A note on perturbed fixed slope iterations

Ahués, Mario

doi:10.1016/j.aml.2004.03.010

Cited by 2 publications

(2 citation statements)

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“…Kantorovich's genius appears when he gets the correspondence between 1 A (x) in the real case and (A (x)) −1 in the case of a Banach space, where the first derivative is in the classical sense and the second is a Fréchet derivative (see [1,4,6,9]). For the equation…”

Section: Introductionmentioning

confidence: 99%

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New linearization method for nonlinear problems in Hilbert space

2022

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In this paper, we build a Newton-like sequence to approach the zero of a nonlinear Fréchet differentiable function defined in Hilbert space. This new iterative sequence uses the concept of the adjoint operator, which makes it more manageable in practice compared to the one developed by Kantorovich which requires the calculation of the inverse operator in each iteration. Because the calculation of the adjoint operator is easier compared to the calculation of the inverse operator which requires in practice solving a system of linear equations, our new method makes the calculation of the term of our new sequence easier and more convenient for numerical approximations. We provide an a priori convergence theorem of this sequence, where we use hypotheses equivalent to those constructed by Kantorovich, and we show that our new iterative sequence converges towards the solution.

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Section: Introductionmentioning

confidence: 99%

“…which is very hard to do and usually ξ k+1 is numerically approximated in each iteration (see [1,6,9]).…”

Section: Introductionmentioning

confidence: 99%