1 Introduction
Root-finding method | Derivative | Order of convergence |
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Newton [20] | Classical | 2 |
Fractional | 1 | |
Damped Newton [20] | Classical | 2 |
Fractional damped Newton [3] | Fractional | 2ν |
Fractional Newton-like [10] | Fractional | ν + 1 |
Chebyshev [26] | Classical | 3 |
Fractional Chebyshev [14] | Fractional | 2ν |
Fractional Chebyshev-like [14] | Fractional | 3ν |
Damped Traub [13] | Classical | 3 |
Fractional damped Traub [10] | Fractional | 2ν + 1 |
2 Fractional derivatives
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Γ(1) = Γ(2) = 1,
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Γ(z + 1) = zΓ(z),
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Γ(n) = (n − 1)! for \(n \in \mathbb {N}\),
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\({\varGamma }(1 / 2) = \sqrt {\pi }\).
3 Iterations
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The Picard iteration (introduced in 1890) [33]:$$ u_{k+1} = w(u_{k}), \quad k = 0, 1, 2, \ldots, $$(14)
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The Mann iteration (defined in 1953) [29]:where αk ∈ (0,1].$$ u_{k+1} = \alpha_{k} w(u_{k}) + (1 - \alpha_{k}) u_{k}, \quad k = 0, 1, 2, \ldots, $$(15)
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The Khan iteration (defined in 2013) [27]:where αk ∈ (0,1].$$ \begin{array}{ll} u_{k+1} &= w(v_{k}),\\ v_{k}& = \alpha_{k} w(u_{k}) + (1 - \alpha_{k}) u_{k}, \quad k = 0, 1, 2, \ldots, \end{array} $$(16)
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The Ishikawa iteration (introduced in 1974) [23]:where αk ∈ (0,1] and βk ∈ [0,1].$$ \begin{array}{ll} u_{k+1} &= \alpha_{k} w(v_{k}) + (1 - \alpha_{k}) u_{k},\\ v_{k} &= \beta_{k} w(u_{k}) + (1 - \beta_{k}) u_{k}, \quad k=0, 1, 2, \ldots, \end{array} $$(17)
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The S iteration (defined in 2007) [2]:where αk ∈ (0,1] and βk ∈ [0,1].$$ \begin{array}{ll} u_{k+1} &= \alpha_{k} w(v_{k}) + (1 - \alpha_{k}) w(u_{k}),\\ v_{k} &= \beta_{k} w(u_{k}) + (1 - \beta_{k}) u_{k}, \quad k=0, 1, 2, \ldots, \end{array} $$(18)
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The Mann iteration for αn = 0 is the Picard iteration,
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The Ishikawa iteration for βn = 0 is the Mann iteration, and for βn = 0 and αn = 1 is the Picard iteration,
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The S iteration for αn = 1 and βn≠ 0 is the Khan iteration, and for αn = 1 and βn = 0 is the Picard iteration.
