|
16 | 16 | For example, |
17 | 17 | \input{sections/figures/05_fig_coefficients_a} |
18 | 18 |
|
19 | | -Essentially, the polynomial $P(m, X, N)$ is derived from a rearrangement of Faulhaber's formula. |
20 | | -It was inspired by Knuth's \textit{Johann Faulhaber and sums of powers}, see~\cite{knuth1993johann}. |
| 19 | +Essentially, the polynomial $P(m, X, N)$ is derived from a rearrangement of Faulhaber's formula |
| 20 | +in the context of Knuth's work entitled \textit{Johann Faulhaber and sums of powers}, see~\cite{knuth1993johann}. |
21 | 21 | In particular, the polynomial $P(m, X, N)$ yields an identity for odd powers |
22 | 22 | \begin{align*} |
23 | 23 | P(m, X, X) = X^{2m+1} |
|
28 | 28 | \end{align*} |
29 | 29 | The exact relation between Faulhaber's formula and $P(m,X,N)$ is shown by~\cite{kolosov2025unexpected}. |
30 | 30 |
|
31 | | -However, apart from the polynomial identity for odd powers, I've discovered several approximation properties of $P(m,X,N)$. |
| 31 | +However, apart from the polynomial identity for odd powers, a few approximation properties of $P(m,X,N)$ |
| 32 | +were discovered in addition. |
32 | 33 | Therefore, in this manuscript we explore the approximation properties of the polynomial $P(m,X,N)$. |
33 | | -I use a few well-known criteria to measure and estimate error of approximation: Absolute error, Relative error and |
34 | | -Percentage error. |
| 34 | +During our discussion, we utilize the following well-known criteria to measure and estimate |
| 35 | +the error of approximation: Absolute error, Relative error and Percentage error. |
35 | 36 | Assume that the function $f_2(x)$ approximates the function $f_1 (x)$, then errors are given by |
36 | 37 | \begin{align*} |
37 | 38 | \mathrm{Absolute \; Error} &= \lvert f_1(x) - f_2(x) \rvert \\ |
38 | 39 | \mathrm{Relative \; Error} &= \frac{\lvert f_1(x) - f_2(x) \rvert}{\lvert f_1(x) \rvert} \\ |
39 | 40 | \mathrm{Percentage \; Error} &= \frac{\lvert f_1(x) - f_2(x) \rvert}{\lvert f_1(x) \rvert} \times 100\% |
40 | 41 | \end{align*} |
41 | 42 |
|
42 | | -Diving straight into the point, we switch our focus to the previously mentioned polynomial |
| 43 | +Diving straight into the point, we switch our focus to the partial case of polynomial |
43 | 44 | $P(2,X,4) = 900X^2 - 6000X + 10624$ |
44 | 45 | to show the first example of how it approximates the odd power function $X^5$. |
45 | | -In fact, we approximate the polynomial $X^{2m+1}$ using a lower-degree polynomial of degree $m$, |
| 46 | +In general, we approximate the polynomial $X^{2m+1}$ using a lower-degree polynomial of degree $m$, |
46 | 47 | as shown in the following image |
47 | 48 | \begin{figure}[H] |
48 | 49 | \centering |
|
52 | 53 | Convergence interval is $4.0 \leq X \leq 5.1$ with percentage error $E < 1\%$. |
53 | 54 | }\label{fig:03_plots_polynomial_p2_n4_with_fifth} |
54 | 55 | \end{figure} |
55 | | -As observed, the polynomial $P(2, X, 4)$ approximates $X^5$ in the neighborhood of $N=4$ with |
56 | | -the convergence interval $4.0 \leq X \leq 5.1$ where the percentage error is less than $1\%$ which is quite remarkable. |
| 56 | +As observed, the polynomial $P(2, X, 4)$ approximates $X^5$ in a certain neighborhood of $N=4$ with |
| 57 | +the convergence interval $4.0 \leq X \leq 5.1$ such that the percentage error is less than $1\%$ which is quite remarkable. |
57 | 58 | The following table presents specific values of absolute, relative, and percentage errors for this approximation |
58 | 59 | \input{sections/figures/032_polynomials_p2_table_n4} |
59 | 60 |
|
|
86 | 87 | It approximates the odd power function $X^{2m+1}$ within a specific neighborhood of $N$. |
87 | 88 | The length $L$ of the convergence interval between $X^{2m+1}$ and $P(m,X,N)$ increases as $N$ grows. |
88 | 89 |
|
89 | | -For the sake of clear and precise verification of results, I attach mathematica programs to generate |
| 90 | +For the sake of clear and precise verification of results, consider the Mathematica programs to generate |
90 | 91 | plots and data tables, so that reader is able to verify the main results of current part of manuscript, |
91 | | -see the \href{https://gist.github.com/kolosovpetro/2b5c55094c66b8d6a97b9798be9a8dec}{\texttt{link}}. |
| 92 | +see~\cite{kolosovpetro_gist}. |
92 | 93 |
|
93 | 94 | So far we have discussed approximation of odd power function $X^{2m+1}$, now we focus on its even case $X^{2m+2}$ |
94 | 95 | which is quite straightforward. |
|
104 | 105 | Therefore, we have reached the statement that |
105 | 106 | the polynomial $P(m,X,N)$ is an $m$-degree polynomial in $X$, having fixed non-negative |
106 | 107 | integers $m$ and $N$. |
107 | | -It approximates the power function $X^{j}$ in some neighborhood of fixed $N$. |
108 | | -The length of convergence interval between power function $X^j$ and $P(m,X,N) \cdot X^K$ rises as $N$ rise. |
| 108 | +It approximates the power function $X^{j}$ in a certain neighborhood of fixed $N$. |
| 109 | +The length of convergence interval between the power function $X^j$ and $P(m,X,N) \cdot X^K$ increases as $N$ grow. |
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