| Summary: | In the recent literature, some fractional one-point Newton-type methods have been proposed in order to find roots of nonlinear equations using fractional derivatives. In this paper, we introduce a new fractional Newton-type method with order of convergence <inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> and compare it with the existing fractional Newton method with order <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2</mn> <mi>α</mi> </mrow> </semantics> </math> </inline-formula>. Moreover, we also introduce a multipoint fractional Traub-type method with order <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2</mn> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> and compare its performance with that of its first step. Some numerical tests and analysis of the dependence on the initial estimations are made for each case, including a comparison with classical Newton (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> of the first step of the class) and classical Traub’s scheme (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> of fractional proposed multipoint method). In this comparison, some cases are found where classical Newton and Traub’s methods do not converge and the proposed methods do, among other advantages.
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