.  A Method for Tracking Singular Paths with Application to the Numerical Irreducible Decomposition , 2002 .

[44]  G. Reddien,et al.  Characterization and computation of singular points with maximum rank deficiency , 1986 .

[45]  Takuya Tsuchiya,et al.  Enlargement procedure for resolution of singularities at simple singular solutions of nonlinear equations , 1988 .

[46]  Jean Charles Faugère,et al.  A new efficient algorithm for computing Gröbner bases without reduction to zero (F5) , 2002, ISSAC '02.

[47]  S. Smale.  The fundamental theorem of algebra and complexity theory , 1981 .

[48]  Wolfgang Middelmann,et al.  Augmented Systems for the Computation of Singular Points in Banach Space Problems , 1998 .

[49]  Wilhelm Werner,et al.  On the accurate determination of nonisolated solutions of nonlinear equations , 1981, Computing.

[50]  T. Ojika,et al.  Deflation algorithm for the multiple roots of a system of nonlinear equations , 1983 .

[51]  Stephen Smale,et al.  Computational Complexity: On the Geometry of Polynomials and a Theory of Cost: II , 1986, SIAM J. Comput..

[52]  Peter Kravanja,et al.  A Modification of Newton's Method for Analytic Mappings Having Multiple Zeros , 1999, Computing.

[53]  Arnold Neumaier,et al.  Existence Verification for Singular Zeros of Complex Nonlinear Systems , 2000, SIAM J. Numer. Anal..

[54]  N. Yamamoto,et al.  Regularization of Solutions of Nonlinear Equations with Singular Jacobian Matrices , 1984 .

[55]  Z. Mei.  A special extended system and a Newton-like method for simple singular nonlinear equations , 2005, Computing.