Merge pull request #2 from kolosovpetro/MATH-99

MATH-99

Author: kolosovpetro

CHANGELOG.md

@@ -5,6 +5,20 @@ 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).
 
+## [1.0.1] - 2025-02-21
+
+### Changed
+
+- Update figure captions
+- Equation labeling change
+- Add coefficients U bibliography
+- Fix grammar abstract
+- Fix grammar introduction
+- Fix grammar generalizations
+- Fix grammar use cases
+- Fix grammar use conclusions
+- Update abstract
+
 ## [1.0.0] - 2025-02-21
 
 ### Changed

mathematica/MathematicaPrograms.txt

@@ -1,4 +1,4 @@
-(* Requires package: PlotsOfClosedForms.m, see https://github.com/kolosovpetro/AnEfficientMethodOfSplineApproximation*)
+(* Requires package: PlotsOfClosedForms.m, see https://github.com/kolosovpetro/AnEfficientMethodOfSplineApproximation*)
 
 f1[X_] := X^7;
 f2[X_] := P[3, X, 50];

mathematica/figures_notebook.cdf

@@ -1,4 +1,4 @@
-\begin{thebibliography}{1}
+\begin{thebibliography}{10}
 
 \bibitem{alekseyev2018mathoverflow}
 {Alekseyev, Max}.
@@ -47,4 +47,22 @@ Petro Kolosov.
 \newblock
   \url{https://kolosovpetro.github.io/pdf/UnexpectedPolynomialIdentity.pdf}.
 
+\bibitem{oeis_coefficients_u_m_l_k_defined_by_polynomial_identity_1}
+Petro Kolosov.
+\newblock {The coefficients U(m, l, k), m = 1 defined by the polynomial
+  identity}, 2018.
+\newblock \url{https://oeis.org/A320047}.
+
+\bibitem{oeis_coefficients_u_m_l_k_defined_by_polynomial_identity_2}
+Petro Kolosov.
+\newblock {The coefficients U(m, l, k), m = 2 defined by the polynomial
+  identity}, 2018.
+\newblock \url{https://oeis.org/A316349}.
+
+\bibitem{oeis_coefficients_u_m_l_k_defined_by_polynomial_identity_3}
+Petro Kolosov.
+\newblock {The coefficients U(m, l, k), m = 3 defined by the polynomial
+  identity}, 2018.
+\newblock \url{https://oeis.org/A316387}.
+
 \end{thebibliography}

out/AnEfficientMethodOfSplineApproximation.bbl

@@ -5,44 +5,44 @@ Reallocating 'name_of_file' (item size: 1) to 6 items.
 The style file: unsrt.bst
 Reallocating 'name_of_file' (item size: 1) to 39 items.
 Database file #1: AnEfficientMethodOfSplineApproximation.bib
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out/AnEfficientMethodOfSplineApproximation.blg

@@ -34,7 +34,7 @@
 \newtheorem{conjecture}[theorem]{Conjecture}
 \newtheorem{definition}[theorem]{Definition}
 
-\numberwithin{equation}{section}
+%\numberwithin{equation}{section}
 
 \title[An efficient method of spline approximation for power function]
 {An efficient method of spline approximation for power function}

out/AnEfficientMethodOfSplineApproximation.pdf

@@ -1,13 +1,10 @@
 Let $P(m, X, N)$ be an $m$-degree polynomials in $X\in\mathbb{R}$
 having fixed non-negative integers $m$ and $N$.
-Essentially, the polynomial $P(m, X, N)$ is a result of rearrangement inside Faulhaber's formula
-in context of Knuth's work \textit{Johann Faulhaber and sums of powers}.
-
-In this manuscript we discuss approximation properties of polynomial $P(m,X,N)$.
-In particular, the polynomial $P(m,X,N)$ approximates odd power function $X^{2m+1}$ in certain neighborhood
-of fixed non-negative integer $N$ with percentage error lesser than $1\%$.
-
-By increasing the value of $N$ the length of convergence interval with odd-power $X^{2m+1}$ increasing as well.
-
-Furthermore, this approximation technique is generalized for arbitrary non-negative exponent power function $X^j$
+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 of the power function $X^j$
 by using splines.

