Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · cs/9904006

Jacobian matrix: a bridge between linear and nonlinear polynomial-only problems

W. Chen
View original source ↗Read source document ↗Export metadata ↓

Abstract

By using the Hadamard matrix product concept, this paper introduces two generalized matrix formulation forms of numerical analogue of nonlinear differential operators. The SJT matrix-vector product approach is found to be a simple, efficient and accurate technique in the calculation of the Jacobian matrix of the nonlinear discretization by finite difference, finite volume, collocation, dual reciprocity BEM or radial functions based numerical methods. We also present and prove simple underlying relationship (theorem (3.1)) between general nonlinear analogue polynomials and their corresponding Jacobian matrices, which forms the basis of this paper. By means of theorem 3.1, stability analysis of numerical solutions of nonlinear initial value problems can be easily handled based on the well-known results for linear problems. Theorem 3.1 also leads naturally to the straightforward extension of various linear iterative algorithms such as the SOR, Gauss-Seidel and Jacobi methods to nonlinear algebraic equations. Since an exact alternative of the quasi-Newton equation is established via theorem 3.1, we derive a modified BFGS quasi-Newton method. A simple formula is also given to examine the deviation between the approximate and exact Jacobian matrices. Furthermore, in order to avoid the evaluation of the Jacobian matrix and its inverse, the pseudo-Jacobian matrix is introduced with a general applicability of any nonlinear systems of equations. It should be pointed out that a large class of real-world nonlinear problems can be modeled or numerically discretized polynomial-only algebraic system of equations. The results presented here are in general applicable for all these problems. This paper can be considered as a starting point in the research of nonlinear computation and analysis from an innovative viewpoint.

Explore related subjects

cs.CEcs.NAmath.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

W. Chen. 1999-04-15. Jacobian matrix: a bridge between linear and nonlinear polynomial-only problems. https://arxiv.org/abs/cs/9904006

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

SabreAgent: Language Models at Design Time for Lost-Sales Inventory Control

SabreAgent uses a language model at design time to construct two components for lost-sales inventory control: a product-specific seasonal prior and a validation-selected capped base-stock policy family. During operation, statistical forecasting and inventory optimization use these frozen artifacts to determine orders, with zero language-model calls. We evaluate the approach on the $1{,}320$ instances of InventoryBench. Under the benchmark's cost assumptions, the operations-research core draws on a zero-lead-time optimality result and a projected-inventory rule for positive deterministic lead times. The latter computes replenishment shortfalls by propagating inventory using sales along simulated demand paths. The seasonal prior adds forecast variants alongside the original forecaster, and the selected policy family handles stochastic lead times with order destruction. SabreAgent scores $0.6311$, compared with $0.5380$ for the strongest published baseline, and ranks first in all six benchmark cells. Ablations attribute most of the gain to the OR core. In the paired analysis, the seasonal component adds $1.79\%$ across the three real-data cells, and the search component adds $2.3\%$ across the two stochastic-lead-time cells. These results demonstrate how model-generated priors and policy structure can improve an OR controller through design-time use.

cs.CE↗
arXiv2026

Hierarchical Multi-Task Learning with Liquidity-Aware Signals for Stock Forecasting

Stock price forecasting is a long-standing challenge in computational finance, driven by the inherent randomness of markets and complex temporal patterns. While recent deep-learning models have raised forecasting accuracy by jointly modeling inter-stock and temporal price dynamics, they conflate inter-stock relationships with intra-stock temporal dependencies and focus solely on the univariate objective of price movement. To address these limitations, we propose LiMT, a Hierarchical Multi-Task Learning framework that integrates liquidity-aware signals for stock price forecasting. LiMT employs a Market Regime Encoder (MRE) module that first extracts contemporaneous cross-stock dependencies, then models each stock's temporal dynamics, yielding a unified latent state. Building on this latent state, we introduce a Liquidity-Driven Learning (LDL) module, a mixture-of-experts architecture that features cross-task gating mechanisms to jointly predict price movement, volatility, and trading volume. We further design an Adaptive Portfolio Optimization (APO) mechanism that converts multi-task forecasts into executable portfolio weights under transaction-cost and liquidity constraints. Extensive experiments on the CSI300 and CSI500 benchmarks show that LiMT performs best among strong neural and tree-based baselines across the reported metrics. In realistic CSI300 backtests, APO improves annualized return from 3.99% to 10.01% and Sharpe ratio from 1.22 to 1.86 over equal weighting, showing that the multi-task forecasts translate into deployable portfolio gains.

cs.CE↗
arXiv2026

Scientific capabilities and deployment sustainability of small-scale LLMs in biological wastewater treatment

Large language models (LLMs) are emerging as scientific assistants, yet their computational demands and limited domain specialization constrain sustainable deployment in environmental engineering. Here, we investigate whether domain-specialized small-scale LLMs can combine scientific capability with sustainable deployment in biological wastewater treatment. We developed a benchmark evaluating three scientific capabilities of LLMs: retrospective cognition, comprehension fidelity, and prospective extrapolation. BioWater (8 billion parameters, fine-tuned on specialized domain knowledge) achieved higher comprehension-fidelity scores than participating human experts and performance comparable to a 397-billion-parameter general-purpose LLM in retrospective cognition and prospective extrapolation. Human-BioWater collaboration generated a scientific hypothesis that was subsequently supported by laboratory experiments, demonstrating its potential to contribute to prospective scientific research. We further evaluated the economic and environmental implications of LLM deployment across global wastewater treatment plants (WWTPs). Locally deployed small-scale LLMs became more sustainable than cloud-based large-scale LLMs as inference demand increased in intelligent WWTPs. These findings highlight domain-specialized small-scale LLMs as a promising pathway towards scientifically capable, computationally efficient, and sustainably deployable artificial intelligence for wastewater treatment.

cs.CE↗