4 Newton’s method with fractional derivatives
5 Fractional Newton method with different iterations
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Fractional Newton’s method with the Mann iteration:where α ∈ (0,1].$$ z_{k+1} = (1 - \alpha) z_{k} + \alpha N_{\nu}(z_{k}), \quad k = 0, 1, 2, \ldots, $$(23)
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Fractional Newton’s method with the Khan iteration:where α ∈ (0,1].$$ \left\{\begin{array}{ll} z_{k+1} = N_{\nu}(v_{k}), \\ v_{k} = (1 - \alpha) z_{k} + \alpha N_{\nu}(z_{k}), \quad k = 0, 1, 2, \ldots, \end{array}\right. $$(24)
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Fractional Newton’s method with the Ishikawa iteration:where α ∈ (0,1] and β ∈ [0,1].$$ \left\{\begin{array}{ll} z_{k+1} = (1 - \alpha) z_{k} + \alpha N_{\nu}(v_{k}), \\ v_{k} = (1 - \beta) z_{k} + \beta N_{\nu}(z_{k}), \quad k = 0, 1, 2, \ldots, \end{array}\right. $$(25)
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Fractional Newton’s method with the S iteration:where α ∈ (0,1] and β ∈ [0,1].$$ \left\{\begin{array}{ll} z_{k+1} = (1 - \alpha) N_{\nu}(z_{k}) + \alpha N_{\nu}(v_{k}), \\ v_{k} = (1 - \beta) z_{k} + \beta N_{\nu}(z_{k}), \quad k = 0, 1, 2, \ldots, \end{array}\right. $$(26)
6 Numerical and graphical results
6.1 Stability
6.1.1 Mann iteration
6.1.2 Khan iteration
6.1.3 Ishikawa iteration
6.1.4 S iteration
6.2 Dynamics
6.2.1 Mann iteration
6.2.2 Khan iteration
6.2.3 Ishikawa iteration
6.2.4 S iteration
6.3 Numerical results
6.3.1 Mann iteration
6.3.2 Khan iteration
6.3.3 Ishikawa iteration
6.3.4 S iteration
6.4 Comments
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The stability of the considered methods and iterations highly depends on the order of the fractional derivative and less on the parameters of the iterations. The overall behaviour for all the methods is similar. For ν < 1, the vertical border between the basins of attraction wraps up to the right and it becomes more rough. For ν > 1, the basin for − 1 shrinks and forms a closed shape that depends on the iteration. The best stability compared with the most stable case presented in Fig. 1 is obtained near ν = 1, which corresponds to the classical derivative.
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For ν-order approaching to 1.0 from above or from below, the fractional Newton method is equivalent to the classical Newton root-finding method modified by α or α and β parameters in the used iterations. For the classical Newton method with various iterations, one can observe symmetric polynomiographs having three braids separating cube roots of unity. The width of those braids depends on the kind of iteration used and its parameters. A more detailed discussion on that case can be found in [18].
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With ν decreasing below 1.0 or ν increasing above 1.0, one can observe that the polynomiographs lose their rotational symmetry, but they still possess an axial symmetry with only one axis of symmetry located on the real axis. Moreover, their shapes are changing quickly, especially in their left parts, where areas of chaos and non-convergence are getting larger and are more visible. Fractional Newton’s method gradually loses possibility of finding complex cube roots of unity, but the real root (1) is still found.
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The use of various iterations in fractional Newton’s method has influence on the convergence of the processes (measured by ANI and CAI). Namely, Khan assures better convergence than Mann. Unfortunately, both Mann and Khan are worse in comparison with Picard iteration for most of the parameters’ values. In the neighbourhood of α = 1, Khan iteration obtains better (lower) values of ANI in comparison with the Picard iteration. Similarly, S iteration is better than Ishikawa, but both are worse in comparison with Picard iteration for most of the parameters’ values. The S iteration obtains lower values of ANI than Picard iteration for values in the neighbourhood of the (1,1) point in the parameters’ space. In each case, the size of the neighbourhood depends on the value of the derivative order ν used in Newton’s method. Moreover, the dependency of the two measures (ANI, CAI) on the parameters of the methods (derivative order and the iteration parameters) is a highly nonlinear one. The minimal and maximal values of the ANI and CAI for the considered methods and iterations are gathered in Table 2.
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For each of the considered iterations (Picard, Mann, Khan, Ishikawa and S), one can observe that the obtained generation times of polynomiographs are much longer than those in the case of the classical derivative. This is an obvious observation when one looks at the complexity of computations needed in each case. To compute fractional derivative, we need more computations and in consequence one iteration of Newton’s method has greater computational complexity. Moreover, from the generation time plots, we can observe that the dependency of this measure on the parameters (derivative order and the iteration parameters) is a nonlinear one. The minimal and maximal values of the time (in seconds) for the considered methods and iterations are gathered in Table 2.
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In general, C-derivative assures better behaviour of the considered root-finding processes in comparison with RL-derivatives.
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Two-parameter iterations allow obtaining more ample collection of different images in comparison with those generated with the help of one-parameter iterations.
