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REVIEW 4 major objections 8 minor 35 references

Neural Network Acceleration of Iterative Methods for Nonlinear Schr\"odinger Eigenvalue Problems

T0 review · 4 major / 8 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A neural network inserted mid-iteration into the energy-adaptive Riemannian conjugate gradient solver for the rotating Gross-Pitaevskii equation cuts iteration counts by 22% on average and wall-clock time by 14.5% over 500 test cases…

desk verdict A useful empirical hybrid acceleration for a hard 2D GPE solver, with an honestly reported but partly inflated headline speed-up that needs a same-minima-only breakdown before the exact numbers should be trusted. read the letter →

arxiv 2507.16349 v1 pith:BRIUYKVM submitted 2025-07-22 math.NA cs.NA

Daniel Peterseim , Jan-F. Pietschmann , Jonas Püschel , Kilian Ruess This is my paper · ORCID
classification math.NAcs.NA MSC 65H1765N2581Q1035Q5558C40
keywords nonlinearSchrödingereigenvalueproblemGross-PitaevskiiequationBose-EinsteincondensateRiemannianoptimizationenergy-adaptiveconjugategradientneuralnetworkaccelerationU-Netnormalizationerror
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 establish that a trained neural network can accelerate a classical iterative solver for nonlinear Schrödinger eigenvalue problems rather than replace it. The testbed is the rotating Gross-Pitaevskii equation for Bose-Einstein condensates, where the energy-adaptive Riemannian conjugate gradient (EARCG) method can stagnate for thousands of iterations while a vortex lattice slowly rotates into alignment with a slightly asymmetric trap. The accelerator is a U-Net that sees the current state and gradient and proposes an improved state; the proposal is accepted only when its $L^2$ normalization error is small, then renormalized and handed back to EARCG. Across 500 randomized test cases this strategy saves 22.06% of iterations and 14.51% of wall-clock time on average, and reduces density error by 46.20% on average, far better than applying the same network at a random iteration. If correct, the template extends to other Riemannian solvers used in electronic structure and multicomponent condensates.

What carries the argument

The load-bearing object is the EARCG iteration on the $L^2$ unit sphere $\mathbb{S} = \{\varphi \in H^1 : \|\varphi\|_{L^2} = 1\}$, equipped with the energy-adaptive metric $g^{\mathrm{a}}_{\varphi}(v,w) = a_{\varphi}(v,w)$ and the normalization retraction. The accelerator is a U-Net, a convolutional encoder-decoder originally built for image segmentation, with 31 million parameters: it takes an $(n,n,4)$ tensor holding the real and imaginary parts of the current state and the current gradient, and outputs an $(n,n,2)$ enhanced state. The mechanism that carries the argument is the acceptance gate: because the network has no normalization layer, the scalar $e = |1 - \|\tilde{\varphi}\|_{L^2}|$ serves as a cheap proxy for proposal quality, and only proposals with $e < e_0 = 5 \times 10^{-3}$ are normalized and fed back into EARCG. This keeps the network a proposal generator inside the original solver rather than an end-to-end surrogate.

What would settle it

Compute the rank correlation between the normalization error $e$ and the true density error $\|\tilde{\rho} - \rho^*\|_{L^1}$ across the 40,000 training pairs; if the correlation is near zero, the acceptance gate is not selecting good proposals and the reported speedups would need a different explanation.

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Extended reading notes

Core claim

The central claim is that a single, well-timed neural network proposal can bypass the slowest phase of EARCG for rotating condensates. The network is trained offline on 40,000 pairs gathered from EARCG runs: each pair contains an intermediate state and energy-adaptive gradient at a tolerance in $[10^{-4}, 10^{-1}]$ together with the converged state. When the running solver reaches that tolerance window, the U-Net proposes a new state, and the proposal is accepted only if $|1 - \|\tilde{\varphi}\|_{L^2}| < 5\times 10^{-3}$; the accepted state is normalized and the solver continues to tolerance $10^{-8}$. On 500 test problems this reduces iterations by 22.06% on average (median 19.89%) and wall time by 14.51% (median 13.92%), with density error reduced by 46.20% on average; iteration counts improve in 91.8% of cases and wall time in 75%. The strongest gains occur in near-symmetric traps ($v_1 \approx 1.1$, $v_2 = 1$), where the network learns to rotate the vortex lattice into the preferred orientation, cutting runs such as 25,285 iterations to 267.

