{"id":"157984ad-289b-421d-96b3-5b5479e17582","arxiv_id":"1908.01453","paper_version":6,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Fractional Newton methods with a derivative order that changes during iteration find multiple real and complex roots of nonlinear systems from real initial guesses.","lead":"This paper proposes three variants of Newton's method that use fractional derivatives with changing orders to find complex solutions of nonlinear equations from real starting points. The variants differ in cost and speed, and the authors demonstrate them on polynomials and small systems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The tables assume a complex extension of the fractional derivative that the paper never defines; without a branch or contour convention, the residuals in Examples 3.2-3.7 are not well-defined.","rationale":"The reader's weakest assumption is close to the real issue, but slightly mislocated. The functions f in Examples 3.2-3.7 are entire or meromorphic, so passing complex arguments into sin, exp, sinh, or cosh is harmless; the undefined object is the fractional derivative of those functions at complex points, which requires a branch and contour convention not given in Section 2. This is the most load-bearing concern because every numerical table beyond the polynomial example depends on it, and the tables are the only evidence supporting the central claim. The quadratic-order claim is less fragile: near a simple root, α_f(x)=1 by (28), so the final iterates follow classical Newton; the open question is whether the iteration reaches that neighborhood, which depends on the undefined fractional derivative branch. A single comparative implementation with two explicit branch conventions would settle whether this is a notational gap or a substantive flaw. I keep the reader's CONDITIONAL verdict rather than hardening to REJECT, because the gap is a missing definition rather than an internal contradiction, and it is plausibly addressable by stating a branch rule, providing code, and recomputing the tables.","tokens_in":16477,"tokens_out":7485,"duration_ms":86548,"concrete_test":"Implement (29) with Riemann-Liouville derivatives using an explicit convention: expand f in its Taylor series, apply (22) termwise with the principal branch of x^(μ−α), branch cut along the negative real axis, and reproduce the first row of Table 2 (f(x)=sin(x)−3/(2x), x0=0.26, α=−1.92915) together with one complex row. Then repeat the same computation with the same series but with the branch cut rotated by π/2. If the iterates or residuals change materially, the published tables are branch-dependent and the central claim is not well-defined; if they match, the missing convention is only notational.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central empirical claim is that iteration (29), using the fractional derivative (21), reliably produces the complex roots reported in Tables 2-7 from real starting values. For that claim to hold, the expression f^(α)(x) must be an actually defined function of complex x for each f used. Section 2 defines Riemann-Liouville and Caputo operators for real functions on real intervals, and (22) gives fractional derivatives of monomials x^μ. For non-integer α and complex x, x^(μ−α) is multi-valued, and for non-polynomial f the fractional integral (15) from 0 to x is path-dependent unless a branch and contour are specified. The paper supplies no such convention. The functions sin, exp, sinh, and cosh are entire and extend to complex arguments uniquely, but their fractional derivatives are not simply the same real formulas evaluated at complex points; they require analytic continuation or a termwise series with an explicit branch prescription. The tables therefore do not define a reproducible numerical method. The examples from 3.2 onward are where this breaks; even Example 3.1 has branch ambiguity through (22) for complex x.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":16694,"tokens_out":5353,"duration_ms":58887,"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":[{"comment":"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":"Section 3 and Examples 3.2–3.7"},{"comment":"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":"Section 3, Eqs. (28)–(29)"},{"comment":"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).","section":"Section 3.1, Eq. (39)"}],"minor_comments":[{"comment":"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.","section":"Section 3.2, Eq. (48)"},{"comment":"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":"Table 6, Example 3.6"},{"comment":"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":"Section 1.2, p. 4"},{"comment":"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.","section":"Section 3 and Tables 1–7"}],"recommendation":"major_revision","confidential_remarks":"The main gap is the undefined complex extension of the fractional derivative, which is fixable but requires substantial additions to the definitions and to the numerical section. I see no circularity or data-fitting concern. The paper relies heavily on the authors' own prior work, but the proposed variants should be evaluated on their own merits."