Scalable higher-order nonlinear solvers via higher-order automatic differentiation

Alan Edelman; Christopher Rackauckas; Keming Miao; Songchen Tan

REVIEW 3 major objections 4 minor 52 references

Scalable higher-order nonlinear solvers via higher-order automatic differentiation

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Halley's method can be made as cheap per step as Newton's method by computing directional derivatives via Taylor-mode automatic differentiation, and it outperforms Newton on large dense and sparse nonlinear systems.

desk verdict Useful implementation results, but Eq. (4.1) is not the classical multivariate Halley update and the cubic convergence claim is unsupported. read the letter →

arxiv 2501.16895 v1 pith:MKKUZFEP submitted 2025-01-28 math.NA cs.NA

Songchen Tan , Keming Miao , Alan Edelman , Christopher Rackauckas This is my paper · ORCID

classification math.NAcs.NA MSC 65H1065H0565L04

keywords nonlinearsolversHalley'smethodHouseholder'sautomaticdifferentiationTaylor-modedirectionalderivativesstiffODEshigher-orderconvergence

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 tries to show that higher-order nonlinear solvers are not intrinsically too expensive for large problems, provided the higher derivatives are computed as directional derivatives rather than full tensors. Its central case is Halley's method (cubic convergence): because Taylor-mode automatic differentiation computes the directional second derivative $D^2f(x)[a,a]$ at roughly twice the cost of evaluating $f$, and because the two linear solves in each Halley step can share one factorization of the Jacobian, each Halley step can be asymptotically as cheap as one Newton step. If true, this makes Halley's method a direct third-order competitor to Newton's method wherever direct factorization is available. The authors demonstrate the point on a dense radiative-transfer problem, a large sparse ill-conditioned reaction-diffusion problem, and inside implicit stiff ODE solvers, reporting speedups over Newton in each setting.

What carries the argument

The machinery is Taylor-mode automatic differentiation, which propagates a truncated Taylor series $(h_0,h_1,\ldots,h_p)$ through each elementary operation using Faà di Bruno's formula, so that the $p$-th directional derivative $D^pf(x)[v,\ldots,v]$ is obtained in $O(p)$ times the cost of evaluating $f$. In Algorithm 4.1 the bundle is initialized as $(x,a,0)$ and pushed through $f$ to produce $(f(x),Df(x)[a],D^2f(x)[a,a])$; this single pushforward replaces the full Hessian computation. The second ingredient is reuse of the Jacobian factorization: both linear solves use the same LU/QR factorization, so the extra solve is $O(n^2)$; for sparse problems, sparsity detection and graph coloring keep the factorization scalable.

What would settle it

Compare the step produced by equation (4.1) with the step of the standard matrix-inverse multivariate Halley iteration on a small non-diagonal system such as $f(x_1,x_2)=(x_1^2+x_2-1,\,x_1-x_2^2)$; if the two updates differ at a generic point, the iteration studied is not Halley's method.

Watch

Extended reading notes

Core claim

The paper's central claim is that the multivariate Halley update $$x_{n+1}=x_n+(a_n\odot a_n)\oslash\left(a_n+\frac{1}{2}[Df(x_n)]^{-1}$D^{2}$f(x_n)[a_n,a_n]\right),\quad Df(x_n)a_n=-f(x_n),$$ can be executed without ever forming the full Hessian. The vector $D^2f(x_n)[a_n,a_n]$ is obtained from a Taylor bundle $(x_n,a_n,0)$ pushed through $f$, which costs about two function evaluations; then a second linear solve with the already-factorized Jacobian yields the correction. Thus each Halley step costs one Jacobian factorization plus $O(n^2)$ work, the same asymptotic cost as Newton's step, while converging cubically instead of quadratically. The paper reports that on dense problems this gives about 40% savings over Newton at $n=128$, on a sparse ill-conditioned Brusselator steady-state problem about 25% savings at $n=2048$, and 5--10% savings for the whole stiff ODE solve across several implicit integrators.

Load-bearing premise

The load-bearing premise is that equation (4.1) really is the multivariate Halley method as derived from abstract rational approximation in [13,15]; if that identification is wrong, the reported speedups would still be empirical facts about a third-order iteration, but the paper's theoretical framing as Halley's method would not hold.

Editorial extensions

If this is right

If the central claim is right, Halley's method can replace Newton's method as a practical third-order solver for any problem where a direct Jacobian factorization is affordable, with expected savings that grow with problem size.
Large dense systems, such as those from integral equations, can be solved to the same tolerance with roughly 40% less time at $n\approx 128$ than Newton.
Large sparse ill-conditioned systems from PDE discretizations can be solved with about 25% less time at $n\approx 2048$ when sparsity detection is used.
Replacing Newton with Halley inside implicit stiff ODE integrators yields 5--10% total speedups across a range of methods, because the nonlinear solve is the bottleneck.
The paper's trade-off study suggests the biggest gain comes from moving from first to second order; orders above Halley are postulated to add little, though the paper does not implement multivariate $p\ge 3$ variants.