Load-bearing premise

The entire acceptance strategy depends on the assumption that a small normalization error $|1 - \|\tilde{\varphi}\|_{L^2}|$ reliably means the neural network's proposal is a good approximation to the ground state; the paper gives no proof or independent validation of that correlation, so a well-normalized but physically wrong proposal would be accepted and a poorly normalized good proposal rejected.

Editorial extensions

If this is right

  • A single well-timed network proposal cuts iterations in about 92% of test cases and wall-clock time in 75%, with mean savings of 22.06% and 14.51% respectively across 500 runs.
  • The largest gains occur in the near-degenerate trap regime ($v_1 \approx 1.1$, $v_2 = 1$) where EARCG stagnates; reported examples shrink runs from 25,285 to 267 iterations, a 98.94% reduction.
  • The normalization-error gate makes acceleration reliable: only 5.3% of runs worsen the density under the strategy, versus 40% under random application of the same network.
  • Because Hartree-Fock, Kohn-Sham, and multicomponent BEC problems share the same Riemannian solver structure, the same acceleration workflow transfers with minimal adaptation.
  • Allowing multiple accelerations per run and moving inference to a GPU are stated as likely further improvements to wall-time savings.

Reading between the lines

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

  • One could isolate the contribution of the acceptance gate by replacing the U-Net proposals with random states whose normalization errors match the network's; if the speedup persists, the gate rather than the learned content is doing the work.
  • The 46% average density-error improvement suggests the network has learned the rotation-alignment step, so an architecture with built-in rotational equivariance might generalize better outside the trained parameter box.
  • Because the training data comes entirely from EARCG trajectories, the acceleration is tied to that solver's convergence bias; a network trained on trajectories from a different Riemannian solver may not transfer, which is a natural open question.
  • The reported wall-time saving is a CPU-only number; the difference between iteration savings and time savings quantifies inference overhead, and GPU inference is an obvious untested lever.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 8 minor

Summary. The paper proposes a neural-network accelerator for the energy-adaptive Riemannian conjugate gradient (EARCG) method applied to ground-state computations of the rotating Gross-Pitaevskii equation. A U-Net is trained offline on EARCG trajectories to map mid-iteration states and gradients to the solver's converged state, and it is inserted once during a prescribed acceleration window. The acceptance of the network output is decided by the L2 normalization error of the prediction, and the algorithm then continues with EARCG to the final tolerance. Numerical experiments on 500 test cases report mean reductions of 22.06% in iteration count, 14.51% in wall-clock time, and 46.20% in density error, together with histograms, best/worst case studies, and a comparison against random application of the network.

Significance. If the reported gains are robust, this is a useful hybrid contribution to nonlinear eigenvalue solvers, complementing end-to-end neural approaches by retaining the classical solver's convergence guarantees while reducing cost in challenging regimes. Strengths include the reproducible open-source repository, a large test set of 500 cases, explicit disclosure of failure cases, and a sensible baseline comparison against random network application. The main weaknesses are statistical: the headline means pool runs that are not like-for-like (different local minima and non-converged classical baselines), and the acceptance heuristic is asserted but not independently validated. These issues are fixable in revision, and the paper's core idea deserves further consideration.

