An Algorithm for Approximating Implicit Functions by Polynomials without Higher-Order Differentiability

2023/10/23 by Kyung Soo Rim, Rim, Kyung Soo
Computer Science · Engineering · Mathematics · #26B10 #Advanced Control Systems Optimization #Analysis of PDEs (math.AP) #FOS: Mathematics #G.1.2 #Numerical Analysis (math.NA) #Numerical methods for differential equations #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2310.14787

openalex publication_date 2023/10/23 · openalex created_date 2023/10/25 · openalex updated_date 2026/07/28

Abstract

We consider an equation of multiple variables in which a partial derivative does not vanish at a point. The implicit function theorem provides a local existence and uniqueness of the function for the equation. In this paper, we propose an algorithm to approximate the function by a polynomial without using higher-order differentiability, which depends essentially on integrability. Moreover, we extend the method to a system of equations if the Jacobian determinant does not vanish. This is a robust method for implicit functions that are not differentiable to higher-order. Additionally, we present two numerical experiments to verify the theoretical results.

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