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Third-order Halley-type iterative method with positive and bounded correction function

Hirai Mukasa

arXiv:2609.08110v1math.NAcs.NA

TL;DR

Halley’s correction can become singular or reverse direction, complicating its behavior away from a root. This paper proposes a positive, bounded correction that matches Halley’s locally. The method retains at least third-order local convergence and shows strong convergence performance across the tested initial guesses, including near Halley’s singularity.

  • Problem

    Halley’s correction has a pole at Lf(x) = 2 and becomes negative for Lf(x) > 2, potentially producing excessively large corrections or reversing the Newton direction.

    Source: §1 · p. 2

  • Method

    The method replaces Halley’s correction with a positive, bounded correction whose local expansion matches Halley’s near a simple root.

    Source: §1 · p. 3, §5 · p. 19

  • Results

    Across seven test functions, the method had the highest average CSR and lowest average ANI, with CSR of at least 96% on every test function.

    Source: §4.1 · p. 16

  • Takeaways & Limitations

    In the van der Waals equation application, convergence performance was comparable to Halley’s method, including stable behavior from an initial guess with large |Lf(x0)|.

    Source: §5 · p. 19

    The stated global-convergence result is conditional, including a simple root in (a, b) and f ∈ C3([a, b]).

    Source: §3 · p. 12

Abstract

from arXiv · show

In this paper, we propose a new one-point iterative method for finding simple roots of nonlinear equations by modifying Halley's method. Its correction function is positive and bounded on $\mathbb{R}$ and matches the local expansion of Halley's correction function near a simple root. The method avoids the singularity and sign reversal of Halley's correction function while retaining at least third-order local convergence. We also establish sufficient conditions for global convergence. Numerical experiments show that the proposed method achieves high convergence success rates and favorable iteration counts over a wide range of initial guesses. Further experiments near the singularity of Halley's correction function demonstrate robust convergence behavior of the proposed method. An application to the van der Waals equation illustrates the effectiveness of the proposed method.

1 Introduction

Halley’s method has cubic local convergence, but its correction has a pole and reverses the Newton direction for Lf(x) > 2. The proposed method targets these global issues with a positive, bounded correction that matches Halley’s local behavior.

1 Introduction

  • Newton’s method converges quadratically near a simple root but may fail when the initial guess is far from it.Halley’s method is introduced as a one-point method with higher local convergence order than Newton’s.Source: §1 · p. 2
  • A pole at Lf(x) = 2 and a negative correction for Lf(x) > 2 can cause excessively large steps or reverse the Newton direction.These global features do not affect Halley’s cubic local convergence, but may adversely affect behavior when iterates are far from the root.Source: §1 · p. 2
  • Table 1 compares correction-function properties, and none of the considered methods satisfies all four properties.The properties include real-valuedness on R, positivity, boundedness, and Halley matching.Source: Table 1 · p. 3, §1 · p. 3
  • Earlier alternatives include an Extended Newton correction and a Padé-based correction, but both remain unbounded on R.The Extended Newton method was proposed to address Halley’s correction behavior for Lf(x) > 2.Source: §1 · p. 3
  • The proposed correction is positive and bounded on R, matches Halley’s local expansion, and is designed to improve robustness for initial guesses far from the root.The authors state that this local matching preserves Halley’s cubic convergence behavior.Source: §1 · p. 3

2 Proposed method

The method constructs a positive, bounded correction that matches Halley’s correction locally. Its rational-function design meets the stated conditions and includes an alternative evaluation intended to improve numerical stability.

2 Proposed method

  • The design conditions ensure 0 < C*(Lf(x)) < 2 on R, local matching with Halley’s correction, and C*(Lf(x))/CH(Lf(x)) → 1 as Lf(x) → −∞.The boundedness and positivity follow from condition (2.1), local matching from (2.2), and the asymptotic relation from (2.3).Source: §2 · p. 4
  • A positive quadratic family supplies the initial form, then a rational factor is introduced to match the required local expansion.The rational function is chosen as the lowest-degree form satisfying the stated condition and positivity on R.Source: §2 · pp. 4–5
  • The correction function is used to define the corresponding one-point iterative method.The paper first derives the correction function and then defines the iteration using it.Source: §2 · pp. 4, 7
  • For large positive Lf(x), an equivalent expression for R(Lf(x)) is offered to reduce loss of significance during direct evaluation.The two evaluation expressions are mathematically equivalent.Source: §2 · p. 7

3 Theoretical properties

The proposed correction function is positive and bounded on R while matching Halley’s correction near a simple root. Under sufficient conditions, the resulting method has at least third-order local convergence and monotone global convergence.

