Numerical continuation methods: an introduction
Numerical continuation methods: an introduction
Numerical analysis: mathematics of scientific computing
Numerical analysis: mathematics of scientific computing
A construction procedure of iterative methods with cubical convergence
Applied Mathematics and Computation
Third-order iterative methods for operators with bounded second derivative
ICCAM '96 Proceedings of the seventh international congress on Computational and applied mathematics
On a convex acceleration of Newton's Method
Journal of Optimization Theory and Applications
Modification of the Kantorovich assumptions for semilocal convergence of the Chebyshev method
Journal of Computational and Applied Mathematics
An acceleration of Newton's method: Super-Halley method
Applied Mathematics and Computation
Journal of Computational and Applied Mathematics
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In this paper, the semilocal convergence of a continuation method combining the Chebyshev method and the convex acceleration of Newton's method used for solving nonlinear equations in Banach spaces is established by using recurrence relations under the assumption that the second Frechet derivative satisfies the Holder continuity condition. This condition is mild and works for problems in which the second Frechet derivative fails to satisfy Lipschitz continuity condition. A new family of recurrence relations are defined based on two constants which depend on the operator. The existence and uniqueness regions along with a closed form of the error bounds in terms of a real parameter @a@?[0,1] for the solution x^* is given. Two numerical examples are worked out to demonstrate the efficacy of our approach. On comparing the existence and uniqueness regions for the solution obtained by our analysis with those obtained by using majorizing sequences under Holder continuity condition on F^'', it is found that our analysis gives improved results. Further, we have observed that for particular values of the @a, our analysis reduces to those for the Chebyshev method (@a=0) and the convex acceleration of Newton's method (@a=1) respectively with improved results.