Merge pull request #4 from kolosovpetro/MATH-94

MATH-94

Author: kolosovpetro

CHANGELOG.md

@@ -5,6 +5,17 @@ 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).
 
+## [1.0.3] - 2025-02-24
+
+### Changed
+
+- Update readme
+- Improve the flow
+- Update figure labels
+- Fix typos
+- Update keywords
+- Grammar
+
 ## [1.0.2] - 2025-02-24
 
 ### Changed

README.md

@@ -1,15 +1,14 @@
-# An efficient method of spline approximation for power function
-
-Let $P(m, X, N)$ be an $m$-degree polynomials in $X\in\mathbb{R}$
+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 rearrangement inside Faulhaber's formula
-in context of Knuth's work 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 $1\%$.
+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 $P(m,X,N)$.
 
-By increasing the value of $N$ the length of convergence interval with odd-power $X^{2m+1}$ increasing as well.
+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 $1\%$.
 
-Furthermore, this approximation technique is generalized for arbitrary non-negative exponent power function $X^j$
+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.

out/AnEfficientMethodOfSplineApproximation.pdf

@@ -45,7 +45,7 @@
 \keywords{
     Approximation,
     Power function,
-    Splines,
+    Spline approximation,
     Polynomials,
     Polynomial identities,
     Binomial coefficients
@@ -71,7 +71,8 @@
         Bernoulli polynomials,
         Combinatorics,
         Approximation,
-        Splines
+        Splines,
+        Spline approximation
     },
     pdfauthor={Petro Kolosov},
     pdfkeywords={
@@ -91,7 +92,8 @@
         Bernoulli polynomials,
         Combinatorics,
         Approximation,
-        Splines
+        Splines,
+        Spline approximation
     }
 }
 \begin{document}

src/AnEfficientMethodOfSplineApproximation.tex

@@ -1,8 +1,8 @@
-Use case scenarios of the approximation technique we discuss have their own constraints and limitations.
+The approximation technique above has its own constraints and limitations.
 For instance, approximation requirements should have precisely specified exponent $j$ in $X^j$ because
 for each $j$ there is a matching polynomial $P(m,X,N)$.
 Perfectly, there should be a set precompiled polynomials $P(m,X,N)$ matching precise exponent $j$ in $X^j$ over
-precisely defined approximation range with required error of approximation $E$ as a constraint.
+exactly defined approximation range with required error of approximation $E$ as a constraint.
 Generally, the approximation of power function $X^j$ by $P(m,X,N)$ can be broken down into the following steps
 \begin{enumerate}
     \item Define the exponent $j$ in $X^j$
@@ -31,7 +31,7 @@
 \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 $X^j \leq 100$.
-I have chosen the approximation range $10 \leq X \leq 15$ and $j=3$ intentionally to show the spline approximation with
+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
 \begin{align}
@@ -51,22 +51,22 @@
     \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 $E\leq 1\%$.
+        Convergence interval is $10 \leq X \leq 15$ with percentage error $E < 1\%$.
     }
     \label{fig:08_plots_of_cubes_power_with_p_2_10_15}
 \end{figure}
 where the spline knots are integers in the range $10 \leq N \leq 14$.
 
 The same principle applies for even exponent $j=4$ in $X^j$ with the same convergence interval $10 \leq X \leq 15$
-and approximation error under $1\%$ constraints
+and approximation error under $1\%$
 \begin{align}
     X^4 \approx
     \begin{cases}
         P(1,X,10) \cdot X = -2300X + 330X^2, & 10 \leq X < 11 \\
-        P(1,X,10) \cdot X = -3025X + 396X^2, & 11 \leq X < 12 \\
-        P(1,X,10) \cdot X = -3888X + 468X^2, & 12 \leq X < 13 \\
-        P(1,X,10) \cdot X = -4901X + 546X^2, & 13 \leq X < 14 \\
-        P(1,X,10) \cdot X = -6076X + 630X^2, & 14 \leq X \leq 15
+        P(1,X,11) \cdot X = -3025X + 396X^2, & 11 \leq X < 12 \\
+        P(1,X,12) \cdot X = -3888X + 468X^2, & 12 \leq X < 13 \\
+        P(1,X,13) \cdot X = -4901X + 546X^2, & 13 \leq X < 14 \\
+        P(1,X,14) \cdot X = -6076X + 630X^2, & 14 \leq X \leq 15
     \end{cases}
     \label{eq:spline_approximation_fourth_power}
 \end{align}
@@ -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 $E\leq 1\%$.
+        Convergence interval is $10 \leq X \leq 15$ with percentage error $E < 1\%$.
     }
     \label{fig:09_plots_of_fourth_power_with_p_2_10_15_times_x}
 \end{figure}

src/sections/04_use_cases.tex

@@ -24,4 +24,4 @@
     P(m,X, X+1) &= (X+1)^{2m+1} - 1
 \end{align*}
 This can be further optimized depending on the value of $N$ in $P(m,X,N)$ because the convergence interval
-with the power function $X^j$ increases as $N$ rises.
+with the power function $X^j$ increases as $N$ grows.