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<p>This paper presents an explicit formula based on reparameterization technique for progressively computing a simple root of a smooth function, which may have wide applications in robotics, geomagnetic navigation, geometric processing and computer graphics. Comparing with Newton-like method, it can achieve convergence rate 2 by adding one more functional evaluation, improve the computational stability and ensure the convergence, and also obtain higher convergence rate and higher efficiency index. Compared with clipping methods for polynomials, it doesn’t need to bound the polynomials, directly bound the roots and can also work well for non-polynomial functions with much higher computational efficiency. Comparing with previous progressive methods, it achieves a much higher computational efficiency and is extended to solve bivariate equation system. Numerical examples show its much better performance on approximation error, computational efficiency and computational stability.</p>

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Progressive explicit formulae for root-finding problems based on reparameterization

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • Hui Wang,
  • Cheng Qian,
  • Xiao-diao Chen

摘要

This paper presents an explicit formula based on reparameterization technique for progressively computing a simple root of a smooth function, which may have wide applications in robotics, geomagnetic navigation, geometric processing and computer graphics. Comparing with Newton-like method, it can achieve convergence rate 2 by adding one more functional evaluation, improve the computational stability and ensure the convergence, and also obtain higher convergence rate and higher efficiency index. Compared with clipping methods for polynomials, it doesn’t need to bound the polynomials, directly bound the roots and can also work well for non-polynomial functions with much higher computational efficiency. Comparing with previous progressive methods, it achieves a much higher computational efficiency and is extended to solve bivariate equation system. Numerical examples show its much better performance on approximation error, computational efficiency and computational stability.