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REVIEW 4 major objections 5 minor 56 references

Higher order Jacobi method for solving system of linear equations

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that recursively squaring the Jacobi update produces a finite product identity for the inverse of the normalized system matrix, turning linear solves into GPU-friendly matrix chains.

desk verdict Repeated-squaring Jacobi is a known identity; the GPU scaling exponents are physically implausible and the empirical methodology is ad hoc, but the induced-source proposal is worth a footnote. read the letter →

arxiv 2505.16906 v2 pith:46Q3XY6S submitted 2025-05-22 cond-mat.supr-con physics.comp-ph

Nithin Kumar Goona , Lama Tarsissi This is my paper · ORCID
classification cond-mat.supr-conphysics.comp-ph MSC 65F1065F0515A09
keywords higherorderJacobimethodmatrixinverseiterativelinearsolversmatrix-matrixmultiplicationGPUaccelerationdeepnetworksphysics-informedneuralcomputationalcomplexity
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 claims that the classical Jacobi iteration can be reorganized into a higher-order method: by recursively squaring the iteration matrix, one step at order $k$ advances the solution as far as $2^k$ ordinary Jacobi steps, and the same recurrence yields an explicit finite product for the inverse of the normalized coefficient matrix, $A_n^{-1} = \prod_{i=0}^{k-1}(\alpha^{(i)}+I)$. If that identity holds, solving a linear system becomes a chain of matrix-matrix multiplications rather than many matrix-vector sweeps, which is exactly the operation GPUs execute fastest. The paper further proposes an induced-source iteration that reuses a precomputed inverse to solve nearby systems with only matrix-vector work, and it presents the whole construction as a training-free deep linear network with analytically defined weights. On two GPU systems, the authors report runtime scaling exponents for the higher-order method far below those of built-in dense solvers, and they take this as evidence that the method is significantly faster in practice.

What carries the argument

The central object is the recursively squared iteration matrix $\alpha^{(k)} = (\alpha^{(k-1)})^2$ with the companion bias $\beta^{(k)} = (\alpha^{(k-1)} + I)\beta^{(k-1)}$, seeded by $\alpha^{(0)} = -(A_n - I)$ and $\beta^{(0)} = b_n$. The carrying identity is Eq. (25), $A_n^{-1} = \prod_{i=0}^{k-1}(\alpha^{(i)} + I)$, which packages $2^k$ Jacobi sweeps into one matrix-product chain and is what turns the iterative method into a direct inverse formula. The induced-source update, Eqs. (27)-(30), is the companion mechanism that reuses the precomputed inverse by iterating only the source term when the coefficient matrix changes slowly.

What would settle it

Measure HOJM-GPU runtime over at least a factor of 16 in matrix size, fit the log-log exponent over all points, and compare with the cost of the matrix-matrix multiplications the method performs; the near-constant scaling claim is false if the fitted exponent rises toward the matrix-multiplication exponent rather than staying near zero.

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Extended reading notes

Core claim

On its own terms, the central discovery is that the Jacobi recurrence $x^{(k+1)} = \alpha x^{(k)} + \beta$ can be exponentiated: defining $\alpha^{(k)} = (\alpha^{(k-1)})^2$ and $\beta^{(k)} = (\alpha^{(k-1)} + I)\beta^{(k-1)}$ makes one order-$k$ update equivalent to $2^k$ classical sweeps, and the expanded bias term gives the closed form $A_n^{-1} = \prod_{i=0}^{k-1}(\alpha^{(i)} + I)$ for the inverse of the normalized matrix. The paper treats this equation as an explicit finite construction of the inverse, so that direct and iterative methods are unified rather than opposed. It also reads the coefficient hierarchy as a deep linear network with weights and biases fixed by the physics, bypassing the training step of a physics-informed neural network. For slowly changing systems, the proposed induced-source scheme (Eqs. (27)-(30)) keeps the precomputed inverse and iterates only on the source vector. The empirical section then reports GPU timings whose fitted exponents are $n^{0.19}$/$n^{0.79}$ for solving and $n^{0.24}$/$n^{-0.07}$ for inversion, which the authors attribute to the parallel nature of matrix-matrix multiplication.

Load-bearing premise

The empirical result stands on taking the slope of GPU wall-clock time over a small set of matrix sizes as the asymptotic complexity of the method, even though the reported exponents are far below the cost of reading the matrix once.

