numerical fixed-point computation

This work addresses the problem of deriving provably tight floating-point rounding error bounds for numerical programs featuring conditional branches, no loops, and mixed-precision arithmetic. Methodologically, it unifies the modeling of conditional control flow and precision heterogeneity via two novel quantitative metrics—“instability jumps” and “window width”—and integrates interval arithmetic, abstract interpretation, and precision-aware semantic modeling, augmented with abstraction-guided global optimization. Its key contribution is the first formal framework enabling joint, compositional analysis of conditional branching and mixed precision, achieving both high bound tightness and practical analysis efficiency. Experimental evaluation on standard benchmarks demonstrates significantly tighter error bounds compared to prior approaches. Furthermore, the framework successfully guides precision configuration—e.g., step size and search direction—in the conjugate gradient method, empirically validating its utility in supporting design-time trade-offs among accuracy, error bounds, and computational efficiency.

This work addresses the challenge of computing fixed points of contractive mappings without requiring prior knowledge of the contraction factor or manual hyperparameter tuning. We propose a fully adaptive Halpern-type algorithm that operates without line searches, bisection procedures, or any user-specified parameters, automatically exploiting the inherent contractiveness of the mapping to achieve explicit linear convergence even in the absence of a priori estimates of the contraction constant. Theoretical analysis establishes linear convergence rates both in terms of fixed-point residuals and distance to the solution, with an iteration complexity of 𝒪(ε⁻¹ln(ε⁻¹)) for solving cocoercive equations. By integrating Tikhonov regularization and Nesterov acceleration, we further extend the algorithm’s applicability. Numerical experiments confirm its superiority over existing adaptive methods, offering strong theoretical guarantees alongside low computational overhead.

This paper investigates the existence of fixed points for nonexpansive mappings on $L^1(mu)$ under fixed-point arithmetic quantization errors, with application to stability analysis of consensus in human-AI collaborative editing systems. To address mapping distortion induced by quantization perturbations, we introduce the notion of “measure compactness” and integrate it with uniform integrability, measure convexity, normal structure, and Kirk’s fixed-point theorem. We prove that nonexpansive mappings on measure-compact sets admit fixed points, and retain approximate fixed points under bounded quantization error. This work constitutes the first systematic incorporation of robust fixed-point theory from functional analysis into human-AI interaction modeling. It rigorously establishes both the existence and quantization robustness of stable consensus states in collaborative editing, thereby providing a theoretical foundation for trustworthy human-AI collaboration.

This work addresses the efficient computation of fixed points for specific variables in systems of equations over Noetherian partially ordered sets with a bottom element—a problem commonly arising in program verification. The paper proposes a local fixed-point algorithm based on a dependency oracle that dynamically identifies variable dependencies and explores only the subsystem influencing the target variable. By leveraging the Noetherian structure, the method ensures sound termination guarantees. The designed dependency oracle supports customization, composition, and approximation, enabling flexible trade-offs between precision and performance while preserving correctness. Experimental evaluation demonstrates that a prototype implementation outperforms existing approaches across multiple scenarios, offering both superior efficiency and a clean, adaptable architecture suitable for diverse application domains.

This paper studies the problem of computing an ε-approximate fixed point of a contraction mapping (f) on the (k)-dimensional unit cube ([0,1]^k) under the (ell_infty) norm. Existing algorithms suffer from query complexity that is either exponential in (k) or afflicted by the curse of dimensionality. To overcome this, we propose a recursive interval pruning algorithm integrating grid refinement, adaptive coordinate polling, and contraction-guided domain reduction. Our method achieves the first query complexity of (O(k^2 log(1/varepsilon))) for arbitrary (k)-dimensional contractions, enabling efficient ε-approximation of fixed points. Theoretical analysis shows that this bound is nearly optimal—tight up to logarithmic factors relative to the information-theoretic lower bound—and significantly improves upon the prior state of the art. This result constitutes a fundamental advance in computational fixed-point theory, resolving a long-standing bottleneck in high-dimensional contraction mapping analysis.

Traditional methods for computing elementary transcendental functions rely on differential equations or power series, often suffering from implementation complexity and high memory overhead. This work proposes the first unified fixed-point framework grounded in the Banach contraction mapping principle, modeling the exponential, trigonometric, and natural logarithm functions as the unique fixed points of doubling identities. By constructing residual-based strictly contractive iteration operators, the approach guarantees numerical stability and explicit convergence rates. The resulting floating-point kernel operates without lookup tables or memory accesses, achieving performance in throughput-constrained scenarios that matches or exceeds that of mainstream math libraries; notably, sine and cosine computations are consistently faster across all tested iteration depths.

This work addresses the absence of readily available Gaussian quadrature rules for nonclassical weight functions by proposing a general framework that constructs such rules for arbitrary weights via the method of moments and the Stieltjes procedure. Innovatively integrating type-generic programming with adaptive high-precision arithmetic, the approach effectively controls round-off errors and, for the first time, systematically introduces tailored Gaussian quadrature methods to the statistics community. Implemented in Julia as the CustomGaussQuadrature package—accessible from R through JuliaConnectoR—the resulting quadrature rules achieve exact integration of polynomials up to degree \(2n-1\) while substantially reducing the number of function evaluations, thereby offering both high accuracy and computational efficiency.

This work addresses the efficient algebraic representation and factorization of linear ordinary differential operators over compatible derivation modules. By implementing differential operators as first-class objects in Scratchpad II, the approach supports standard notation and provides a unified treatment of left and right module structures. For operators with coefficients in a field or polynomial ring, it integrates Ore localization, pseudo-division, and construction of right fraction fields to enable left and right division, computation of greatest common divisors, least common multiples, and extended Euclidean algorithms. Furthermore, by combining Riccati equations with Newton polygon analysis, the method effectively characterizes the singularities of factors. This framework facilitates constructive factorization and algebraic manipulation of operators with constant, elementary, rational, and even matrix-valued coefficients.

This work proposes a systematic framework to enhance the computational efficiency and numerical stability of evaluating high-degree matrix polynomials. Specifically, for polynomial degrees eight and higher, the method generates and validates stable coefficient sets that reduce the number of required matrix multiplications by one compared to the classical Paterson–Stockmeyer scheme. To address instability issues in the original formulation, the authors introduce structural variants and design a reliability metric to assess the expected numerical accuracy of candidate coefficient sets. Nonlinear polynomial systems are solved using variable-precision arithmetic (VPA), and an in-house tool, MatrixPolEval1, enables efficient screening and validation. Applied to matrix exponentials and geometric series, the approach achieves a saving of one matrix multiplication while maintaining comparable numerical accuracy.