perform root finding

This paper addresses the fundamental symbolic computation problem of real root isolation for univariate integer polynomials. It introduces smoothed analysis—previously absent in symbolic computation—to bridge the gap between practical efficiency and worst-case complexity bounds. Methodologically, it integrates Descartes’ rule of signs, Sturm sequences, the hybrid ANewDsc algorithm, and sparse polynomial techniques to establish a smoothed complexity framework. Key contributions include: (i) the first proof that Descartes’ method achieves quasilinear bit complexity in both expectation and under smoothing; (ii) the first non-worst-case theoretical guarantees for Sturm’s method, ANewDsc, and sparse symbolic solvers; and (iii) a systematic explanation of why classical algorithms consistently outperform their worst-case bounds in practice. This work fills a long-standing gap in the average-case and smoothed complexity theory of symbolic 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 paper addresses the forward error analysis of sum-product algorithms under stochastic rounding (SR). We propose a probabilistic error bounding method grounded in martingale theory. Our key contributions are threefold: (1) We introduce the first automated martingale construction framework tailored to multilinear computational structures—encompassing addition, subtraction, multiplication, and intermediate result reuse; (2) We extend SR error analysis to algorithms with structural reuse, notably Karatsuba polynomial multiplication—previously unaddressed in SR literature; (3) Leveraging the Azuma–Hoeffding inequality, we derive a tight probabilistic error bound of $O(sqrt{n},u)$, markedly improving upon the classical worst-case bound $O(n,u)$. Our framework uniformly recovers known error guarantees for pairwise summation and Horner’s method, and—crucially—provides the first rigorous SR error guarantee for Karatsuba multiplication.

This work addresses longstanding circuit complexity bottlenecks for fundamental polynomial algebra problems—namely, computing greatest common divisors (GCDs), discriminants, resultants, Bézout coefficients, square-free factorizations, and inverses of Sylvester/Bézout matrices—in the AC⁰_F model. Prior to this work, only superpolynomial-size arithmetic circuits were known for these tasks. We present the first polynomial-size, constant-depth AC⁰_F arithmetic circuits for all these problems. Our method introduces a novel algorithmic paradigm that avoids explicit root access; instead, it implicitly handles root multiplicities and symmetric functions via structured matrix algebraic transformations and constant-depth evaluation of symmetric polynomials. Consequently, problems long believed “non-AC⁰-computable,” such as GCD computation, are now shown to reside in the class of polynomial-size, constant-depth circuits. The approach naturally extends to multivariate polynomials and multiple inputs, substantially enhancing parallelism and hardware feasibility in algebraic computation.

This work proposes an efficient method for computing Rational Univariate Representations (RUR) of zero-dimensional polynomial systems by leveraging dense linear algebra and Gaussian elimination. Building upon classical FGLM-type algorithms, the approach replaces conventional steps with Gaussian elimination, thereby significantly enhancing computational efficiency for large-scale systems while rigorously preserving theoretical correctness. Experimental results demonstrate that the proposed method correctly parameterizes zero-dimensional ideals with thousands of solutions in just a few seconds. The implementation is publicly available as the open-source Julia package RationalUnivariateRepresentation.jl.

Floating-point round-off errors are ubiquitous in numerically intensive programs arising in fields such as scientific computing and optimization. As floating-point errors potentially lead to unexpected and catastrophic program failures, one must derive guaranteed round-off thresholds to ensure the correctness of these programs. However, deterministic round-off thresholds tend to be too conservative to be usable in practice, since they often involve large round-off errors that occur with small probability. Probabilistic thresholds relax deterministic ones by specifying that the probability of the round-off error exceeding a threshold is below a given confidence. In this work, we propose a novel approach to probabilistic round-off analysis, by applying concentration inequalities over the Taylor expansion from FPTaylor (TOPLAS 2018). A major obstacle in applying concentration inequalities is that the Taylor expansion involves absolute value operators that make the calculation of the expected values of the first order partial differential terms difficult. Our first step to overcome this obstacle is a sound over-approximation that removes the absolute value operators in polynomial expressions. Then, we show how to handle fractional expressions by a transformation into polynomial case. Finally, we show how to improve our approach with range partitioning. Our approach is scalable since the key computational part is the calculation of expected values of polynomial expressions with independent variables, for which the linear and independence properties of expectation boost the computation. Experimental results show that our approach is orders of magnitude more time efficient, while producing thresholds with comparable precision against the state of the art.

This study addresses the problem of accurately computing the output distributions of small-scale programs—such as those processing GPS or inertial sensor data—that involve random inputs. To this end, it introduces cylindrical algebraic decomposition (CAD) into probabilistic program analysis for the first time, combining symbolic and numerical integration to effectively handle conditional branches and nonlinear operations. The approach is grounded in a rigorous semantic model of probabilistic programs and has been validated on both floating-point arithmetic benchmarks and representative programs from open-source sensor libraries, demonstrating its feasibility and effectiveness in deriving exact output distributions.

This work addresses the problem of computing sample points in each connected component of a semi-algebraic set defined by inequalities involving real-coefficient polynomials. Under generic smoothness assumptions on the input polynomials, the authors propose a probabilistic algorithm that characterizes connected components via critical points and reduces the problem to solving zero-dimensional polynomial systems. For the first time, the algorithm leverages the actual degree structure of both the input polynomials and their partial derivatives to deliver a refined bit complexity analysis based on Bézout bounds: parameterization of sample points achieves cubic complexity, while rational approximation incurs quartic complexity. Experimental results demonstrate that the method efficiently handles previously intractable instances, including random dense systems and those arising from practical applications.

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.