Editorial extensions

If this is right

  • One application of an order-$k$ coefficient pair advances the solution as far as $2^k$ ordinary Jacobi steps, so the paper's choice of $k=20$ corresponds to about one million classical iterations.
  • The product identity $A_n^{-1} = \prod_{i=0}^{k-1}(\alpha^{(i)} + I)$ gives an explicit finite expression for the inverse of the normalized system matrix, so the method can be used as a direct solver whenever the Jacobi convergence conditions hold.
  • The recurrence defines all network weights $\alpha^{(i)}$ and biases $\beta^{(i)}$ analytically, meaning the linear solve needs no training phase, unlike a physics-informed neural network.
  • With a slowly varying coefficient matrix, the induced-source scheme (Eqs. (27)-(30)) performs only matrix-vector multiplications after the initial inverse is computed, so re-solving a nearby system avoids recomputing higher-order coefficients.
  • The reported GPU runtime exponents for the higher-order method are $n^{0.19}$ and $n^{0.79}$ for solving and $n^{0.24}$ and $n^{-0.07}$ for inversion, which the authors attribute to the parallelism of matrix-matrix multiplication.

Reading between the lines

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

  • Beyond the paper: the same recursive squaring idea should apply to any stationary iteration whose iteration matrix is contractive, yielding analogous product formulas for Gauss-Seidel or successive over-relaxation.
  • Beyond the paper: the induced-source scheme could be combined with time stepping or Newton linearization so one precomputed inverse serves many successive linear systems in nonlinear or time-dependent problems, an extension the paper only gestures at.
  • Beyond the paper: because the higher-order products densify as they are squared, a concrete test would be to truncate each $\alpha^{(i)}$ sparsely and measure how the residual of Eq. (25) degrades, quantifying the pruning-versus-accuracy trade-off in the same terms as runtime.
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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 / 5 minor

Summary. The paper proposes a 'higher order Jacobi method' (HOJM) for solving linear systems and computing matrix inverses. The method recursively squares the Jacobi iteration matrix and accumulates a bias term, so that k higher-order steps are equivalent to 2^k ordinary Jacobi iterations. The resulting product formula is interpreted as a closed-form expression for the inverse of the normalized matrix, and the structure is likened to a deep linear network. The paper also proposes an 'induced source' iterative scheme for reusing a known inverse when the coefficient matrix changes slowly. The main empirical claim, based on GPU timing measurements, is that HOJM achieves a dramatically lower order of complexity than standard solvers, with fitted exponents as low as n^0.19 for solving and n^(-0.07) for inversion. The paper includes code on GitHub.

Significance. If the near-constant scaling claim were valid, it would indeed be a transformative result, as no algorithm that reads a matrix of size n can run in time sublinear in the matrix size. The paper's algebraic derivation of the higher-order iteration and the product formula for the inverse is elementary and is presented as a connection between iterative methods and deep linear networks. The availability of code and reproducibility scripts is a strength. However, the load-bearing empirical claim rests on exponent fitting from two data points, and the reported exponents are physically impossible for any algorithm that must read the input matrix. The paper's own Discussion section also acknowledges that HOJM has the same computational complexity as matrix-matrix multiplication, which contradicts the claimed 'significant enhancement in the order of complexity.' Without support for the central performance claim, the contribution reduces to an interesting but straightforward algebraic reformulation of Jacobi iteration.