src/AnEfficientMethodOfSplineApproximation.tex

@@ -16,57 +16,57 @@
 For example,
 \input{sections/figures/05_fig_coefficients_a}
 
-Essentially, the polynomial $P(m, X, N)$ is a result of rearrangement inside Faulhaber's formula.
+Essentially, the polynomial $P(m, X, N)$ is derived from a rearrangement of Faulhaber's formula.
 It was inspired by Knuth's \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*}
     P(m, X, X) = X^{2m+1}
 \end{align*}
-In extended form
+In its extended form
 \begin{align*}
     X^{2m+1} = \sum_{r=0}^{m} \sum_{k=1}^{X} \coeffA{m}{r} k^r (X-k)^r
 \end{align*}
-Precisely, the relation between Faulhaber's formula and $P(m,X,N)$ is shown by~\cite{kolosov2025unexpected}.
+The exact relation between Faulhaber's formula and $P(m,X,N)$ is shown by~\cite{kolosov2025unexpected}.
 
-However, apart polynomial identity for odd powers, I've spotted several approximation properties of $P(m,X,N)$.
-Therefore, in this manuscript we discuss approximation properties of polynomial $P(m,X,N)$.
+However, apart from the polynomial identity for odd powers, I've discovered several approximation properties of $P(m,X,N)$.
+Therefore, in this manuscript we explore the approximation properties of the polynomial $P(m,X,N)$.
 I use a few well-known criteria to measure and estimate error of approximation: Absolute error, Relative error and
 Percentage error.
-Assume that function $f_2(x)$ approximates the function $f_1 (x)$ then the errors are
-
+Assume that the function $f_2(x)$ approximates the function $f_1 (x)$, then errors are given by
 \begin{align*}
     \mathrm{Absolute \; Error}   &= \lvert f_1(x) - f_2(x) \rvert \\
     \mathrm{Relative \; Error}   &= \frac{\lvert f_1(x) - f_2(x) \rvert}{\lvert f_1(x) \rvert} \\
     \mathrm{Percentage \; Error} &= \frac{\lvert f_1(x) - f_2(x) \rvert}{\lvert f_1(x) \rvert} \times 100\%
 \end{align*}
 
-Diving straight to the point, we switch our focus to already mentioned polynomial $P(2,X,4) = 900X^2 - 6000X + 10624$
+Diving straight into the point, we switch our focus to the previously mentioned 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 fact, we approximate the polynomial $X^{2m+1}$ by lower degree polynomial $X^m$ as the following image presents
+In fact, we approximate the polynomial $X^{2m+1}$ using a lower-degree polynomial of degree $m$,
+as shown in the following image
 \begin{figure}[H]
     \centering
     \includegraphics[width=1\textwidth]{sections/images/03_plots_polynomial_p2_n4_with_fifth}
-    ~\caption{Polynomial plot $P(2, X, 4)$ with fifth power $X^5$.
+    ~\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: $4.0 \leq X \leq 5.1$ with percentage error $E < 1\%$.
+    Convergence interval is $4.0 \leq X \leq 5.1$ with percentage error $E < 1\%$.
     }\label{fig:03_plots_polynomial_p2_n4_with_fifth}
 \end{figure}
-As we see, polynomial $P(2, X, 4)$ approximates $X^5$ in a neighborhood of $N=4$ with
-the convergence interval $4.0 \leq X \leq 5.1$ that has percentage error lesser than $1\%$ which is quite impressive.
-%Therefore, having fixed $N=4$ the polynomial $P(2, X, 4)$ approximates odd power in neighborhood of $N=4$
-%which is $3.9 \leq X \leq 5.3$.
-Consider the table below to showcase the concrete values of absolute, relative and percentage errors of this approximation
+As observed, the polynomial $P(2, X, 4)$ approximates $X^5$ in the neighborhood of $N=4$ with
+the convergence interval $4.0 \leq X \leq 5.1$ where the percentage error is less than $1\%$ which is quite remarkable.
+The following table presents specific values of absolute, relative, and percentage errors for this approximation
 \input{sections/figures/032_polynomials_p2_table_n4}
 
