Fractional Newton-Raphson Method and Some Variants for the Solution of Non-linear Systems

A. Torres-Hernandez; F. Brambila-Paz

REVIEW 3 major objections 4 minor 13 references

Fractional Newton-Raphson Method and Some Variants for the Solution of Non-linear Systems

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Varying the fractional derivative order lets Newton-type iterations leave the real line and land on complex roots.

desk verdict Interesting new fractional Newton variants, but the numerical tables rest on an undefined complex extension of the fractional derivative, so the central evidence is not yet reproducible. read the letter →

arxiv 1908.01453 v6 pith:43STF3VI submitted 2019-08-05 math.NA cs.NAphysics.app-ph

A. Torres-Hernandez , F. Brambila-Paz This is my paper · ORCID

classification math.NAcs.NAphysics.app-ph MSC 65H1026A33

keywords fractionalNewtonmethodnonlinearsystemscomplexrootsderivativeorderofconvergencequasi-Newtonpseudo-NewtonRiemann-Liouville

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes Newton-type iterations in which the order of the derivative is a function of the current point rather than a fixed integer. Its central aim is to show that sweeping this fractional order allows a single real initial condition to produce both real and complex solutions of nonlinear systems. The first method, the fractional Newton method (29), is argued to converge with order at least quadratic near simple roots; the fractional quasi-Newton method (39) and pseudo-Newton method (47) trade that speed for cheaper derivative matrices and avoid the discontinuity caused by fractional derivatives of constants. If the central claim is correct, these are usable algorithms for collecting many roots of polynomials and of systems with large or infinite solution sets.

What carries the argument

The carrying object is the fractional Jacobian matrix $f^{(\alpha)}(x)=\left(\partial_j^\alpha f_k(x)\right)$, where $\partial_j^\alpha$ is a Riemann-Liouville or Caputo-type derivative acting only on $x_j$ and continuous in $\alpha$ at $\alpha=1$. The fractional Newton method is $x_{i+1}=x_i-\left(f^{(\alpha_f(x_i))}(x_i)\right)^{-1}f(x_i)$, with $\alpha_f$ as in (28); the variants exchange $f^{(\alpha)}$ for $Q_{g,\beta}$ (built from a linearization) or $P_{\epsilon,\beta}$ (a diagonal matrix). The variable order is what lets the iteration leave the real line: for a monomial $x^m$, a fractional Riemann-Liouville derivative produces $x^{m-\alpha}$ with a rational exponent, so real inputs can become complex, while switching the order to 1 near the root recovers the quadratic convergence condition.

What would settle it

Recompute the iteration (29) for f(x)=sin(x)-3/(2x) with x0=0.26 and the $\alpha$ values of Table 2, recording residuals under the principal branch of the Riemann-Liouville formula; if the reported residuals appear only after silently using an analytic continuation, the real-domain formulation is incomplete. Separately, near a simple root compute the ratios $|x_{k+1}-\xi|/|x_k-\xi|^2$ for the fractional Newton method; if these ratios do not remain bounded, the claimed quadratic order fails.

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Extended reading notes

Core claim

The central claim is that replacing the fixed integer order 1 in Newton-Raphson by a variable fractional order gives iterations whose trajectories can enter the complex plane even when $x_0$ is real. In the fractional Newton method (29), the order is $\alpha_f(x)$, which equals a chosen non-integer $\alpha$ while $\|f(x)\|\ge\delta$ and switches to 1 near a root; the paper concludes that near a simple root the iteration behaves like classical Newton and has order at least quadratic. The quasi-Newton method (39) applies fractional derivatives to a fixed linearization $g$ rather than to $f$, and the pseudo-Newton method (47) replaces the inverse Jacobian by a diagonal fractional-derivative matrix, with $\beta(\alpha,(x_i)_j)$ set to 1 at zero components to avoid the nonzero fractional derivative of a constant; both are claimed to have order at most linear. The reported tables show complex roots of a degree-16 polynomial and of several transcendental and multivariate systems obtained from real starting points.

Load-bearing premise

The methods rely on functions introduced on real space being evaluated at the complex iterates the algorithm produces, and the paper never specifies which complex extension or branch convention is used for transcendental functions in Examples 3.2 through 3.7.