major comments (4)
  1. [Section 5.2 and 5.4] The headline improvements of 22.06% mean iteration saving and 14.51% mean wall-time saving are computed over all 500 test cases, as stated in Section 5.2 (Figure 8b and Figure 9b), yet the histograms color-code cases where the accelerated and classical runs converge to different local minima (orange). Section 5.4 explicitly excludes such cases from the iteration best/worst analysis because 'local convergence behavior may vary significantly, skewing the result,' which applies equally to the aggregate means. The pooled number also includes 12 runs where the classical method did not converge within 30000 iterations; for those runs the baseline is a stopping cap rather than a converged cost (Figure 19). For a like-for-like acceleration claim, report the mean and median savings restricted to runs where both methods converge to the same local minimum, and report the non-converged-classical cases separately. As written, the 22.06% figure conflates acceleration with changing the target solution (for example, Figure 15 reports a 61.57% iteration saving while the density deteriorates by 67.7%).
  2. [Section 5.4 and Figure 19] In the best-iteration examples (Section 5.4), the text states that 'in all cases, the classical and enhanced algorithm eventually converged to the same local minimum,' but Figure 19 is included as a best case even though its caption states the classical algorithm 'did not converge within 30000 steps' and the saving is quoted as 98.38% relative to that non-converged cap. This is internally inconsistent. A stopping cap is not an iteration count, and using it as the denominator inflates the saving. Either remove Figure 19 from the best-case analysis or provide a converged classical baseline; the claim of same-minimum convergence must then be adjusted accordingly.
  3. [Section 4.4] The acceptance strategy 'crucially relies' on the normalization error e = |1 - ||φ̃||_{L2}| as a quality indicator for the network output (Section 4.4). No theoretical justification, calibration, or independent validation of this correlation is provided; the only support is the aggregate improvement of the full strategy over random application. Because the acceptance threshold e0 controls exactly when the accelerator intervenes, a bad proxy could reject the best proposals or accept bad ones, directly affecting the reported speed-up. Provide a validation plot (e.g., e versus the density/energy error on a hold-out set) or an ablation showing the results are stable over a range of e0, and state the proportion of proposals accepted by the threshold in the 500 test runs.
  4. [Section 5.2] The density-improvement measure imprρ in Section 5.2 is defined relative to ρ∗, 'the reference solution (i.e., the EARCG converged state),' which is also the target the network was trained to predict (Section 4.2). The metric therefore rewards agreement with the classical solver's local minimum rather than with the physical ground state. This is visible in Figure 15, where the accelerated run attains a lower energy than the classical result yet is reported as 67.7% density deterioration. The accuracy claim should be split into (i) agreement with the classical converged state and (ii) comparison of the final energies of the two runs; at present the density-improvement statistic is partially circular and can mislead.
minor comments (8)
  1. [Abstract and Section 1] The phrase 'promising an order-of-magnitude speedup across a broad class of applications' in the introduction is not supported by the reported average savings (22% iterations, 14.5% wall time); only a few best cases approach an order of magnitude. Qualify this claim.
  2. [Section 5.2] Report standard deviations or confidence intervals for the mean savings; the histograms show substantial spread (e.g., for random application the mean 5.52% differs markedly from the median 12.62%).
  3. [Section 5.2] Define what 'correct energy' means in the sentence 'random acceleration converged to the correct energy in 439 cases'; clarify whether this refers to the same local minimum as the classical method or to a global-energy criterion.
  4. [Section 5.4 and Figure 19] Beyond the major inconsistency, the caption of Figure 19 states 'no estimation of the density improvement can be given,' which contradicts the general exclusion rule in Section 5.3; cross-check all figure captions for consistency with the methodology.
  5. [Section 4.3] The observation that the non-phase-invariant loss L(φ̂, φ) outperforms the invariant loss L̂(φ̂, φ) = ||ρ̂ - ρ||_{L1} is reported without quantitative evidence; include the training or validation losses for both losses.
  6. [Section 4.2] The phrase '20 log-equidistantly spaced tolerances' should be '20 logarithmically equidistant tolerances' for clarity.
  7. [Keywords] The keyword 'Riemmanian' is misspelled; it should be 'Riemannian'.
  8. [References] Reference [30] is listed as 2025 without a journal or preprint identifier; since Algorithm 1 and the step-size strategy depend on it, provide a stable reference or summarize the relevant details in the text.

Circularity Check

1 steps flagged · score 4.0 of 10

Density-improvement metric is measured against the same EARCG converged states used as training targets, so the 46% figure partly restates the training objective; the 22% iteration and 14.5% wall-time savings remain independent empirical claims.

