https://github.com/kolosovpetro/anefficientmethodofsplineapproximation

An efficient method of spline approximation for power function
https://github.com/kolosovpetro/anefficientmethodofsplineapproximation

approximation approximation-algorithms math mathematics

Last synced: 4 months ago
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An efficient method of spline approximation for power function

• Host: GitHub
• URL: https://github.com/kolosovpetro/anefficientmethodofsplineapproximation
• Owner: kolosovpetro
• License: gpl-3.0
• Created: 2025-01-20T22:56:21.000Z (5 months ago)
• Default Branch: main
• Last Pushed: 2025-02-21T22:27:12.000Z (4 months ago)
• Last Synced: 2025-02-21T22:29:35.950Z (4 months ago)
• Topics: approximation, approximation-algorithms, math, mathematics
• Language: Mathematica
• Homepage: https://kolosovpetro.github.io/pdf/AnEfficientMethodOfSplineApproximation.pdf
• Size: 5.57 MB
• Stars: 0
• Watchers: 1
• Forks: 0
• Open Issues: 1
• Metadata Files:
  • Readme: README.md
  • Changelog: CHANGELOG.md
  • License: LICENSE

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README

        
# An efficient method of spline approximation for power function

Let $P(m, X, N)$ be an $m$-degree polynomial in $X\in\mathbb{R}$

having fixed non-negative integers $m$ and $N$.

Essentially, the polynomial $P(m, X, N)$ is a result of a rearrangement inside Faulhaber's formula

in the context of Knuth's work entitled "Johann Faulhaber and sums of powers".

In this manuscript we discuss the approximation properties of polynomial $P(m,X,N)$.

In particular, the polynomial $P(m,X,N)$ approximates the odd power function $X^{2m+1}$ in a certain neighborhood

of a fixed non-negative integer $N$ with a percentage error less than $1\%$.

By increasing the value of $N$ the length of convergence interval with odd-power $X^{2m+1}$ also increases.

Furthermore, this approximation technique is generalized for arbitrary non-negative exponent $j$ of the power function $X^j$

by using splines.