Perturbation Bounds for Nonlinear Equations
David Michael Gay · SIAM Journal on Numerical Analysis · 1981
In various contexts, such as when dealing with a simulation model, one is concerned with the solution to a system of nonlinear algebraic equations. Often the system involves measured data (and estimated coefficients), and it is of interest to find bounds on the extent to which errors in the data (and coefficients) can affect the solution. Given $\mathcal{D}$, an interval vector (i.e., Cartesian product of compact intervals) in which the data (and coefficients) are assumed to lie, this paper proposes a procedure that seeks to compute an interval vector X containing a solution to the system for any choice of data from $\mathcal{D}$. The procedure employs straightforward generalizations of a test suggested by Moore and of one suggested by Moore and Kioustelidis [SIAM J. Numer. Anal., 17 (1980), pp. 521–5291 to tests which, if passed, assure that a candidate for X is acceptable, and it uses a scalar “majorizing equation” that helps select candidates for X and sometimes guarantees that a candidate is acceptable.