  1. fitted input called prediction [Section 4.2 (Training data) and Section 5.2 (Benchmarking, imprρ definition)]
    "Each training data point then consists of a pair ((φj, gj), φ∗), where φ∗ denotes the converged solution of the EARCG method. ... imprρ = ∥ρ − ρ∗∥L1(Ω) − ∥˜ρ − ρ∗∥L1(Ω) / ∥ρ − ρ∗∥L1(Ω), where φ is the input to the network, ˜φ is its output, φ∗ is the reference solution (i.e., the EARCG converged state) and ρ = |φ(·)|², ˜ρ = |˜φ(·)|², ρ∗ = |φ∗(·)|²."

    The reported 46.20% mean density-error reduction measures L1-density distance from the network output to φ*, the EARCG converged state. That φ* is the same object used as the supervised regression target in training: each training datum is ((φj,gj), φ*), with φ* 'the converged solution of the EARCG method'. Hence 'density improvement' is, by construction, a test-set measure of how well the network reproduces the baseline solver's own fixed point. A positive average imprρ is the expected outcome of a successfully trained regressor, so this headline metric partially restates the training objective rather than providing an independent accuracy benchmark.

full rationale

The paper's main contribution is an empirical acceleration claim: neural-network proposals embedded in EARCG reduce iterations by ~22% and wall time by ~14.5% over 500 test cases, with the network trained offline on EARCG trajectories. This central claim is not circular: the network is trained to output the baseline solver's converged state from early iterates, and the benchmark compares total iteration and runtime counts of accelerated vs. unaccelerated runs on fresh random cases, where the result could have been negative (indeed, some worst-case examples show iteration increases). The baseline EARCG is cited from the authors' own prior work [30], but it is a concrete, independently implementable algorithm and its use is transparent, not a load-bearing unverified self-citation. The only construction-level circularity is the density-error metric: imprρ is defined relative to φ*, the same EARCG converged state that serves as the training label, so the 46% 'density improvement' is partly a restatement of training success rather than an independent accuracy measure. Additionally, the paper itself flags two limitations that are correctness concerns, not circularity: the acceptance strategy 'crucially relies' on the normalization error as a quality indicator (Section 4.4) with no theoretical justification, and Section 5.4 excludes different-minima cases from best/worst iteration analysis because 'local convergence behavior may vary significantly, skewing the result,' while the headline means pool those cases. These weaken the statistical interpretation of the speed-up but do not reduce it to its own inputs. Overall circularity score: 4, reflecting one partially self-referential evaluation metric while the central speed-up claim retains independent empirical content.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central claim rests on several hand-chosen hyperparameters (acceleration window, acceptance threshold, application frequency) and on assumptions that the reference solutions, discretization, and acceptance heuristic are reliable. No new physical entities are introduced; the neural network is a trained computational component. The most questionable entry is the normalization-error quality indicator, which is load-bearing for the acceptance strategy but not independently validated.

free parameters (6)
  • acceleration window lower bound eps1_min = 1e-4
    Hand-chosen threshold defining when the neural network is allowed to act (Section 4.4 and Section 5).
  • acceleration window upper bound eps1_max = 1e-1
    Hand-chosen threshold for the start of the neural acceleration phase.
  • final tolerance eps2 = 1e-8
    Convergence tolerance for the energy norm of the gradient; standard but chosen.
  • network application frequency ne = 5
    Number of EARCG iterations between neural network evaluations; chosen by numerical experience.
  • acceptance threshold e0 = 5e-3
    Maximum allowed normalization error for accepting a neural network output; critical to the acceptance strategy.
  • data augmentation flip probability p = 0.5
    Probability of horizontal and vertical flips during training (Section 5.1).
assumptions (6)
  • domain assumption The rotating Gross-Pitaevskii energy minimization on the L2 unit sphere has well-defined ground states and the Euler-Lagrange formulation (7) characterizes critical points.
    Used implicitly in Section 2 to set up the benchmark problem.
  • domain assumption Zero Dirichlet boundary conditions on a sufficiently large square introduce negligible truncation error.
    Stated in Section 2 after Eq. (1).
  • domain assumption Plane wave discretization with kinetic energy cutoff Ecut = 100 is accurate enough for the reported comparisons.
    Fixed in Section 5.1; no convergence study is presented in the paper.
  • ad hoc to paper The normalization error e = |1 - ||phi_tilde||_{L2}| is a reliable quality indicator for the neural network output.
    Central acceptance criterion introduced in Section 4.4; no theoretical proof or independent validation.
  • domain assumption Random initial states sampled from a normal distribution and normalized cover the relevant basin structure.
    Data generation in Section 4.2; affects both training and test distributions.
  • domain assumption The classical EARCG converged state is a valid reference solution for measuring density improvement.
    Used in the definition of impr_rho in Section 5.2; some accelerated runs reach lower energies than the reference.

