It is known that the critical condition which guarantees quadratic convergence of approximate Newton methods is an approximation of the identity condition. This requires that the composition of the numerical inversion of the Fréchet derivative with the derivative itself approximate the identity to an accuracy calibrated by the residual. For example, the celebrated quadratic convergence theorem of Kantorovich can be proven when this holds, subject to reg ularity and stability of the derivative map. In this paper, we study what happens when this condition is not evident "a priori" but is observed "a posteriori". Through an in-depth example involving a semilinear elliptic boundary value problem, and some general theory, we study the condition in the context of dual norms, and the effect upon convergence. We also discuss the connection to Nash iteration.
- Approximation of the identity
- Nash iteration
- Newton methods
- Partial differential equations
ASJC Scopus subject areas
- Applied Mathematics