Editorial extensions

If this is right

From a single real seed, sweeping the derivative orders $\alpha_m$ over $(-2,2)$ can recover multiple real and complex zeros; the degree-16 polynomial example in Table 1 shows this explicitly.
The fractional Newton method requires only a few iterations per $\alpha$, so the order sweep is affordable when the fractional Jacobian can be computed and inverted.
The fractional quasi-Newton method uses a cheaper derivative matrix and behaves like fractional Newton when the initial guess is close to a solution, but its linear order demands more iterations.
The fractional pseudo-Newton method avoids inverting any matrix, at the cost of slower linear convergence and more $\alpha$ values.
None of the methods claims to find all solutions: they return subsets of the solution set, and more zeros require changing $x_0$, refining the $\alpha_m$ partition, or raising the iteration limit.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: The same order-switching idea could turn other iterative solvers into parameter-swept root finders, treating the derivative order as a continuation parameter rather than a fixed convergence accelerator.
Editorial inference: For transcendental functions with branch ambiguity, the method's complex trajectory depends on the chosen analytic continuation; specifying that continuation gives a direct way to compare the fractional Newton iterations against classical Newton with complex initial conditions.
Editorial inference: The convergence claims depend on the threshold $\delta$ in $\alpha_f(x)$, and the paper does not quantify how $\delta$ trades off between basin coverage and convergence speed; this is a testable extension.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things about this paper. First, the algorithmic idea is genuinely new: instead of fixing the derivative order alpha, they let alpha depend on x, with a threshold function alpha_f(x) that switches to ordinary Newton near a root. That is a clean trick and it gives a plausible route to the claimed quadratic convergence for their fractional Newton method (29). The quasi-Newton (39) and pseudo-Newton (47) variants are also new and sensibly motivated by avoiding fractional derivatives of constants. Second, the numerical tables do show small residuals and complex roots from real starting points, which is what makes the paper worth reading.

The stress-test note is correct, and it lands hard. The paper defines Riemann-Liouville and Caputo derivatives for real functions on real intervals, then uses them at complex iterates without ever specifying a branch or contour. For polynomials, the monomial formula (22) is ambiguous for non-integer alpha and complex x. For sin, exp, sinh, cosh, and cos in Examples 3.2-3.7, the fractional derivative is not simply the real formula evaluated at a complex point; it needs analytic continuation or a termwise series with an explicit convention. As written, the residuals in Tables 2-7 are not well-defined numerical facts. This is fixable, but it is a load-bearing gap.

The convergence-order claims are also informal. The quadratic claim for (29) is argued by reduction to ordinary Newton when ||f(x)|| < delta, which is plausible, but it is not stated as a theorem with the required regularity conditions. The linear-order claims for the quasi-Newton and pseudo-Newton methods are plausible but equally unproven. No code is shipped and there is no comparison with classical Newton using complex starting values, which would be the obvious baseline.

Credit where due: the core idea is simple and potentially useful, the examples cover scalar and multi-dimensional systems, and the self-citations to the authors' own fractional Newton-Raphson papers are appropriate rather than suspicious. The reference list otherwise looks normal.

Who is this for? Numerical analysts who work on iterative methods, especially fractional-derivative-based root finding. It deserves a serious referee, but the referee should send it back for major revision: define the complex fractional derivative, prove or at least carefully state the convergence-order conditions, ship code, and benchmark against classical Newton with complex starts. The paper is not a dead end; it just is not ready as written.

Referee Report

3 major / 4 minor

Summary. The paper defines a fractional Newton method (27) built on a fractional Jacobian (25), then introduces a switching function alpha_f(x) in (28) so that the iteration (29) reduces to ordinary Newton-Raphson near a root, aiming at quadratic convergence. It also proposes a fractional quasi-Newton method (39), based on a linearized function g, and a fractional pseudo-Newton method (47), with a diagonal matrix, and claims for these variants at most linear convergence. Examples 3.1 through 3.7 apply the methods to a polynomial, scalar transcendental equations, and several nonlinear systems, reporting both real and complex roots obtained from real initial conditions, together with residual norms for the computed roots.

Significance. The proposed iteration schemes are genuine algorithmic variants of fractional Newton-Raphson, and the idea of switching the fractional order to recover classical quadratic convergence near a simple root is attractive. The paper reports residual norms in all tables and does not fit parameters to data, which strengthens the empirical claims. If the complex-domain definition of the fractional derivative is supplied and the convergence-order statements are proved, the methods could be a useful tool for finding multiple real and complex roots of nonlinear systems from real starting values. As it stands, however, the manuscript does not provide the analytical foundation needed to validate the tables or the stated orders of convergence.

major comments (3)

