Extending the applicability of high-order iterative schemes under Kantorovich hypotheses and restricted convergence regi
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Extending the applicability of high-order iterative schemes under Kantorovich hypotheses and restricted convergence regions Ioannis K. Argyros1 · Santhosh George2 · Shobha M. Erappa3 Received: 5 April 2019 / Accepted: 9 October 2019 / Published online: 11 October 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract We use restricted convergence regions to locate a more precise set than in earlier works containing the iterates of some high-order iterative schemes involving Banach space valued operators. This way the Lipschitz conditions involve tighter constants than before leading to weaker sufficient semilocal convergence criteria, tighter bounds on the error distances and an at least as precise information on the location of the solution. These improvements are obtained under the same computational effort since computing the old Lipschitz constants also requires the computation of the new constants as special cases. The same technique can be used to extend the applicability of other iterative schemes. Numerical examples further validate the new results. Keywords Banach space · High convergence order schemes · Semi-local convergence Mathematics Subject Classification 65J20 · 49M15 · 74G20 · 41A25
1 Introduction Let E 1 , E 2 be Banach spaces and D ⊂ E 1 be a non-empty convex set. Consider F : D → E 2 to be a twice continously differentiable operator in the sense of Fréchet. There is a plethora of problems from many diverse disciplines such as mathematics, computational sciences,
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Shobha M. Erappa [email protected] Ioannis K. Argyros [email protected] Santhosh George [email protected]
1
Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA
2
Department of Mathematical and Computational Sciences, NIT Karnataka, Mangalore 575 025, India
3
Department of Mathematics, Manipal Institute of Technology, Manipal, Karnataka 576104, India
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physics and also engineering that can be reduced to solving a non-linear equation of the form F(x) = 0,
(1.1)
using mathematical modeling [1–17]. A locally unique solution x∗ in closed form is desirable. But this can rarely be achieved. That is why most researchers and practitioners develop iterative schemes which converges to x∗ , if certain convergence criteria are satisfied. Newton’s scheme defined for each n = 0, 1, 2, . . . by xn+1 = xn − F (xn )−1 F(xn )
(1.2)
is the most popular iterative scheme which converges quadratically under certain conditions. To produce even higher convergence order the following scheme has been considered [1], yn = xn − F (xn )−1 F(xn ) xn+1 = yn − (I + L F (xn ) + L 2F (xn )G F (xn ))F (xn )−1 F(yn ),
(1.3)
where L F (x) = F (x)−1 F (x)F (x)−1 F(x) and G F : D → Ł(E 1 , E 1 ), where Ł(E 1 , E 1 ) stands for the space of bounded linear operators from E 1 into E 1 . The operator G F depends on F and its derivatives. The computation of F is expensive in general. As an example for a system of i equations with i unknowns, the Fréchet derivative
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