Many physics and mathematics problems generate linear equation systems, and solving linear equation systems has become an important proposition. Therefore, combined with the characteristics of modern computers, various methods for solutions need to be sought. This article introduces applications of linear equation systems in different fields, as well as representative figures and works with outstanding achievements. It focuses on providing formulas and corresponding Matlab codes for Gauss elimination, Jacobi iteration, and G-S iteration to solve equation systems, and rigorously proves the sufficient and necessary condition for convergence of iterative formula and also proves the convergence of different iteration methods under different types of coefficient matrices. Based on these solving methods, two examples are practiced in Matlab, the running time and iteration times of different methods are comprehensively compared. Thus, the superiority of the G-S iterative method is obtained. Finally, when there are zero elements in the diagonal elements of the coefficient matrix, the article proposes an improved method to solve this problem.

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With the improvement of people’s living standards in the new era, tourism consumption has gradually become a hotspot of popular entertainment. Beijing faces the challenges of high tourist carrying capacity at attractions and uneven distribution of tourism resources. There is a growing need for personalised travel path planning. This study aims to develop a one-stop personalised intelligent recommendation model for tourist attractions in the Beijing area to enhance tourists’ travel experience. By integrating data from mainstream travel websites such as Ctrip, Tongcheng, and Qunar, the paper uses natural language processing (NLP) technology to conduct analyses of online reviews to derive user sentiment and personalisation indicators. The entropy weight method is used to comprehensively consider the user’s personalised travel preferences, combined with the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) method to scientifically rank the attractions and select the candidate set. Finally, the path planning algorithm with distance factor is implemented based on a greedy algorithm to optimise the travel path according to the user’s interest and achieve the recommendation of personalised travel routes. The model proposed in this study shows high accuracy and user satisfaction in empirical tests, which strengthens the user information processing support and personalisation needs in the era of big data, and contributes new solutions to the field of travel path recommendation.

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This paper investigates black hole spin using the relativistic reflection method, which analyses distorted X-ray spectra to understand accretion dynamics near the event horizon. The RELXILL series of models are very popular in anaylising refletion features. These Models enable a detailed and consistent understanding of the interaction between the accretion disk and relativistic effects. The paper shows the spin distribution of 33 black holes, including 14 stellar-mass black holes and 19 supermassive black holes. [1] High-spin (≥0.8) systems dominate, highlighting their association with intense relativistic effects near the innermost stable circular orbit. The high spin in stellar-mass black holes is driven by binary accretion, while the high spin in suppermassive black holes is driven by prolonged disk accretion and aligned mergers. [2] These results confirm the reliability of the method and provide insights into black hole evolution and accretion physics. Future research should explore how evolving models can refine spin predictions for complex systems.

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Large Language Models (LLMs) have demonstrated powerful capabilities in the field of code generation, with a deep understanding of the semantics and functionality of code. Building websites is one of the most important tasks in software development, as it utilizes rich frontend displays and backend processing to achieve various service functions. It is one of the most widely used interactive software models. Although there have been some efforts in Web website generation, these efforts have been limited to the automation of generating Web pages. The advent of LLMs provides a new approach to Web site generation tasks. However, there is currently a lack of comprehensive evaluation of the generation performance of LLMs in this context, making it difficult to optimize and improve the generated results in a targeted manner. To address this issue, this paper conducts a multi-angle investigation and analysis of the performance of LLMs in Web site generation tasks. Firstly, Web generation requirements are collected, and effective prompt engineering is designed. These prompts are then input into different LLMs to initiate the self-iteration process. Next, the generated code is fed back into the LLM for security self-iteration, where the model performs vulnerability detection and repair on the code it has generated. The security-enhanced code is subsequently subjected to manual review, where it is evaluated using predefined quantitative metrics to generate indicator values. Finally, through testing, the quality, security, and code defects of the generated front-end and back-end Web code across different LLMs are analyzed, providing a comprehensive evaluation of the generation results. The experiments demonstrate that LLM systems perform well in completing and implementing the functions and layouts of pages in prompts for Web generation tasks, but there remains room for improvement in the security of the Web code.

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