Verifying Topological Indices for Higher-Order Rank Deficiencies

It has been known how to use computational fixed point theorems to verify existence and uniqueness of a true solution to a nonlinear system of equations within a small region about an approximate solution. This can, be done in O(n3) operations, where n is the number of equations and unknowns. However, these standard techniques are only valid if the Jacobi matrix for the system is nonsingular at the solution. In previous work and a dissertation (of Dian), we have shown, both theoretically and practically, that existence and multiplicity can be verified in a complex setting, and in the real setting for odd multiplicity, when the rank defect of the Jacobi matrix at an isolated solution is 1. Here, after reviewing work to date, we discuss the case of higher rank defect. In particular, it appears that p-dimensional searches are required if the rank defect is p, and that the work increases exponentially in p.

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

[2]  R. B. KEARFOTTy,et al.  EXISTENCE VERIFICATION FOR SINGULAR ZEROS OF NONLINEAR SYSTEMS∗ , 1999 .

[3]  Jon G. Rokne,et al.  New computer methods for global optimization , 1988 .

[4]  R. B. Kearfott.  Rigorous Global Search: Continuous Problems , 1996 .

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

[6]  Gerd Bohlender,et al.  Bibliography on Enclosure Methods and Related Topics , 1993 .

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

[8]  Andrew J. Sommese,et al.  Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components , 2000, SIAM J. Numer. Anal..

[9]  Frank Stenger.  An algorithm for the topological degree of a mapping in $n$-space , 1975 .

[10]  Willy Govaerts,et al.  Numerical Methods for the Generalized Hopf Bifurcation , 2000, SIAM J. Numer. Anal..

[11]  Eldon Hansen,et al.  Global optimization using interval analysis , 1992, Pure and applied mathematics.

[12]  R. Baker Kearfott,et al.  Algorithm 763: INTERVAL_ARITHMETIC: a Fortran 90 module for an interval data type , 1996, TOMS.

[13]  H. Peitgen,et al.  Topological Perturbations in the Numerical Study of Nonlinear Eigenvalue and Bifurcation Problems , 1980 .

[14]  Oliver Aberth.  Computation of topological degree using interval arithmetic, and applications , 1994 .

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

[16]  Christopher A. Sikorski.  Optimal solution of nonlinear equations , 1985, J. Complex..

[17]  G. Alefeld,et al.  Introduction to Interval Computation , 1983 .

[18]  S. M. Robinson.  Analysis and computation of fixed points , 1980 .

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

[20]  V. Kreinovich.  Computational Complexity and Feasibility of Data Processing and Interval Computations , 1997 .

[21]  Baker Kearfott,et al.  An efficient degree-computation method for a generalized method of bisection , 1979 .

[22]  R. Baker Kearfott,et al.  Existence Verification for Higher Degree Singular Zeros of Nonlinear Systems , 2003, SIAM J. Numer. Anal..

[23]  A. Neumaier.  Interval methods for systems of equations , 1990 .

[24]  R. B. Kearfott,et al.  Interval Computations: Introduction, Uses, and Resources , 2000 .