Convergence behavior of Gauss-Newton's method and extensions of the Smale point estimate theory

The notions of Lipschitz conditions with L average are introduced to the study of convergence analysis of Gauss-Newton's method for singular systems of equations. Unified convergence criteria ensuring the convergence of Gauss-Newton's method for one kind of singular systems of equations with constant rank derivatives are established and unified estimates of radii of convergence balls are also obtained. Applications to some special cases such as the Kantorovich type conditions, @c-conditions and the Smale point estimate theory are provided and some important known results are extended and/or improved.

[1]  Michael Shub,et al.  Newton's method for overdetermined systems of equations , 2000, Math. Comput..

[2]  José Antonio Ezquerro,et al.  Generalized differentiability conditions for Newton's method , 2002 .

[3]  Chong Li,et al.  Newton's method on Riemannian manifolds: Smale's point estimate theory under the γ-condition , 2006 .

[4]  P. Deuflhard,et al.  A Study of the Gauss-Newton Method for the Solution of Nonlinear Least Squares Problems , 1980 .

[5]  Chong Li,et al.  Convergence and uniqueness properties of Gauss-Newton's method , 2004 .

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

[7]  Pengyuan Chen.  Approximate zeros of quadratically convergent algorithms , 1994 .

[8]  Myong-Hi Kim,et al.  Newton's Method for Analytic Systems of Equations with Constant Rank Derivatives , 2002, J. Complex..

[9]  Xu,et al.  CONVERGENCE OF NEWTON'S METHOD FOR SYSTEMS OF EQUATIONS WITH CONSTANT RANK DERIVATIVES * , 2007 .

[10]  Chong Li,et al.  Majorizing Functions and Convergence of the Gauss--Newton Method for Convex Composite Optimization , 2007, SIAM J. Optim..

[11]  Xinghua Wang,et al.  Convergence of Newton's method and inverse function theorem in Banach space , 1999, Math. Comput..

[12]  Xinghua Wang,et al.  Convergence of Newton's method and uniqueness of the solution of equations in Banach space , 2000 .

[13]  Xinghua Wang,et al.  Local and global behavior for algorithms of solving equations , 2001 .

[14]  Wang Xinghua,et al.  Convergence of Newton's method and inverse function theorem in Banach space , 1999 .

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

[16]  W. Häuβler,et al.  A Kantorovich-type convergence analysis for the Gauss-Newton-Method , 1986 .

[17]  Chong Li,et al.  Kantorovich's type theorems for systems of equations with constant rank derivatives , 2008 .

[18]  Chong Li,et al.  Newton's Method for Underdetermined Systems of Equations Under the γ-Condition , 2007 .

[19]  P. Bahr,et al.  Sampling: Theory and Applications , 2020, Applied and Numerical Harmonic Analysis.

[20]  Henryk Wozniakowski,et al.  Convergence and Complexity of Newton Iteration for Operator Equations , 1979, JACM.

[21]  Chong Li,et al.  Convergence of Newton's Method and Uniqueness of the Solution of Equations in Banach Spaces II , 2003 .

[22]  José Antonio Ezquerro,et al.  On an Application of Newton's Method to Nonlinear Operators with w-Conditioned Second Derivative , 2002, BIT Numerical Mathematics.

[23]  Richard E. Ewing,et al.  "The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics" , 1986 .

[24]  S. Smale,et al.  Complexity of Bezout's theorem IV: probability of success; extensions , 1996 .

[25]  Chong Li,et al.  Convergence criterion of Newton's method for singular systems with constant rank derivatives☆ , 2008 .

[26]  R. Tapia,et al.  Optimal Error Bounds for the Newton–Kantorovich Theorem , 1974 .

[27]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

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

[29]  Adi Ben-Israel,et al.  Generalized inverses: theory and applications , 1974 .

[30]  P. Deuflhard,et al.  Affine Invariant Convergence Theorems for Newton’s Method and Extensions to Related Methods , 1979 .

[31]  M. A. Hernández.  The Newton Method for Operators with Hölder Continuous First Derivative , 2001 .

[32]  A. Ostrowski.  Solution of equations in Euclidean and Banach spaces , 1973 .

[33]  W. Deren,et al.  The theory of Smale's point estimation and its applications , 1995 .

[34]  Jose M. Gutikez.  A new semilocal convergence theorem for Newton's method , 1997 .

[35]  Ioannis K. Argyros,et al.  On the Newton-Kantorovich hypothesis for solving equations , 2004 .

[36]  Steve Smale,et al.  Complexity theory and numerical analysis , 1997, Acta Numerica.

[37]  X. Uan.  RANDOM POLYNOMIAL SPACE AND COMPUTATIONAL COMPLEXITY THEORY , 1987 .

[38]  Miguel Ángel Hernández,et al.  A note on a modification of Moser's method , 2008, J. Complex..