Reading between the lines

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

My inference: the same two-linear-solve trick should extend to multivariate Householder methods of order $p\ge 3$, since Taylor-mode AD still supplies all needed directional derivatives at $O(p)$ cost; the paper postulates but does not build this.
My inference: the matrix-free limitation the paper flags is the main practical boundary; a meaningful extension would be to keep a preconditioner or Krylov subspace across the two linear solves so the second solve costs much less than a full Newton solve.
My inference: the elementwise update (4.1) may be a variant rather than the classical matrix-inverse Halley iteration, so a direct convergence analysis of this specific update, and a numerical comparison on non-diagonal test problems, would settle whether the Halley name is essential to the speedups.

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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

The short version: this paper has a genuinely useful implementation idea, but the central mathematical claim about multivariate Halley's method does not hold up as stated. The authors take the elementwise update x + a⊙a⊘(a+b/2), call it Halley's method 'derived from abstract rational approximation', and assert cubic convergence. Neither is established. The classical Cuyt–Rall multivariate Halley iteration involves a matrix inverse, not an elementwise quotient; the two coincide only in scalar or diagonal cases. A Taylor expansion about the root shows the elementwise update's error after one step is e_new,i ≈ u_i^2/[4(u_i - e_i)], where u_i is the i-th component of the second-order term. For a component that is already O(||e||^2), e_new,i stays O(||e||^2), so the infinity norm is only quadratically reduced. So the method is at best Q-quadratically convergent in general, not cubically. The paper never proves otherwise.

What is new and good: the implementation strategy is the real contribution. Using Taylor-mode AD to compute the directional second derivative D2f(x)[a,a] at about twice the cost of f, and reusing the Jacobian factorization for the two linear solves, is a clean idea. The scaling benchmarks on dense and sparse problems and inside stiff ODE solvers are reasonable, and the speedups (25–40% on selected problems) are consistent, even if modest. The paper is clearly written and points to working Julia packages.

Soft spots: besides the convergence order issue, the benchmarks lack error bars or repeated-run statistics, and the comparison to 'naive Halley' computing the full Hessian is expected to be slow. The attribution of the update formula to [13,15] needs checking; those references likely contain the matrix-inverse form.

Bottom line: I'd send this to a serious referee. The implementation and empirical results are worth publishing, but the authors need to either derive Eq (4.1) from the cited literature, prove its convergence order, or re-frame the paper as a study of a related third-order-style iteration. As it stands, the phrase 'cubic-convergence Halley's method' is likely incorrect.

Recommendation: major revision, with the math checked carefully.

Referee Report

3 major / 4 minor

Summary. The paper proposes using Taylor-mode automatic differentiation to compute higher-order directional derivatives efficiently and thereby implement higher-order nonlinear solvers. It first implements univariate Householder's method of arbitrary order and then focuses on the second-order variant, Halley's method, generalized to systems of nonlinear equations through the componentwise update in Eq. (4.1). The central claims are that each Halley step can be asymptotically as cheap as a Newton step because the two linear solves share one Jacobian factorization, and that numerical experiments on a dense (Chandrasekhar H-function) problem, a sparse ill-conditioned Brusselator steady-state problem, and stiff ODE integration show that Halley's method can outperform Newton's method. The paper concludes with the stated limitation that matrix-free or iterative linear solves are not considered.

Significance. If the central claims hold, the paper offers a practical and interesting contribution: Taylor-mode AD is used to avoid full higher-order derivative tensors, the linear-algebra cost model (one factorization plus two triangular solves per step) is plausible, and the numerical demonstrations span relevant problem classes. The released Julia packages (TaylorDiff.jl, SimpleNonlinearSolve.jl) and the explicit discussion of scalability and limitations are strengths. However, the identification of Eq. (4.1) with the classical multivariate Halley method is not derived, and the asserted cubic convergence of that update is not established. The empirical speedup claims also lack repeated-run statistics. The significance is therefore real but conditional on clarifying what iteration is actually implemented and proving or tempering the convergence-order claim.

major comments (3)

[§4, Eq. (4.1)] Equation (4.1) is presented as the multivariate Halley method 'derived from abstract rational approximation' citing [13, 15], but no derivation is shown. The classical multivariate Halley iteration, as given in the cited Cuyt and Rall paper [15], is of the form x + [I - 0.5 J^{-1} D^2 f(x)[a, .]]^{-1} a, which is not equal to the elementwise quotient a⊙a ⊘ (a + b/2) unless the operator J^{-1}D^2f(x)[a, .] is diagonal. The paper must either derive Eq. (4.1) from the cited abstract rational approximation theory or explicitly state that it is a new componentwise third-order variant; the current wording misidentifies the iteration and makes the subsequent 'Halley's method' nomenclature and convergence claims unsupported.
[§4, convergence of Algorithm 4.1] The paper asserts that Algorithm 4.1 has cubic convergence, but no proof is provided for the componentwise update. Expanding about a root with a = -e + h/2 + O(3) and b = h + O(3) gives e_new,i = h_i^2/[4(h_i - e_i)] + O(||e||^3); if e_i is O(||e||^2), the first step reduces that component only quadratically. Thus uniform cubic Q-convergence does not follow from a naive Taylor argument. This is load-bearing because the claimed advantage over Newton's method is premised on the higher convergence order. The authors should either prove convergence under explicit conditions, or replace the third-order claim with a numerically verified statement about observed order and clarify the asymptotic nature of the convergence.
[§4.1–§4.3, benchmarks] The performance claims (e.g., about 40% faster for n=128 in Section 4.1, about 25% faster in Section 4.2, and 5–10% average speedup in Section 4.3) are reported without repeated runs, variance estimates, or a complete statement of the solver configurations (tolerances, linear solver algorithms, number of runs). Since the central contribution is empirical, the paper should state how many repetitions were performed and report the observed spread, or otherwise make the benchmark scripts and machine configuration available in a way that allows independent reproduction. This is needed to support the 'outperform Newton's method' conclusion.