Method | ANI (min) | ANI (max) | CAI (min) | CAI (max) | Time (min) | Time (max) |
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Newton | 5.056 | 5.056 | 1.0 | 1.0 | 0.747 | 0.747 |
(Picard) | ||||||
F-N RL | 4.918 | 40.0 | 0.0 | 1.0 | 1.698 | 11.058 |
(Picard) | (1.02) | (0.02) | (0.02) | (0.86) | (0.98) | (0.72) |
F-N C | 5.093 | 40.0 | 0.0 | 1.0 | 1.108 | 7.275 |
(Picard) | (1.02) | (0.02) | (0.02) | (0.96) | (1.04) | (0.02) |
Newton | 5.056 | 40.0 | 0.0 | 1.0 | 0.634 | 4.386 |
(Mann) | (1.0) | (0.02) | (0.02) | (0.6) | (0.98) | (0.06) |
F-N RL | 4.918 | 40.0 | 0.0 | 1.0 | 1.797 | 16.990 |
(Mann) | (1.02, 1.0) | (0.02, 0.04) | (0.02, 0.04) | (0.84, 0.72) | (1.02, 1.0) | (1.12, 0.2) |
F-N C | 5.093 | 40.0 | 0.0 | 1.0 | 1.164 | 10.501 |
(Mann) | (1.02, 1.0) | (0.02, 0.02) | (0.02, 0.02) | (0.96, 0.9) | (1.02, 1.0) | (0.04, 0.24) |
Newton | 2.281 | 4.668 | 1.0 | 1.0 | 0.537 | 1.298 |
(Khan) | (1.0) | (0.02) | (0.02) | (0.02) | (0.96) | (0.02) |
F-N RL | 2.232 | 40.0 | 0.0 | 1.0 | 1.733 | 24.156 |
(Khan) | (1.02, 1.0) | (0.02, 0.02) | (0.02, 0.02) | (0.76, 0.28) | (1.02, 1.0) | (0.02, 0.82) |
F-N C | 2.317 | 40.0 | 0.0 | 1.0 | 1.123 | 13.907 |
(Khan) | (1.02, 1.0) | (0.02, 0.02) | (0.02, 0.02) | (0.82, 0.72) | (1.02, 0.96) | (0.02, 0.22) |
Newton | 2.281 | 40.0 | 0.0 | 1.0 | 0.583 | 8.057 |
(Ishikawa) | (1.0, 1.0) | (0.02, 0.02) | (0.02, 0.02) | (0.4, 0.6) | (1.0, 1.0) | (0.02, 0.02) |
F-N RL | 2.619 | 40.0 | 0.0 | 1.0 | 1.996 | 28.237 |
(Ishikawa) | (1.1, | (0.5, | (0.5, | (0.9, | (1.1, | (0.7, |
1.0, 1.0) | 0.02, 0.02) | 0.02, 0.02) | 0.34, 0.18) | 1.0, 1.0) | 0.02, 0.66) | |
F-N C | 2.757 | 40.0 | 0.0 | 1.0 | 1.332 | 21.171 |
(Ishikawa) | (1.1, | (0.5, | (0.5, | (0.9, | (1.1, | (1.3, |
1.0, 1.0) | 0.02, 0.02) | 0.02, 0.02) | 0.56, 0.84) | 1.0, 1.0) | 0.02, 0.34) | |
Newton | 2.281 | 5.053 | 1.0 | 1.0 | 0.552 | 1.266 |
(S) | (1.0, 1.0) | (0.02, 0.04) | (0.02, 0.02) | (0.02, 0.02) | (1.0, 1.0) | (0.02, 0.02) |
F-N RL | 2.619 | 40.0 | 0.0 | 1.0 | 2.061 | 26.241 |
(S) | (1.1, | (0.5, | (0.5, | (0.9, | (1.1, | (0.5, |
1.0, 1.0) | 0.02, 0.04) | 0.02, 0.02) | 0.02, 0.02) | 1.0, 1.0) | 0.02, 0.44) | |
F-N C | 2.757 | 36.066 | 0.211 | 1.0 | 1.319 | 16.716 |
(S) | (1.1, | (0.5, | (0.5, | (0.9, | (1.1, | (0.5, |
1.0, 1.0) | 0.06, 0.02) | 0.06, 0.02) | 0.02, 0.92) | 1.0, 1.0) | 0.02, 0.72) |