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Chebyshev filter matlab code4/8/2023 The test shows that, for some problems, the methods using fractional derivatives reach different solutions with the same initial approximation. Additionally, a better performance of iterative methods based on Caputo derivatives, with respect to Riemann–Liouville-based schemes, has been observed. Also, Traub type procedures improve Newton type ones and Chebyshev type methods with order 3 α improve those with order 2 α, not only because they require fewer iterations, but also because they have higher percentages of convergence. It can be concluded that Chebyshev type methods show a better behavior for polynomials, whereas Newton and Traub type methods show a better behavior for transcendent functions. Although the interest in this kind of method is mainly theoretical, some numerical tests have been done, and the dependence on the initial estimation has been analyzed. The convergence properties of procedures CFN 2, R-LFN 2 and CFT, R-LFT imply always (at least) linear convergence, reaching order 1 + α and 1 + 2 α, respectively. Vassileva, in Fractional-Order Modeling of Dynamic Systems with Applications in Optimization, Signal Processing and Control, 2022 5.5 Concluding remarksįour fractional Newton type methods, six fractional Chebyshev type methods and two fractional Traub type schemes have been designed by using Caputo and Riemann–Liouville derivatives. Iterative processes with fractional derivatives
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