-One more interesting observation can be done by increasing the value of $N$ in $P(m, X, N)$ having fixed $m$, it
-follows that by increasing $N$ the length of convergence interval with odd-power $X^{2m+1}$ increasing as well.
+One more interesting observation arises by increasing the value of $N$ in $P(m, X, N)$ while keeping $m$ fixed.
+As $N$ increases, the length of the convergence interval with the odd-power $X^{2m+1}$ also increases.
 For instance,
 \begin{itemize}
-    \item Having $P(2, X, 4)$ and $X^5$ the convergence interval with percentage error lesser than $1\%$ is $4.0 \leq X \leq 5.1$ with lenght of interval $L=1.1$
-    \item Having $P(2, X, 20)$ and $X^5$ the convergence interval with percentage error lesser than $1\%$ is $18.7 \leq X \leq 22.9$ with lenght of interval $L=4.2$
-    \item Having $P(2, X, 120)$ and $X^5$ the convergence interval with percentage error lesser than $1\%$ is $110.0 \leq X \leq 134.7$ with lenght of interval $L=24.7$
+    \item For $P(2, X, 4)$ and $X^5$, the convergence interval with a percentage error less than $1\%$ is $4.0 \leq X \leq 5.1$, with a length $L=1.1$
+    \item For $P(2, X, 20)$ and $X^5$, the convergence interval with a percentage error less than $1\%$ is $18.7 \leq X \leq 22.9$, with a length $L=4.2$
+    \item For $P(2, X, 120)$ and $X^5$, the convergence interval with a percentage error less than $1\%$ is $110.0 \leq X \leq 134.7$, with a length $L=24.7$
 \end{itemize}
-The reason why the lenght of convergence interval rises as $N$ rise lays beneath the implicit form of polynomial $P(m,X,N)$
+
+The reason behind this behavior lies in the implicit form of the polynomial $P(m,X,N)$,
 meaning that
 \begin{align*}
     P(m,X,N) = \sum_{r=0}^{m} (-1)^{m-r} U(m, N, r) \cdot X^{r}
@@ -75,13 +75,16 @@
 \begin{align*}
     U(m, N, r) = (-1)^m \sum_{k=1}^{N} \sum_{j=r}^{m} \binom{j}{r} \coeffA{m}{j} k^{2j-r} (-1)^j
 \end{align*}
-which rises as $N$ rise.
+which grows as $N$ increases.
+Few cases of coefficients $U(m, N, r)$ are registered as OEIS sequences~\cite{
+    oeis_coefficients_u_m_l_k_defined_by_polynomial_identity_1,
+    oeis_coefficients_u_m_l_k_defined_by_polynomial_identity_2,
+    oeis_coefficients_u_m_l_k_defined_by_polynomial_identity_3}.
 
-To wrap up the current state of the manuscript, refresh the key facts and finding we got so far.
-Therefore, the polynomial $P(m,X,N)$ is an $m$-degree polynomial in $X \in \mathbb{R}$, having fixed non-negative
-integers $m$ and $N$.
-It approximates odd power function $X^{2m+1}$ in some neighborhood of fixed $N$.
-The length $L$ of convergence interval between $X^{2m+1}$ and $P(m,X,N)$ rises as $N$ rise.
+To summarize, let us recap the key findings so far.
+The polynomial $P(m,X,N)$ is an $m$-degree polynomial in $X \in \mathbb{R}$ with fixed non-negative integers $m$ and $N$.
+It approximates the odd power function $X^{2m+1}$ within a specific neighborhood of $N$.
+The length $L$ of the convergence interval between $X^{2m+1}$ and $P(m,X,N)$ increases as $N$ grows.
 