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Cite this review

Pith. "Pith review of Neural Network Acceleration of Iterative Methods for Nonlinear Schr\"odinger Eigenvalue Problems." pith.science (2026). https://pith.science/paper/BRIUYKVM

@misc{pith2026250716349,
  author       = {Pith},
  title        = {Pith review of: Neural Network Acceleration of Iterative Methods for Nonlinear Schr\"odinger Eigenvalue Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BRIUYKVM}},
  note         = {Machine review of arXiv:2507.16349}
}
read the original abstract

We present a novel approach to accelerate iterative methods to solve nonlinear Schr\"odinger eigenvalue problems using neural networks. Nonlinear eigenvector problems are fundamental in quantum mechanics and other fields, yet conventional solvers often suffer from slow convergence in extreme parameter regimes, as exemplified by the rotating Bose- Einstein condensate (BEC) problem. Our method uses a neural network to predict and refine solution trajectories, leveraging knowledge from previous simulations to improve convergence speed and accuracy. Numerical experiments demonstrate significant speed-up over classical solvers, highlighting both the strengths and limitations of the approach.

Figures

Figures reproduced from arXiv: 2507.16349 by the authors.

Figure 1
Figure 1. Local minimizer densities for a = 20, v = (1, 1)T [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Local minimizer densities for a = 20, v = (1.1, 1)T 3. Energy-adaptive Riemannian conjugate gradient descent Riemannian optimization is one well-established methodology for solving (3). Here, as a prime representative of more general Riemannian methods, we introduce the Energy-Adaptive Riemannian Conjugate Gradient (EARCG) method; later we will also highlight its limitations [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Local minimizer densities for a = 20, v = (2, 1)T in extreme parameter regimes. For a more detailed description of the method on the more general Stiefel manifold as well as some numerical improvements, the interested reader is referred to [30]. Recall the coercive, bounded, symmetric bilinear form aφ from (4), for φ ∈ S, and the respective operator Aφ from (6). Then the energy-adaptive metric at φ ∈ S is given as (… view at source ↗
Figures from the paper (19 more)
Figure 4
Figure 4. Figure 4: Density plots from EARCG iteration for parameters a = 20, v = (1.1, 1), ω = 1.4 and κ = 1000. The final vortex configuration is reached after around 500 iterations, convergence with the correct orientation is reached after 12140 iterations. As shown in [PITH_FULL_IMAG…
Figure 5
Figure 5. Figure 5: Network architecture of the U-Net with a 4-channel input and a 2-channel output. We use a standard U-Net architecture with five blocks in the contracting and expanding paths respectively. Every block consists of two 3 × 3 padded convolution layers using ReLU activation…
Figure 6
Figure 6. Figure 6: Density plots from the neural network accelerated EARCG iteration for parame￾ters a = 20, v = (1.1, 1), ω = 1.4 and κ = 1000. Convergence is reached after 10152 iterations, saving almost 2000 iterations. 5. Numerical Experiments The numerical experiments use the EARCG …
Figure 7
Figure 7. Figure 7: Training and validation loss for the UNet training. axis through the origin, this allows the network to learn the symmetry. We train 100 epochs of the network, and the losses are shown in [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: Histograms of the percentage of iterations saved by the neural network-accelerated algorithms, evaluated on all 500 test cases. In blue are the cases, where the enhanced and classical algorithm converged to the same local minimum, in orange the cases of different minim…
Figure 9
Figure 9. Figure 9: Histograms of the reduction in wall time by the neural network-accelerated al￾gorithms, evaluated on all 500 test cases. In blue are the cases, where the enhanced and classical algorithm converged to the same local minimum, in orange the cases of different minima. The …
