New computer methods for global optimization
New computer methods for global optimization
Computation of topological degree using interval arithmetic, and applications
Mathematics of Computation
Algorithm 763: INTERVAL_ARITHMETIC: a Fortran 90 module for an interval data type
ACM Transactions on Mathematical Software (TOMS)
Computation of Singularities in Large Nonlinear Systems
SIAM Journal on Numerical Analysis
Optimal solution of nonlinear equations
Optimal solution of nonlinear equations
Existence Verification for Singular Zeros of Complex Nonlinear Systems
SIAM Journal on Numerical Analysis
Numerical Methods for the Generalized Hopf Bifurcation
SIAM Journal on Numerical Analysis
Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components
SIAM Journal on Numerical Analysis
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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.