Merge pull request #5 from kolosovpetro/MATH-94

MATH-94

Author: kolosovpetro

.gitattributes

@@ -8,4 +8,5 @@
 *.xml linguist-detectable=false
 *.css linguist-detectable=false
 *.m linguist-detectable=false
-*.nb linguist-detectable=false
+*.nb linguist-detectable=false
+*.cdf linguist-detectable=false

CHANGELOG.md

@@ -5,6 +5,15 @@ All notable changes to this project will be documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/),
 and this project adheres to [Semantic Versioning v2.0.0](https://semver.org/spec/v2.0.0.html).
 
+## [1.0.4] - 2025-02-24
+
+### Changed
+
+- Improve reasoning
+- Update abstract
+- Git attributes
+- Improve the flow
+
 ## [1.0.3] - 2025-02-24
 
 ### Changed

out/AnEfficientMethodOfSplineApproximation.pdf

@@ -1,10 +1,10 @@
 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
+The polynomial $P(m, X, N)$ is derived from a rearrangement of 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 $1\%$.
+of a fixed non-negative integer $N$ with a percentage error under $1\%$.
 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.

src/sections/01_abstract.tex

@@ -16,7 +16,7 @@
 For example,
 \input{sections/figures/05_fig_coefficients_a}
 
-Essentially, the polynomial $P(m, X, N)$ is derived from a rearrangement of Faulhaber's formula
+The polynomial $P(m, X, N)$ is derived from a rearrangement of Faulhaber's formula
 in the context of Knuth's work entitled \textit{Johann Faulhaber and sums of powers}, see~\cite{knuth1993johann}.
 In particular, the polynomial $P(m, X, N)$ yields an identity for odd powers
 \begin{align*}
@@ -43,14 +43,14 @@
 Diving straight into the point, we switch our focus to the partial case of polynomial
 $P(2,X,4) = 900X^2 - 6000X + 10624$
 to show the first example of how it approximates the odd power function $X^5$.
-In general, we approximate the polynomial $X^{2m+1}$ using a lower-degree polynomial of degree $m$,
-as shown in the following image
+In general, we approximate the polynomial $X^{2m+1}$ using a lower-degree polynomial of degree $m$.
+The following image demonstrates the approximation of fifth power $X^5$ by
+$P(2,X,4) = 900X^2 - 6000X + 10624$
 \begin{figure}[H]
     \centering
     \includegraphics[width=1\textwidth]{sections/images/03_plots_polynomial_p2_n4_with_fifth}
     ~\caption{Approximation of fifth power $X^5$ by $P(2, X, 4)$.
-    Points of intersection $X=4$, $X=4.42472$, $X=4.99181$.
-    Convergence interval is $4.0 \leq X \leq 5.1$ with percentage error $E < 1\%$.
+    Convergence interval is $4.0 \leq X \leq 5.1$ with a percentage error $E < 1\%$.
     }\label{fig:03_plots_polynomial_p2_n4_with_fifth}
 \end{figure}
 As observed, the polynomial $P(2, X, 4)$ approximates $X^5$ in a certain neighborhood of $N=4$ with
@@ -99,7 +99,7 @@
     \centering
     \includegraphics[width=1\textwidth]{sections/images/07_plot_of_6th_power_with_p_2_4_times_x}
     ~\caption{Approximation of sixth power $X^6$ by $P(2, X, 4) \cdot X$.
-    Convergence interval is $3.9 \leq X \leq 5.1$ with percentage error $E < 3\%$.
+    Convergence interval is $3.9 \leq X \leq 5.1$ with a percentage error $E < 3\%$.
     }\label{fig:07_plot_of_6th_power_with_p_2_4_times_x}
 \end{figure}
 Therefore, we have reached the statement that

src/sections/02_introduction.tex

@@ -29,11 +29,11 @@
     P(m,X, X) &= X^{2m+1} \\
     P(m,X, X+1) &= (X+1)^{2m+1} - 1
 \end{align*}
-Therefore, for each two consequential points $N=X, N=X+1$ the absolute difference is 1, making that range
-at least $1\%$ percentage error for $X^j \leq 100$.
+Therefore, for each two consequential points $N=X, N=X+1$ the absolute difference $P(m,X, X) - P(m,X, X+1)$ equals to one,
+consequently, the range $[X, X+1]$ satisfies the $1\%$ error threshold for any pair $j, X$ such that $X^j \leq 100$.
 The approximation range $10 \leq X \leq 15$ and exponent $j=3$ are chosen intentionally to show the spline approximation with
 percentage error threshold less than $1\%$.
-Therefore, to approximate $X^3$ in the range $10 \leq X \leq 15$, we use the following spline function
+Therefore, to approximate the polynomial $X^3$ in the range $10 \leq X \leq 15$, we use the following spline function
 \begin{align}
     X^3 \approx
     \begin{cases}
@@ -51,7 +51,7 @@
     \includegraphics[width=1\textwidth]{sections/images/08_plots_of_cubes_power_with_p_2_10_15}
     ~\caption{
         Approximation of cubes $X^3$ by splines~\eqref{eq:spline_approximation_of_cubes}.
-        Convergence interval is $10 \leq X \leq 15$ with percentage error $E < 1\%$.
+        Convergence interval is $10 \leq X \leq 15$ with a percentage error $E < 1\%$.
     }
     \label{fig:08_plots_of_cubes_power_with_p_2_10_15}
 \end{figure}
@@ -76,7 +76,7 @@
     \includegraphics[width=1\textwidth]{sections/images/09_plots_of_fourth_power_with_p_2_10_15_times_x}
     ~\caption{
         Approximation of $X^4$ by splines~\eqref{eq:spline_approximation_fourth_power}.
-        Convergence interval is $10 \leq X \leq 15$ with percentage error $E < 1\%$.
+        Convergence interval is $10 \leq X \leq 15$ with a percentage error $E < 1\%$.
     }
     \label{fig:09_plots_of_fourth_power_with_p_2_10_15_times_x}
 \end{figure}