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nCredit 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.\n\nWho 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.","headline":"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.","tokens_in":17213,"tokens_out":2294,"would_cite":false,"duration_ms":27500,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65H10","26A33"],"pacs":[],"model":"deepseek-v4-flash","headline":"Varying the fractional derivative order lets Newton-type iterations leave the real line and land on complex roots.","keywords":["fractional Newton method","nonlinear systems","complex roots","fractional derivative","order of convergence","quasi-Newton method","pseudo-Newton method","Riemann-Liouville derivative"],"falsifier":"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.","tokens_in":16254,"feed_emoji":"🧮","tokens_out":11842,"duration_ms":105530,"temperature":0.7,"pith_summary":"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.","feed_headline":"A fractional Newton method starts real and lands on complex roots","feed_subtitle":"By sweeping the derivative order, one real seed can collect many real and complex solutions of nonlinear systems.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the fractional Newton-Raphson method that this paper generalizes by making the derivative order a function.","marker":"[1]"},{"why":"Presents an accelerated fractional Newton-Raphson variant that motivates the family of methods studied here.","marker":"[2]"},{"why":"Supplies the fixed-point and contraction results used to frame convergence and order of convergence.","marker":"[3]"},{"why":"Gives the definition of order of convergence (Definition 1.4) and Theorem 1.5 used to characterize the methods.","marker":"[4]"},{"why":"Provides classical Newton-Raphson and parallel chord methods from which the quasi-Newton and pseudo-Newton variants are built.","marker":"[7]"},{"why":"Supplies the Riemann-Liouville fractional integrals and derivatives used in the iteration formulas.","marker":"[8]"},{"why":"Defines the Caputo derivative and its relation to the Riemann-Liouville derivative used in the continuity condition.","marker":"[9]"}],"fun_headline_variants":["Fractional Newton method finds complex roots from real seeds","Variable-order fractional Newton leaps to complex solutions","Fractional Newton's method: real start, complex roots","Sweeping derivative order unlocks complex roots for Newton"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Fractional Newton method finds complex roots from real seeds","Variable-order fractional Newton leaps to complex solutions","Fractional Newton's method: real start, complex roots","Sweeping derivative order unlocks complex roots for Newton"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000141,"raw_usage":{"total_tokens":1126,"prompt_tokens":867,"completion_tokens":259,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":483,"completion_tokens_details":{"reasoning_tokens":197}},"tokens_in":483,"tokens_out":259,"duration_ms":3394,"temperature":1.0,"reasoning_tokens":197,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:12:12.245166+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Fractional Newton-Raphson Method Accelerated with Aitken's Method","cited_arxiv_id":"1804.08445","evidence_quote":"Presents an accelerated fractional Newton-Raphson variant that motivates the family of methods studied here."},{"cited_title":"Introduction to numerical analysis , volume 12","cited_arxiv_id":null,"evidence_quote":"Supplies the fixed-point and contraction results used to frame convergence and order of convergence."},{"cited_title":"Concise numerical mathematics","cited_arxiv_id":null,"evidence_quote":"Gives the definition of order of convergence (Definition 1.4) and Theorem 1.5 used to characterize the methods."},{"cited_title":"Iterative solution of nonlinear equations in several variables, volume 30","cited_arxiv_id":null,"evidence_quote":"Provides classical Newton-Raphson and parallel chord methods from which the quasi-Newton and pseudo-Newton variants are built."},{"cited_title":"Applications of fractional calculus in physics","cited_arxiv_id":null,"evidence_quote":"Supplies the Riemann-Liouville fractional integrals and derivatives used in the iteration formulas."},{"cited_title":"Theory and Applications of Fractional Differential Equations","cited_arxiv_id":null,"evidence_quote":"Defines the Caputo derivative and its relation to the Riemann-Liouville derivative used in the continuity condition."}],"review_version":1}