P-order: Unified Convergence Analysis for Nonlinear Iterative Methods

TL;DR

This work introduces P-order, a norm-independent convergence framework that uses a power function $\psi(k)$ with $\psi(k)\to\infty$ to classify iteration rates via $\limsup_{k\to\infty} \|x_k - x_\*\|^{1/\psi(k)} = C_\psi$. It defines Quasi-Uniform P-Order (QUP-order) and Uniform P-Order (UP-order) to provide precise rate characterizations under weaker continuity assumptions, connecting naturally to Taylor remainder forms. The framework yields new results, including fractional-power convergence near singularities, and explicit rate characterizations for Newton's method, gradient descent, and $K$-point methods under Hölder continuity $C^{K-1,\nu}$. It also clarifies the relationships among P-, QUP-, and R-orders, and demonstrates norm-independence and fine-grained rate distinctions such as linear, linearithmic, and anti-linearithmic convergence. Together, these contributions offer a sharper, unified toolkit for convergence analysis in modern computational applications where classical assumptions fail.

Abstract

Measuring how quickly iterative methods converge is essential in computational mathematics, but current approaches have significant limitations. Q-order analysis requires strict smoothness conditions, while R-order analysis lacks precision and creates ambiguity, especially when analyzing convergence rates close to linear. We introduce P-order, a new framework that overcomes these limitations by using a power function $ψ(k)$ combined with asymptotic notation ($Θ, o, ω$). Our approach offers two key advantages: it works independently of the chosen norm while providing the precision needed to classify diverse convergence behaviors, including previously hard-to-characterize rates like fractional-power and linearithmic convergence. P-order also systematically accommodates weaker continuity conditions by naturally connecting mathematical assumptions to appropriate Taylor approximation forms. To enhance practical analysis, we develop two important subclasses, QUP-order and UP-order, which work effectively under different smoothness conditions. We demonstrate P-order's practical value through three applications: (1) refining fixed-point iteration analysis with minimal smoothness requirements (mere differentiability suffices where classical analysis required stronger conditions), (2) identifying previously unreported convergence rates for Newton's method and gradient descent algorithms, and (3) providing a unified analysis of $K$-point methods under $C^{K-1,ν}$ (i.e., with Hölder continuous $(K-1)$th derivatives), yielding a new characteristic rate $q_{K}(ν)$. Our P-order framework provides researchers and practitioners with a sharper, more comprehensive toolbox for convergence analysis, particularly valuable when classical assumptions fail or when analyzing complex convergence behaviors in modern computational applications.

Figures (5)

• Figure 1: Comparison of P-order with different $\psi(x)$
• Figure 1: Examples of sublinear fractional-power rates of Newton's method.
• Figure 1: Numerical illustration of varying convergence rates of $K$-point iterative method for secant ($K=2$) and inverse Muller's ($K=3$) methods with different Hölder exponent $\nu$.
• Figure 2: Hierarchy of linear ($\psi=k$) and exponential ($\psi=q^k$, $q>1$) convergence rates. Not to scale (especially for the substantial UP/Q-linear overlap).
• Figure 3: Numerical illustration of QUP-fractional-power convergence ($r=0.25, 0.5, 0.75$) for gradient descent (\ref{['ex:gd_designed_fractional']}).

Theorems & Definitions (34)

• Definition 2.1: Q-Order
• Remark 2.1: Variants of Q-Order
• Definition 2.2: R-Order (Ortega and Rheinboldt ortega1970iterative)
• Remark 2.2: Differences Between R-order-1 and R-linear
• Definition 3.1: P-Order
• Lemma 3.1: Norm Independence
• Proof 1
• Lemma 3.2: P-Sub-$\psi$ and P-Super-$\psi$ Convergence
• Proof 2
• Definition 3.2: Quasi-Uniform P-Order (QUP-Order)