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Efficient and stable derivative-free Steffensen algorithm for root finding

Alexandre Wagemakers, Vipul Periwal

TL;DR

The paper tackles derivative-free root finding by modifying the Steffensen framework with a nonlinear function $g$ to better approximate the derivative. This derivative-approximation strategy preserves second-order convergence near a root while expanding the basin of attraction and reducing iterations, improving stability over the vanilla Steffensen method. Comprehensive numerical experiments across scalar functions, vector-valued maps, and zeros of scalar fields show consistent performance gains and broader convergence domains, with detailed analyses of convergence order and dynamical behavior. The work suggests practical implications for optimization and dynamical-systems applications, and points to promising directions such as annealing-based extensions.

Abstract

We explore a family of numerical methods, based on the Steffensen divided difference iterative algorithm, that do not evaluate the derivative of the objective functions. The family of methods achieves second-order convergence with two function evaluations per iteration with marginal additional computational cost. An important side benefit of the method is the improvement in stability for different initial conditions compared to the vanilla Steffensen method. We present numerical results for scalar functions, fields, and scalar fields. This family of methods outperforms the Steffensen method with respect to standard quantitative metrics in most cases.

Efficient and stable derivative-free Steffensen algorithm for root finding

TL;DR

The paper tackles derivative-free root finding by modifying the Steffensen framework with a nonlinear function to better approximate the derivative. This derivative-approximation strategy preserves second-order convergence near a root while expanding the basin of attraction and reducing iterations, improving stability over the vanilla Steffensen method. Comprehensive numerical experiments across scalar functions, vector-valued maps, and zeros of scalar fields show consistent performance gains and broader convergence domains, with detailed analyses of convergence order and dynamical behavior. The work suggests practical implications for optimization and dynamical-systems applications, and points to promising directions such as annealing-based extensions.

Abstract

We explore a family of numerical methods, based on the Steffensen divided difference iterative algorithm, that do not evaluate the derivative of the objective functions. The family of methods achieves second-order convergence with two function evaluations per iteration with marginal additional computational cost. An important side benefit of the method is the improvement in stability for different initial conditions compared to the vanilla Steffensen method. We present numerical results for scalar functions, fields, and scalar fields. This family of methods outperforms the Steffensen method with respect to standard quantitative metrics in most cases.

Paper Structure

This paper contains 8 sections, 22 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. A family of Steffensen functions for root finding
  3. Order of convergence
  4. Solving roots for functions in $\mathbb{R}^k \to \mathbb{R}^k$
  5. Numerical results
  6. Dynamics on the plane
  7. Zeros of scalar fields
  8. Conclusion

Figures (1)

  • Figure 1: Representation of the basins of attraction of the functions $f_9(z) = (z^2 - 4)(z + 1.5)(z - 0.5)$, and $f_{18}(x,y) = \left( x^2 - 2x - y + 0.5, x^2 + 4y^2 - 4.0 \right)$. The root-finding algorithm has been computed with the functions $g_i$ described in Sec. \ref{['sec:num']}. The classical Steffensen algorithm is simulated in the right column of the panels for the normal and accelerated variations. Plots are computed for a grid of 1000$\times$1000 initial conditions for initial conditions $(x,y)$ in $[-2,2]\times [-2,2]$.