构造分段可微二次型以确定数学运行时间并对非多项式函数建模
Constructing Segmented Differentiable Quadratics to Determine Algorithmic Run Times and Model Non-Polynomial Functions
摘要：
我们提出了一种方法来确定算法效率的持续进展，作为标准计算的时间复杂度的替代方案，可能，但不排除，当处理数据结构与未知的最大索引和算法，是依赖于多个变量，除了输入大小。所提出的方法可以有效地确定在任何给定的索引$x$，以及$\frac{\partial F}{\partial x}$，作为一个功能的只有一个或多个参数，通过结合$\frac{n}{2}$二次段的运行时间行为$F$，基于拉格朗日多项式和它们各自的割线的原则。尽管所使用的方法是为了分析计算算法的功效而设计的，但所提出的方法可以在纯数学领域中用作一种新的方法来构造非多项式函数，例如$\log_2{n}$或$\frac{n+1}{n-2}$，作为一系列分段可微二次方程来建模功能行为和重现的自然模式。经测试，该方法在功能相似度方面的平均准确率在99%以上。
英文摘要：
We propose an approach to determine the continual progression of algorithmic efficiency, as an alternative to standard calculations of time complexity, likely, but not exclusively, when dealing with data structures with unknown maximum indexes and with algorithms that are dependent on multiple variables apart from just input size. The proposed method can effectively determine the run time behavior $F$ at any given index $x$ , as well as $\frac{\partial F}{\partial x}$, as a function of only one or multiple arguments, by combining $\frac{n}{2}$ quadratic segments, based upon the principles of Lagrangian Polynomials and their respective secant lines. Although the approach used is designed for analyzing the efficacy of computational algorithms, the proposed method can be used within the pure mathematical field as a novel way to construct non-polynomial functions, such as $\log_2{n}$ or $\frac{n+1}{n-2}$, as a series of segmented differentiable quadratics to model functional behavior and reoccurring natural patterns. After testing, our method had an average accuracy of above of 99\% with regard to functional resemblance.
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