 For the sake of clear and precise verification of results, I attach mathematica programs to generate
 plots and data tables, so that reader is able to verify the main results of current part of manuscript,
@@ -94,8 +97,8 @@
 \begin{figure}[H]
     \centering
     \includegraphics[width=1\textwidth]{sections/images/07_plot_of_6th_power_with_p_2_4_times_x}
-    ~\caption{Polynomial plot $P(2, X, 4)\cdot X$ with sixth power $X^6$.
-    Convergence interval: $3.9 \leq X \leq 5.1$ with percentage error $E < 3\%$.
+    ~\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 $E < 3\%$.
     }\label{fig:07_plot_of_6th_power_with_p_2_4_times_x}
 \end{figure}
 Therefore, we have reached the statement that

src/sections/01_abstract.tex

@@ -1,8 +1,8 @@
-Previously, we have discussed that polynomial $P(m,X,N)$ approximates power function $X^j$
+Previously, we have discussed that polynomial $P(m,X,N)$ approximates power function $X^j$
 in some neighborhood of fixed non-negative integer $N$.
 Approximation by $P(m,X,N)$ can be adjusted by $X^k$ multiplication for even exponents of power function.
 In general, it is safe to say that power function $X^j$ is approximated by $P(m,X,N) \cdot X^k$
-where $k=0$ for odd exponent $j$ and $k$ either $k=1$ or $k=-1$ for even exponent $j$.
+where $k=0$ for odd exponent $j$ and $k$ is either $k=1$ or $k=-1$ for an even exponent $j$.
 Therefore, for arbitrary exponent $j$ in $X^j$ we have
 \begin{align*}
     X^j \approx
@@ -12,14 +12,14 @@
         P(m,X,N) \cdot X^{-1} \quad & j=2m \\
     \end{cases}
 \end{align*}
-Of course, there are other variations of the value of $k$, we stick to simple case for the moment.
+Of course, there are other variations of the value of $k$, but we will stick to the simple case for the moment.
 
-As we also discussed, the lenght $L$ of convergence interval between $X^j$ and approximation by $P(m,X,N)$
-rises as $N$ rise.
+As we also discussed, the length $L$ of the convergence interval between $X^j$ and its approximation by $P(m,X,N)$
+increases as $N$ grow.
 However, the convergence interval is still bounded, which could not satisfy certain approximation scenarios.
-Depending on approximation requirements in terms of convergence interval length $L$ a single polynomial $P(m,X,N)$
-with fixed $m$ and $N$ can be unfit.
-Here is the place where spline approximation comes to play.
+Depending on the approximation requirements in terms of convergence interval length $L$ a single polynomial $P(m,X,N)$
+with fixed $m$ and $N$ may be unsuitable.
+Here is the place where spline approximation comes into play.
 The spline $S(x)$ is piecewise defined function over the interval $(x_0, \ldots x_n)$
 \begin{align*}
     S(x) &=
@@ -32,9 +32,9 @@
 \end{align*}
 The given points $x_k$ are called \textit{knots}.
 
-Assume that approximation requirement in terms of convergence length $L$ is to approximate the power function $X^j$
+Assume that the approximation requirement in terms of convergence length $L$ is to approximate the power function $X^j$
 bounded by real points $A$ and $B$ such that $A < B$.
-Splines perfectly fits the need to match arbitrary convergence range of power function's $X^j$
+Splines perfectly fit the need to match an arbitrary convergence range for the power function $X^j$ using the
 approximation by $P(m,X,N)$.
 Formally,
 \begin{align*}