[Section 3 and Examples 3.2–3.7] Equations (21), (22), and (25) define fractional derivatives for real-valued functions on real intervals, but the tables report iterates m_xi in C for non-polynomial functions such as f(x)=sin(x)-3/(2x) in Example 3.2 and the exponential/trigonometric systems in Examples 3.5–3.7. No analytic continuation, branch cut, or contour prescription is given for the Riemann-Liouville operator aD^x at complex x, nor for the power x^(mu-alpha) in (22). Consequently the values f^(alpha)(m_xi) and the residuals ||f(m_xi)|| in Tables 2–7 are not mathematically defined as written, and the numerical experiments are not reproducible. The same ambiguity already affects Example 3.1, because the monomial formula (22) is multi-valued for complex x unless a branch is specified.
[Section 3, Eqs. (28)–(29)] The claim that the fractional Newton method (29) has order of convergence at least quadratic is not proved as a theorem. The argument that alpha_f(x)=1 near a root and the iteration reduces to Newton-Raphson is plausible for a simple root, but the usual assumptions (f continuously differentiable, f^(1)(xi) invertible) and a proof that all sufficiently close iterates satisfy ||f(x_i)||<delta should be stated explicitly. For the quasi-Newton and pseudo-Newton methods, the statement that failure of (12) implies "at most an order of convergence (at least) linear" is not logically valid: not satisfying a sufficient condition for quadratic convergence does not exclude superlinear convergence, so a formal statement and proof are needed.
[Section 3.1, Eq. (39)] The function g(x)=f(x0)+f^(1)(x0)x is a linearization at the initial point, not at the current iterate, so Q_g,beta is a fixed-affine slope rather than a standard quasi-Newton update. If this is intentional, the method should be described as a chord method and its convergence properties need separate analysis. If the intent was to linearize at the current iterate x_i, the formula should instead read f(x_i)+f^(1)(x_i)(x-x_i).

minor comments (4)

[Section 3.2, Eq. (48)] The definition of P_epsilon,beta(x_i) is ambiguous: applying the fractional derivative to the Kronecker delta delta_kj is not meaningful as written. The matrix should be defined componentwise, for example by specifying the action of d^beta_j on the components of the identity map and on the constant term separately.
[Table 6, Example 3.6] The parameter line reads "M = +6" where the other tables use "M = e+6"; this appears to be a typographical error and should be corrected.
[Section 1.2, p. 4] The text contains several language errors, including "the competent k-th" for "the component k-th" and "sil" for "if"; these should be corrected throughout.
[Section 3 and Tables 1–7] The selection rule for the alpha_m values is not specified beyond a partition of [-2,2], but the tabulated values appear irregular. To make the experiments reproducible, the exact partition or a deterministic rule for generating the alpha_m grid should be stated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the quadratic-order switch is an explicit construction, and the new variants stand on their own definitions.

full rationale

No circular step is present. The quadratic-convergence claim for method (29) follows by explicit construction: α_f(x) in Eq. (28) is set to 1 whenever ‖f(x)‖ < δ, so once the iterates enter that neighborhood the iteration becomes the classical Newton method whose local quadratic convergence is already established in Section 1.2. This is a design choice, not a hidden fit, and no fitted constant is later renamed as a prediction. The fractional quasi-Newton and pseudo-Newton methods are newly defined iteration functions in Eqs. (39)-(41) and (47)-(48), with their claimed at-most-linear behavior explained from the same classical convergence framework rather than from a parameter fitted to the tabulated roots. The paper does cite the authors' own prior work [1,2] for the baseline fractional Newton-Raphson iteration (27), but that reference merely supplies a starting point; the load-bearing definitions and convergence statements for the new methods are restated in this manuscript and do not reduce to an unverified self-citation. The under-specified evaluation of fractional derivatives at complex iterates in Examples 3.2-3.7 is a reproducibility and correctness concern, not a circularity: it does not make any equation equivalent to its own input. Therefore the derivation chain is self-contained with respect to the circularity criteria.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claims rest on unproved assumptions about the n-dimensional convergence theorem, the continuity condition of the fractional derivative, invertibility of the fractional Jacobian, and analytic continuation of real functions to complex iterates. The main free parameters are algorithmic tolerances and the alpha partition, which are chosen per example rather than fitted to data; they affect which roots are found and what counts as a solution.

free parameters (4)