3 Theoretical properties

  • 0 < C*(Lf(x)) < 2 for every real Lf(x), so the proposed correction is well-defined, positive, and bounded on R.Its range is (0, 2).Source: §3 · pp. 7–8
  • For Lf(x) < 2, C*(Lf(x)) is positive and no larger than Halley’s correction; for Lf(x) ≥ 2, it stays in (0, 2) while Halley’s is undefined at 2 and negative above 2.For Lf(x) < 2, equality holds only at Lf(x) = 0.Source: §3 · p. 8
  • A negative correction factor increases the distance to the root, a sign-reversal effect that the proposed positive correction rules out.Halley’s correction is negative when Lf(x) > 2.Source: §3 · p. 9
  • C*(Lf(x)) and Halley’s correction have identical coefficients through Lf(x)^3 and differ first at order Lf(x)^4.The proposed method has at least third-order convergence for a sufficiently smooth function and an initial guess sufficiently close to a simple root.Source: §3 · p. 10
  • Under Theorem 3.6’s sufficient conditions, qualifying initial guesses in [a, b] generate a monotone sequence converging to the simple root α.The sequence decreases when f(x0)f′(x0) > 0 and increases when f(x0)f′(x0) < 0.Source: §3 · p. 12
  • The proposed method’s sufficient condition imposes no upper bound on Lf(x), whereas the cited Halley result requires Lf(x) < 1/2.Source: §3 · pp. 14–15

4 Numerical results

Across seven test functions, the proposed method led in average CSR and ANI while maintaining at least 96% CSR per function. It also remained stable near Halley’s singularity and showed comparable iteration counts in the van der Waals application.

4.1 Numerical experiments

  • The proposed method had the highest average CSR and lowest average ANI across seven test functions, with CSR at least 96% for each.It achieved the best values for both metrics on f2.Source: Table 2 · p. 16, §4.1 · p. 16
  • The experiments evaluated convergence success rate and average iteration count over broad initial-guess ranges, assigning the maximum iteration count to unsuccessful runs when calculating ANI.The test set comprised seven functions, with convergence criteria and initial-guess intervals specified for the experiments.Source: §4.1 · pp. 15–16
  • Near Lf(x0) = 2, the proposed method kept its correction factor and first iterate finite and converged in four iterations for every tested initial guess.Halley’s correction factor and first iterate grew rapidly near 2, became negative above 2, and were undefined at 2.Source: Table 3 · p. 17, §4.1 · p. 17

4.2 Application

  • For the van der Waals equation, the proposed method matched Halley’s iteration count from x0 = 0.5 and 1.0, and converged from x0 = 0.25 despite |Lf(x0)| ≈ 349.7462.Some other methods used one fewer iteration for x0 = 0.5 and 1.0.Source: §4.2 · p. 18, Table 4 · p. 19

5 Concluding remarks

The proposed method uses a positive, bounded correction function that preserves Halley’s local behavior while avoiding its singularity and sign reversal. The paper proves third-order convergence and global convergence under certain conditions, and reports robust numerical behavior.

5 Concluding remarks

  • The proposed one-point method uses a correction function that is positive and bounded on R while preserving Halley’s local behavior.Its correction matches Halley’s to high order near the root.Source: §5 · p. 19
  • The method avoids Halley’s correction singularity at Lf(x) = 2 and its sign reversal for Lf(x) > 2.Source: §5 · p. 19
  • The paper proves third-order convergence and global convergence under certain conditions; experiments report high success rates and stable behavior near Halley’s singularity.In the van der Waals application, its convergence performance was comparable to Halley’s, including stable behavior from an initial guess with large |Lf(x0)|.Source: §5 · p. 19

A Proof of Lemma 3.5

The proof uses sign and range cases to establish positivity of Φ away from Lf(x) = 0, where Φ equals zero.

A Proof of Lemma 3.5

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