newton-raphson method

This work addresses large-scale nonlinear equation (root-finding) problems. We propose two stochastic block-coordinate optimistic gradient algorithms—non-accelerated and accelerated versions—guaranteeing convergence under weak Minty solution and co-coercivity assumptions, respectively. Our contributions are threefold: (i) the first integration of the optimistic gradient mechanism with stochastic block-coordinate updates; (ii) the first accelerated block-coordinate root-finding algorithm with almost-sure convergence guarantees; and (iii) an extension to finite-sum inclusion problems, yielding a novel federated learning solver. Theoretically, we establish optimal iteration complexities of $O(1/k)$ for the non-accelerated variant and $O(1/k^2)$ for the accelerated one, along with almost-sure convergence and an almost-sure convergence rate of $O(1/sqrt{k})$. Extensive experiments on synthetic and real-world datasets demonstrate that our methods significantly outperform existing state-of-the-art algorithms.

To stably solve the nonlinear equation $g( heta)=0$ for its unique root $ heta^*$, this paper proposes the Up-crossing/Solving (US) algorithm—a novel open-method that requires no initial guess and avoids root isolation. Its core innovation lies in constructing a U-function based on a “direction-switching inequality,” incorporating first- through third-order derivative information; convergence is achieved iteratively via alternating up-crossing steps (U-function construction) and solving steps (analytical solution of the U-equation). Theoretically, the algorithm is proven to be strongly stable and globally convergent. Empirical evaluations demonstrate its superiority over conventional methods in quantile computation, exact p-value calculation for skewed distributions, and maximum likelihood estimation (MLE). Notably, it achieves both high accuracy and efficiency in parameter estimation for continuous and discrete distributions, as well as in robust multi-root scenarios.

For nonconvex optimization problems with nonlinear equality constraints, this paper proposes an inexact augmented Lagrangian method employing a norm penalty with exponent strictly between 1 and 2. The method constructs Hölder-smooth subproblems under convex feasibility and weak regularity assumptions, leveraging the first use of a non-integer-power Euclidean norm as the augmentation term. We establish, for the first time, accelerated first-order algorithm complexity bounds for such subproblems. Theoretically, we reveal an intrinsic trade-off: constraint violation converges faster as the exponent decreases, while dual residual decay remains controllably degraded. Numerical experiments demonstrate that the proposed method achieves superior constraint satisfaction accuracy and iteration efficiency compared to the standard squared-augmented Lagrangian method.

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.

To address the poor scalability of conventional real sum-of-squares (RSOS) methods—which rely on large-scale semidefinite programming (SDP) for computing lower bounds on the minimum of real homogeneous polynomials over the unit sphere—this paper proposes a purely spectral (non-SDP) convergent hierarchy. The key innovation is the first rigorous reduction of real spherical optimization to Hermitian optimization, enabling the construction of a sequence of lower bounds via minimal eigenvalue computations alone, within the Hermitian sum-of-squares (HSOS) framework. This approach naturally extends to estimating the spectral norm of real tensors, thereby opening a new pathway for spectral methods in general constrained real optimization. Numerical experiments and asymptotic analysis demonstrate substantial improvements over RSOS and other baseline methods; moreover, the framework yields a computable, convergent hierarchy for the spectral norm.

This study addresses the problem of efficiently computing \( \lfloor y^{1/m} \rfloor \) for natural numbers \( y > 2 \) and integers \( m > 1 \), which is central to determining whether \( y \) is a perfect \( m \)-th power. While conventional approaches typically rely on binary search—often limited in efficiency—this work innovatively adapts the Newton–Raphson method to integer arithmetic for floor root computation. By integrating techniques from numerical analysis with exact integer operations, the authors devise two stable and highly efficient algorithms. Experimental results demonstrate that the proposed methods not only guarantee correctness but also significantly outperform traditional binary search, offering practical utility in number-theoretic applications involving perfect power detection.

This study addresses the problem of computing the integer $e$-th root $\lfloor N^{1/e} \rfloor$ of a non-negative integer $N$ using only integer arithmetic, without floating-point operations or division. The authors propose a unified digit-by-digit construction framework grounded in the binomial theorem and invariant-based reasoning, which guarantees algorithmic termination, correctness of intermediate results, and enables perfect power detection. A key contribution is a constructive correctness theory proving that each digit in the square root case ($e=2$) is determined precisely and irrevocably during computation. The work further establishes that stable digit-by-digit extraction is feasible only for $e=2$; for higher-degree roots, nonlinear coupling among digits precludes such stability. The method is well-suited for symbolic computation, verified arithmetic, educational contexts, and hardware implementation.

This work addresses the inefficiency of solving structured sparse polynomial systems over finite fields by proposing an efficient resultant-based algorithm. Leveraging the inherent sparsity and structure of the system, the method iteratively computes resultants to eliminate variables and ultimately derive a univariate polynomial, thereby circumventing the high computational complexity of brute-force search and conventional Gröbner basis approaches. The paper presents the first systematic application of resultant techniques to this class of problems and introduces ResultantSolver, a parallelizable algorithmic framework tailored for such systems. Experimental evaluation on benchmark instances from the GMV 2025 competition demonstrates that the proposed method significantly outperforms existing solvers in both speed and scalability, confirming its effectiveness and practical utility.

This paper proposes a unified optimization framework for three integer factorization problems: (1) factoring semiprimes $N = pq$; (2) factoring power sums/differences $N = a^n pm b^n$; and (3) extracting $r$-th power factors (i.e., finding integers $p$ such that $p^r mid N$). Methodologically, it innovatively integrates Coppersmith’s method with rank-3 lattice construction, and—uniquely—exploits the *second shortest vector* in the LLL-reduced basis to circumvent trivial collisions inherent in Baby-step Giant-step algorithms, thereby breaking the conventional reliance on the shortest vector. The framework further combines enhanced LLL reduction, small-prime sieving, and polynomial root-finding techniques. Theoretically, it achieves a deterministic time complexity of $O(N^{1/5}log^{13/5}N / (loglog N)^{3/5})$ for balanced semiprimes; significant speedups are also attained for power-type factorization and $r$-th power factor extraction. This work delivers the first deterministic, efficient solution to these classical number-theoretic problems.

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.