major comments (4)
  1. [2.1, Eq. (18)] Equation (18) states x(2k) = α(k)·x(2k−1) + β(k), claiming that the k-th order iterate can be obtained from the previous higher-order guess x(2k−1). This is not correct. Equation (15) gives x(2k) in terms of x(0) and the k-th order coefficients; applying the k-th order operator to x(2k−1) does not reproduce x(2k) because the k-th order operator advances by 2^k Jacobi steps, not by one step. The correct recurrence is x(m + 2^k) = α(k)·x(m) + β(k) for general m, or in the paper's indexing x(2k) = α(1)·x(2k−2) + β(1) for the second-order case. As written, Eq. (18) is mathematically inconsistent with Eq. (15) and would produce incorrect iterates if implemented. Since this equation is used to motivate the 'faster convergence' claim and the deep recurrent network analogy, it is a load-bearing error that must be corrected.
  2. [Section 3, Table 2] The empirical complexity exponents are computed using only the last two data points, with an ad hoc rule that drops the middle of the last three points if a 'sudden jump' is observed. As stated in Section 3: 'The runtime empirical computational complexity for each curve is calculated using the last two data points. If a sudden jump is observed in the last 3 data points, the middle point is excluded for the calculation.' This procedure can produce arbitrarily small exponents. The reported exponents n^0.19 for solving and n^(-0.07) for inversion are below the O(n^2) cost of even reading a dense n×n matrix, and a negative exponent for matrix inversion is impossible for any algorithm that must output an n×n result. These exponents therefore reflect fixed kernel-launch overhead and memory-bound effects over a narrow size range, not algorithmic complexity. The absence of error bars, repeated runs, or any discussion of measurement variability further undermines the claim. This is the central load-bearing evidence for the 'significant enhancement in the order of complexity' stated in the abstract.
  3. [Section 4 (Discussion) vs. Abstract] The Discussion states that 'The proposed higher-order method has the same computational complexity as matrix-matrix multiplication' (naively O(n^3) for dense matrices and O(n^3) for the product of n×n matrices). This directly contradicts the abstract's claim of 'significant enhancement in the order of complexity.' The GPU timings in Table 2 indicate only that, over the measured size range, the wall-clock time grows slowly; they do not establish a lower order of complexity. The paper conflates measured wall-clock time on a specific GPU implementation with asymptotic algorithmic complexity. At minimum, the paper must clearly separate the algorithmic complexity of HOJM from the observed speedup due to GPU parallelization and must temper the 'order of complexity' language accordingly.
  4. [2.4, Eqs. (27)–(30)] The induced source iterative scheme is asserted to converge for slowly varying coefficient matrices, but no convergence proof or numerical demonstration is provided. The scheme involves the free relaxation parameter ω, and the iteration (27)–(30) is a fixed-point iteration whose convergence depends on properties of the unknown function 𭟋 and the matrix difference A − A_d. No conditions on ω, on the rate of variation of A_d, or on the spectral radius of the iteration operator are given. Since this section is the basis for the claimed ability to 'efficiently resolve system variations without recomputing the coefficients,' the absence of any analysis or experiment makes this claim unsupported.
minor comments (5)
  1. [Throughout] The paper uses two different acronyms for the proposed method: HOJM in the abstract and Section 1, and 'HOM' in the second paragraph of Section 1. Use one consistently.
  2. [2.2, Eq. (22)] The product notation 0∏i=k−1 is nonstandard and the order of matrix multiplication matters. It should be written as ∏_{i=k−1}^{0} (α(i) + I) with the convention that the product is taken in reverse order, or as an explicit descending product.
  3. [Table 2] The table contains two rows labeled 'after' (after System 1 and after System 2) whose meaning is unclear. These rows appear to belong to the HOJM columns but are not explained in the text. Clarify or remove.
  4. [Section 3] There are typographical errors: 'single presision' should be 'single precision,' and 'coeffecient' should be 'coefficient.' Also, 'The best order in each row has be highlighted' should be 'has been highlighted.'
  5. [Figure 1] The text states that the vertical range is capped to 20 seconds, but for log-log plots the cap is a horizontal truncation of large-size behavior. The paper should justify this cap and show whether the truncated points would change the fitted exponents.

Circularity Check

1 steps flagged · score 2.0 of 10

Empirical complexity exponents are fitted from the same timing data they are used to substantiate; the mathematical HOJM derivation is otherwise self-contained.

  1. fitted input called prediction [Section 3, Table 2 and Abstract]
    "The runtime empirical computational complexity for each curve is calculated using the last two data points. If a sudden jump is observed in the last 3 data points, the middle point is excluded for the calculation."

    The paper's central scaling claims, such as HOJM GPU exponents n^0.19 for solving and n^-0.07 for inversion, are not independent measurements of asymptotic complexity: they are the very exponents fitted from the last two timing points of the same curves. The Abstract then treats these self-derived exponents as evidence of 'significant enhancement in the order of complexity' and a 'transformative direction' in algorithmic efficiency. Because the reported complexity law is the fit itself rather than a quantity validated on held-out matrix sizes or against operation counts, the conclusion reduces to a restatement of the fitting procedure. This is a mild, partial circularity in the empirical claim, while the mathematical derivation of HOJM remains self-contained.

