Integrating Analytical and Numerical Approaches for Linear System Solutions

Abstract

The invention integrates analytical and numerical approaches to efficiently solve linear systems of equations. Combining precision with adaptability, it dynamically selects optimal methods based on matrix properties like sparsity and condition number. Analytical preprocessing, including matrix decomposition, is paired with numerical iterative techniques to ensure accuracy and scalability. Error control mechanisms monitor computations, stabilizing poorly conditioned matrices using regularization techniques. Advanced parallel and GPU-based processing enables rapid solutions for large-scale problems. A user-friendly interface offers both automatic and advanced configurations, making it accessible to experts and non-experts alike. Visualization tools enhance understanding and validation of results. The system is applicable across domains such as engineering, machine learning, and data science, addressing challenges in scalability, accuracy, and computational efficiency. This versatile tool empowers users to solve

Specification

Description:The present invention relates to the field of computational mathematics, specifically to methods and systems for solving linear systems of equations. It integrates analytical and numerical approaches to enhance the accuracy, efficiency, and reliability of solutions for large-scale and complex linear systems encountered in engineering, physics, and data science applications. The invention addresses challenges such as computational overhead, rounding errors, and convergence issues by combining traditional analytical methods, like matrix factorization, with advanced numerical techniques, including iterative solvers and parallel computation. It is particularly useful for applications in signal processing, structural analysis, machine learning, and control systems. The system aims to bridge the gap between theoretical solutions and practical implementations, catering to real-world problem-solving scenarios in academia, research, and industry.
Background of the proposed invention:
The invention, "Integrating Analyt , Claims:1. A system for solving linear systems of equations using a hybrid approach that integrates analytical methods, such as matrix decomposition, with numerical iterative techniques to achieve enhanced accuracy and efficiency.
2. The system of claim 1, wherein preprocessing analyzes matrix properties, including sparsity, condition number, and symmetry, to dynamically select optimal solution methods and parameters.
3. The system of claims 1 and 2, incorporating advanced error control mechanisms to monitor and mitigate numerical instabilities during computations, ensuring robust and reliable solutions.
4. A method for solving large-scale linear systems, wherein the system partitions matrices into smaller submatrices for concurrent processing, leveraging parallel or distributed computing architectures.
5. The method of claim 4, further utilizing GPU acceleration to expedite matrix operations, reducing computational overhead for high-dimensional systems.
6. The system of claim 1, wherein regularization techniques, su

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