Figure 10
Figure 10. Figure 10: Histograms of the reduction in density error by the neural network-accelerated algorithms, evaluated on the 488 relevant test cases. In blue are the cases, where the enhanced and classical algorithm converged to the same local minimum, in orange the cases of different…
Figure 11
Figure 11. Figure 11: Neural network enhanced convergence of energy-adaptive RCG for parameters v ≈ (1.212, 1), ω ≈ 1.396, κ = 538. Enhancement resulted in 1046 total steps, while the classical algorithm took 1313 steps. Thus, the amount of steps was reduced by 20.34%. The density improved…
Figure 12
Figure 12. Figure 12: Neural network enhanced convergence of energy-adaptive RCG for parameters v ≈ (1.207, 1), ω ≈ 1.431, κ = 477. Enhancement resulted in 790 total steps, while the classical algorithm took 1047 steps. Thus, the amount of steps was reduced by 24.55%. The density improved …
Figure 13
Figure 13. Figure 13: Neural network enhanced convergence of energy-adaptive RCG for parameters v ≈ (1.187, 1), ω ≈ 1.115, κ = 442. Enhancement resulted in 605 total steps, while the classical algorithm took 737 steps. Thus, the amount of steps was reduced by 17.91%. The density improved b…
Figure 14
Figure 14. Figure 14: Neural network enhanced convergence of energy-adaptive RCG for parameters v ≈ (1.016, 1), ω ≈ 1.485, κ = 357. Enhancement resulted in 474 total steps, while the classical algorithm took 542 steps. Thus, the amount of steps was reduced by 12.55%. The density deteriorat…
Figure 15
Figure 15. Figure 15: Neural network enhanced convergence of energy-adaptive RCG for parameters v ≈ (1.509, 1), ω ≈ 1.111, κ = 468. Enhancement resulted in 216 total steps, while the classical algorithm took 562 steps. Thus, the amount of steps was reduced by 61.57%. The density deteriorat…
Figure 16
Figure 16. Figure 16: Neural network enhanced convergence of energy-adaptive RCG for parameters v ≈ (1.711, 1), ω ≈ 0.896, κ = 938. Enhancement resulted in 911 total steps, while the classical algorithm took 1005 steps. Thus, the amount of steps was reduced by 9.35%. The density deteriorat…
Figure 17
Figure 17. Figure 17: Neural network enhanced convergence of energy-adaptive RCG for parameters v ≈ (1.011, 1), ω ≈ 1.524, κ = 271. Enhancement resulted in 267 total steps, while the classical algorithm took 25285 steps. Thus, the amount of steps was reduced by 98.94%. The density improved…
Figure 18
Figure 18. Figure 18: Neural network enhanced convergence of energy-adaptive RCG for parameters v ≈ (1.020, 1), ω ≈ 1.423, κ = 385. Enhancement resulted in 346 total steps, while the classical algorithm took 26206 steps. Thus, the amount of steps was reduced by 98.68%. The density improved…
Figure 19
Figure 19. Figure 19: Neural network enhanced convergence of energy-adaptive RCG for parameters v ≈ (1.025, 1), ω ≈ 1.207, κ = 725. Enhancement resulted in 485 total steps, while the classical algorithm did not converge within 30000 steps. Thus, the amount of steps was reduced by 98.38%. S…
Figure 20
Figure 20. Figure 20: Neural network enhanced convergence of energy-adaptive RCG for parameters v ≈ (1.970, 1), ω ≈ 0.829, κ = 248. Enhancement resulted in 797 total steps, while the classical algorithm took 577 steps. Thus, the amount of steps was increased by 38.13%. The density deterior…
Figure 21
Figure 21. Figure 21: Neural network enhanced convergence of energy-adaptive RCG for parameters v ≈ (1.092, 1), ω ≈ 1.563, κ = 393. Enhancement resulted in 4525 total steps, while the classical algorithm took 3530 steps. Thus, the amount of steps was increased by 28.19%. The density improv…
Figure 22
Figure 22. Figure 22: Neural network enhanced convergence of energy-adaptive RCG for parameters v ≈ (1.082, 1), ω ≈ 1.219, κ = 599. Enhancement resulted in no convergence within 30000 total steps, while the classical algorithm took 24068 steps. Thus, the amount of steps was increased by 24…

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