Deriving Closed-Form Expressions for Multi-Level Line Series within the Gradual Summation Method Framework via Regularity Theory
DOI:
https://doi.org/10.21512/emacsjournal.v8i2.15322Keywords:
Multi-level Line Series, GSM Table, Theory of Regularity, Computational Algorithm, Gradual SummationAbstract
The present research develops closed-form mathematical models for multi-level line series using the fundamental concepts of the Theory of Regularity. This research focuses on developing general analytical formulas for first-level series in the Gradual Summation Method (GSM) Table and subsequently extending them to higher levels of the series. Using regular pattern analysis derived from the fundamental principles of the Theory of Regularity, we create general notation and precise formulas. Our findings indicate that our proposed multi-level line series model can accurately compute both ordered sequences from the GSM Table and any other real number series using closed-form analytical equations rather than relying on traditional recursive methods. The primary benefit of our research is that we provide a mathematically rigorous model that replaces repeated summations with analytical formulas. Therefore, our model provides a strong theoretical and practical method for computing hierarchical data structures through software implementation. Our model serves as an analytical substitute for traditional loop-based computational algorithms. Additionally, we describe the structural properties, boundary conditions, and operator functions that define multi-level evaluations. Our findings help connect the field of discrete sequence analysis with practical algorithm design. Future applications may utilize the precise formulas we developed to improve multi-level aggregation tasks across many different types of computing environments.
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