Ball convergence for a family of quadrature-based methods for solving equations in Banach space. (English) Zbl 1404.65047
Summary: We present a local convergence analysis for a family of quadrature-based predictor-corrector methods in order to approximate a locally unique solution of an equation in a Banach space setting. In earlier studies such as [C. L. Howk, ”A class of efficient quadrature-based predictor-corrector methods for solving nonlinear systems”, Appl. Math. Comput. 276, 394–406 (2016; doi:10.1016/j.amc.2015.12.032)]. the \(1 + \sqrt{2}\) order of convergence was shown on the \(m\)-dimensional Euclidean space using Taylor series expansion and hypotheses reaching up to the third-order Fréchet-derivative of the operator involved although only the first-order Fréchet-derivative appears in these methods, which restrict the applicability of these methods. In this paper, we expand the applicability of these methods in a Banach space setting and using hypotheses only on the first Fréchet-derivative. Moreover, we provide computable radii of convergence as well as error bounds on the distances involved using Lipschitz constants. Numerical examples are also presented to solve equations in cases where earlier results cannot apply.
MSC:
| 65J15 | Numerical solutions to equations with nonlinear operators |
| 47J05 | Equations involving nonlinear operators (general) |
| 47J25 | Iterative procedures involving nonlinear operators |
Keywords:
weighted-Newton method; Banach space; local convergence; Fréchet-derivative; radius of convergenceReferences:
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