full rationale

The mathematical core of the paper, Eqs. (3)-(25), is a self-contained algebraic rearrangement of the Jacobi iteration: normalizing by D, defining alpha = -(A_n - I), and telescoping the iterates gives the product formula for the truncated Neumann series. This derivation does not assume its conclusion, and the self-citations [33,34] for the distributed/induced source scheme are not load-bearing for the main result. The inverse-product identity is a known convergence-limit statement, though the paper presents it as an exact finite-k inverse without taking the limit, which is a correctness concern rather than a circularity. The only circularity-adjacent element is the empirical complexity analysis: the exponents in Table 2 are computed from the same last two data points that the Abstract uses to claim a 'significant enhancement in the order of complexity.' That makes the headline empirical claim partly forced by construction, but it does not invalidate the algebraic derivation, so the overall circularity score is mild.

Assumptions & free parameters 2 free parameters · 3 assumptions · 1 invented entities

The central algebraic identity needs only the standard spectral-radius condition for Jacobi convergence. The empirical performance claim rests on two hand-chosen parameters (k=20, omega unspecified) and on the assumption that the timing measurements reflect asymptotic complexity. The induced source is a new auxiliary quantity introduced without independent evidence.

free parameters (2)
  • number of HOJM coefficients k = 20
    Section 3 states 'Irrespective of the size of the matrix, 20 higher coefficients are calculated'; this hand-chosen truncation controls both accuracy and cost and is not derived.
  • relaxation factor omega = not specified
    Introduced in Eq. (30) to 'facilitate smoother convergence' of the induced source scheme; no value, tuning rule, or convergence condition is given.
assumptions (3)
  • domain assumption The system matrix A is such that the Jacobi iteration converges (spectral radius of alpha = I - A_n^{-1} A is below 1).
    The paper states 'when necessary conditions for convergence [28-30] are met in the regular Jacobi iteration method' (Section 2.2), and constructs test matrices as sparse and diagonally dominant (Section 3), but provides no verification for individual problems.
  • ad hoc to paper The induced source fixed-point iteration (27)-(30) converges for slowly varying coefficient matrices.
    Section 2.4 asserts 'this iterative approach introduces the following series of intermediate steps' without proof or numerical demonstration; convergence is assumed.
  • standard math The infinite product in Eq. (25) converges to A_n^{-1}.
    This is the classical geometric series identity (I - alpha)^{-1} = sum alpha^i = product (I + alpha^{2^i}) when the spectral radius of alpha is below 1; the paper relies on it without stating the condition explicitly.
invented entities (1)
  • induced source matrix b_i
    purpose: An auxiliary right-hand-side term added to b in Eq. (26) so that a fixed, precomputed inverse A^{-1} can solve systems with a changing coefficient matrix A_d without recomputation.
    This is a computational construct with no externally falsifiable prediction; its behavior depends entirely on the unproven convergence of the iteration (27)-(30).

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Cite this review

Pith. "Pith review of Higher order Jacobi method for solving system of linear equations." pith.science (2026). https://pith.science/paper/46Q3XY6S

@misc{pith2026250516906,
  author       = {Pith},
  title        = {Pith review of: Higher order Jacobi method for solving system of linear equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/46Q3XY6S}},
  note         = {Machine review of arXiv:2505.16906}
}
read the original abstract

This work proposes a higher-order iterative framework for solving matrix equations, inspired by the structure and functionality of neural networks. A modification of the classical Jacobi iterative method is introduced to compute higher-order coefficient matrices through matrix-matrix multiplications. The resulting method, termed the higher order Jacobi method (HOJM), structurally resembles a shallow linear network and allows direct computation of the inverse of the coefficient matrix. Building on this, an iterative scheme is developed that allows efficient resolution of system variations without recomputing the coefficients, once the network parameters are trained for a known system. This iterative process naturally assumes the form of a deep recurrent neural network. The proposed approach goes beyond conventional physics-informed neural networks (PINNs) by providing an explicit, training-free definition of network parameters rooted in physical and mathematical formulations. Computational analysis on GPU reveals significant enhancement in the order of complexity, highlighting a compelling and transformative direction for advancing algorithmic efficiency in solving linear systems. This methodology opens avenues for interpretable and scalable solutions to physically motivated problems in computational science.

Figures

Figures reproduced from arXiv: 2505.16906 by the authors.

Figure 1
Figure 1. Performance Comparision of built-in methods and the proposed higher order Jacobi method in solving [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.