minor comments (4)

[Algorithm 4.1] The line 'Solve b from Df(x)[b] = D2(x)[a, a]' should read D^2 f(x)[a, a]; the 'f' is missing.
[§4.1] The text refers to 'NonlinearProblemLibrary.jl[35]', but reference [35] is the NonlinearSolve.jl paper, not the problem library; please add the correct citation or URL for NonlinearProblemLibrary.jl.
[§4.3] The phrase 'improv ing the nonlinear solver' contains a typo and should be 'improving'.
[Figures 2–5] The figure captions do not state what quantity is plotted (wall-clock time, median, best-of-N, etc.) or the error metric for the work-precision diagrams; please add units and definitions to the captions.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: cost model is an independently documented AD theorem and all performance claims are benchmarked, not fitted.

full rationale

The Taylor-mode AD cost bound is taken from the first author's prior thesis [46] and TaylorDiff.jl [47], but it is a parameter-free complexity theorem with stated assumptions (elementary functions and arbitrary control flow) and does not assume any nonlinear-solver result, so the self-citation is real evidence and not circular. The Householder and Halley update formulas are standard forms taken from the literature [13,15,24]; no update formula is defined in terms of the paper's own conclusions. The central efficiency claims are tested empirically against Newton on dense, sparse, and stiff-ODE benchmarks with no fitted parameters; the reported speedups are measurements, not predictions derived from the method's definition. The possible mismatch between Eq. (4.1) and the classical matrix-inverse multivariate Halley update is a correctness or attribution concern (the paper does not prove Eq. (4.1) follows from [13,15]), not circularity, because the formula is not derived from the paper's own outputs and no derived quantity reduces to an input. The limitation paragraph on matrix-free Jacobians explicitly admits where the cost argument fails, further showing the derivation is not vacuous. Verdict: no significant circularity.

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

The central efficiency claim rests on four imported premises: the O(p) Taylor-mode cost from prior work, standard smoothness and invertibility assumptions, the identification of Eq. (4.1) with the cited multivariate Halley method, and single-run timing representativeness. No free parameters are fit in the paper, and no new entities are introduced.

assumptions (4)

domain assumption Taylor-mode AD computes p-th order directional derivatives with O(p) overhead per function evaluation (cited to [46], not re-derived).
The cost model justifying cheap higher-order derivatives in Section 2 is imported from the authors' prior thesis and TaylorDiff.jl work, not proven here.
standard math Local smoothness and nonsingular Jacobian sufficient for local cubic convergence of Eq. (4.1).
Standard local convergence assumptions for Newton/Halley-type iterations; not stated explicitly in the paper.
domain assumption Equation (4.1) is the multivariate Halley method from [13,15].
The text asserts the update 'could be expressed as' Eq. (4.1) but does not show the reduction from Cuyt's abstract rational approximation; the elementwise formula may be a distinct third-order variant.
domain assumption Timing measurements on a single core, single run are representative of performance.
Figures 2, 4, and 5 lack repeated-run statistics, so the reported speedups assume run-to-run stability.

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Cite this review

Pith. "Pith review of Scalable higher-order nonlinear solvers via higher-order automatic differentiation." pith.science (2026). https://pith.science/paper/MKKUZFEP

@misc{pith2026250116895,
  author       = {Pith},
  title        = {Pith review of: Scalable higher-order nonlinear solvers via higher-order automatic differentiation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MKKUZFEP}},
  note         = {Machine review of arXiv:2501.16895}
}

read the original abstract

This paper demonstrates new methods and implementations of nonlinear solvers with higher-order of convergence, which is achieved by efficiently computing higher-order derivatives. Instead of computing full derivatives, which could be expensive, we compute directional derivatives with Taylor-mode automatic differentiation. We first implement Householder's method with arbitrary order for one variable, and investigate the trade-off between computational cost and convergence order. We find that the second-order variant, i.e., Halley's method, to be the most valuable, and further generalize Halley's method to systems of nonlinear equations and demonstrate that it can scale efficiently to large-scale problems. We further apply Halley's method on solving large-scale ill-conditioned nonlinear problems, as well as solving nonlinear equations inside stiff ODE solvers, and demonstrate that it could outperform Newton's method.

Discussion (0). Continue with ORCID to comment.

Reference graph

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