On Convergence of the Secant Method

TL;DR

This work provides a rigorous treatment of the secant method's convergence order, proving Q-superlinear convergence for simple roots with p = $r_0=\frac{1+\sqrt{5}}{2}$ and an asymptotic constant $c = m_\alpha^{r_0-1}$, where $m_\alpha=|f''(\alpha)/(2f'(\alpha))|$. It naturalizes the Fibonacci-based analysis to avoid vague asymptotics and supplies a detailed comparison to Newton's method, including a time-based criterion for when the secant method can be faster. For multiple roots, the paper proves Q-linear convergence and identifies the limiting constant $c_{m,0}$ as the positive root of $p_m(k)=k^m+k^{m-1}-1=0$ in (0,1), while providing explicit initial-value conditions that guarantee convergence (and illustrating possible divergences). Overall, the results fill gaps in textbook treatments by offering rigorous, condition-checked convergence proofs and practical guidelines for initial values, with implications for selecting methods in root-finding tasks.

Abstract

The secant method, as an important approach for solving nonlinear equations, is introduced in nearly all numerical analysis textbooks. However, most textbooks only briefly address the Q-order of convergence of this method, with few providing rigorous mathematical proofs. This paper establishes a rigorous proof for the Q-order of convergence of the secant method and theoretically compares its computational efficiency with that of Newton's method.

Theorems & Definitions (51)

• Example 1.1
• Example 1.2
• Definition 2.1: Q-order of convergence
• Definition 2.2
• Theorem 2.3: Binet's formula
• Corollary 2.4
• Proof 1
• Theorem 2.7: Taylor's theorem
• Theorem 2.8: Lagrange's mean value theorem
• Corollary 2.9