Switching threshold delta = 0.5 in all examples
In alpha_f(x), Eq. (28), delta controls when the method switches from fractional order alpha to integer order 1. The quadratic convergence claim depends on the sequence entering ||f(x)||<delta near the root; no sensitivity analysis or rule for choosing delta is given.
Partition of alpha values = not specified; a partition of (-2,2) is assumed
The set of roots found depends on which fractional orders alpha_m are tried. The text only says a partition is created, without giving the number of points or spacing.
Stopping and clustering parameters TOL, LIT, M, epsilon = TOL=e-4, LIT=40 or 200, delta=0.5, M=e+6 or e+17, epsilon used in (31)
These user-chosen tolerances define what counts as a found root and which duplicates are merged, so the solutions in Tables 1 through 7 are tolerance-dependent.
Initial condition x0 = per example, e.g., 0.74, 0.26, (0.88,0.88), (4.35,4.35,4.35)
The method is local in practice; the number and type of roots found depend heavily on the chosen real starting point, and the examples select x0 values that produce successful runs.

assumptions (4)

domain assumption An n-dimensional version of Theorem 1.5 is assumed, under which an iteration function with zero first derivative at the fixed point has at least quadratic convergence.
Invoked in Section 1.2, item 1 ('Suppose we can generalize the Theorem 1.5 to the case n-dimensional'), and never proved for the fractional iteration functions used later.
domain assumption Fractional derivatives satisfy the continuity condition (26), lim_{alpha->1} f^(alpha)(x)=f^(1)(x), for the functions considered.
Used to define the fractional Jacobian and to justify switching alpha to 1 near roots. It holds for monomials under (22), but is assumed without proof for the transcendental systems in Examples 3.2 through 3.7.
domain assumption The fractional Jacobian matrices in (29) and (39) are invertible at every iteration.
The methods require inverting these matrices at each step; no condition, safeguard, or pivoting strategy is stated.
ad hoc to paper Real-valued functions in the examples are extended to complex arguments by analytic continuation.
The iteration sequences become complex while the problem statements in (34) through (36), (42), and (49) define f on R or R^n; no analytic continuation or branch convention is stated anywhere.

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Cite this review

Pith. "Pith review of Fractional Newton-Raphson Method and Some Variants for the Solution of Non-linear Systems." pith.science (2026). https://pith.science/paper/43STF3VI

@misc{pith2026190801453,
  author       = {Pith},
  title        = {Pith review of: Fractional Newton-Raphson Method and Some Variants for the Solution of Non-linear Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/43STF3VI}},
  note         = {Machine review of arXiv:1908.01453}
}

read the original abstract

The following document presents some novel numerical methods valid for one and several variables, which using the fractional derivative, allow to find solutions for some non-linear systems in the complex space using real initial conditions. The origin of these methods is the fractional Newton-Raphson method but unlike the latter, the orders of fractional derivatives proposed here are functions. In the first method, a function is used to guarantee an order of convergence (at least) quadratic, and in the others, a function is used to avoid the discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible that the methods have at most an order of convergence (at least) linear.

Figures

Figures reproduced from arXiv: 1908.01453 by the authors.

**Figure 2.** llustrations of some trajectories generated by the fractional Newton method for the same initial condition [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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Fractional Newton-Raphson Method Accelerated with Aitken's Method

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [2]

Fractional Newton-Raphson Method Accelerated with Aitken's Method

F. Brambila-Paz, A. Torres-Hernandez, U. Iturrar´an-Viveros, and R. Caballero-Cruz. Fractional newton-raphson method accelerated with aitken’s method. arXiv preprint arXiv:1804.08445 , 2018. https://arxiv.org/pdf/ 1804.08445.pdf

work page Pith review arXiv 2018

[2] [3]

Introduction to numerical analysis , volume 12

Josef Stoer and Roland Bulirsch. Introduction to numerical analysis , volume 12. Springer Science & Business Media, 2013

work page 2013

[3] [4]

Concise numerical mathematics

Robert Plato. Concise numerical mathematics. Number 57. American Mathematical Soc., 2003

work page 2003

[4] [5]

Numerical analysis: a second course

James M Ortega. Numerical analysis: a second course. SIAM, 1990

work page 1990

[5] [6]

An´alisis num´erico

Richard L Burden and J Douglas Faires. An´alisis num´erico. Thomson Learning,, 2002

work page 2002

[6] [7]

Iterative solution of nonlinear equations in several variables, volume 30

James M Ortega and Werner C Rheinboldt. Iterative solution of nonlinear equations in several variables, volume 30. Siam, 1970

work page 1970

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Applications of fractional calculus in physics

Rudolf Hilfer. Applications of fractional calculus in physics. World Scientiﬁc, 2000

work page 2000

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Theory and Applications of Fractional Diﬀerential Equations

AA Kilbas, HM Srivastava, and JJ Trujillo. Theory and Applications of Fractional Diﬀerential Equations. Elsevier, 2006

work page 2006