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An efficient method of spline approximation for power function

Petro Kolosov

TL;DR

This work introduces $P(m,X,N)$, an $m$-degree polynomial in $X$ derived from a Faulhaber-based rearrangement in Knuth’s framework, which can exactly represent $X^{2m+1}$ when evaluated at $X$ itself ($P(m,X,X)=X^{2m+1}$). It shows that $P(m,X,N)$ can approximate odd powers near a fixed integer $N$ with errors under $1\%$, and that enlarging $N$ increases the convergence interval; it also extends the approach to even powers via multiplication by $X$. The paper generalizes to arbitrary exponents $j$ using spline constructions, yielding piecewise approximations with controllable accuracy over user-defined intervals, demonstrated with explicit $j=3$ and $j=4$ splines on $[10,15]$. This yields an efficient, tunable framework for accurate low-degree polynomial approximations of power functions, with potential benefits for numerical computations requiring controlled interval coverage and error bounds.

Abstract

Let $P(m, X, N)$ be an $m$-degree polynomial in $X\in\mathbb{R}$ having fixed non-negative integers $m$ and $N$. 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 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.

An efficient method of spline approximation for power function

TL;DR

This work introduces , an -degree polynomial in derived from a Faulhaber-based rearrangement in Knuth’s framework, which can exactly represent when evaluated at itself (). It shows that can approximate odd powers near a fixed integer with errors under , and that enlarging increases the convergence interval; it also extends the approach to even powers via multiplication by . The paper generalizes to arbitrary exponents using spline constructions, yielding piecewise approximations with controllable accuracy over user-defined intervals, demonstrated with explicit and splines on . This yields an efficient, tunable framework for accurate low-degree polynomial approximations of power functions, with potential benefits for numerical computations requiring controlled interval coverage and error bounds.

Abstract

Let be an -degree polynomial in having fixed non-negative integers and . The polynomial 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 . In particular, the polynomial approximates the odd power function in a certain neighborhood of a fixed non-negative integer with a percentage error under . By increasing the value of the length of convergence interval with odd-power also increases. Furthermore, this approximation technique is generalized for arbitrary non-negative exponent of the power function by using splines.

Paper Structure

This paper contains 4 sections, 15 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Generalizations
  3. Use cases
  4. Conclusions

Figures (4)

  • Figure 1: Approximation of fifth power $X^5$ by $P(2, X, 4)$. Convergence interval is $4.0 \leq X \leq 5.1$ with a percentage error $E < 1\%$.
  • Figure 2: Approximation of sixth power $X^6$ by $P(2, X, 4) \cdot X$. Convergence interval is $3.9 \leq X \leq 5.1$ with a percentage error $E < 3\%$.
  • Figure 3: Approximation of cubes $X^3$ by splines \ref{['eq:spline_approximation_of_cubes']}. Convergence interval is $10 \leq X \leq 15$ with a percentage error $E < 1\%$.
  • Figure 4: Approximation of $X^4$ by splines \ref{['eq:spline_approximation_fourth_power']}. Convergence interval is $10 \leq X \leq 15$ with a percentage error $E < 1\%$.