The manuscript presents a computational framework, termed the Method of the Periodic Cauchy Integral (MICP), for the evaluation of arbitrary real (n) -th roots (A^1/n) with (A>0) and integer (n 2), grounded in the analytic structure of holomorphic functions and their integral representations. The central conceptual departure from classical root-finding methodologies lies in the replacement of local, iterative fixed-point schemes by a global integral formulation derived from complex analysis. Specifically, the method exploits the Cauchy integral formula, which asserts that the value of a holomorphic function at an interior point is exactly recoverable as a contour integral over any closed curve enclosing that point. By selecting a circular contour centered at the target value (A) and applying this identity to the principal branch of the function (f (z) =z^1/n), the root (A^1/n) is expressed as the mean value of the function over a complex neighborhood of (A). This continuous representation is subsequently discretized via the uniform trapezoidal rule on a periodic domain, yielding a finite summation over equispaced angular samples along the contour. The resulting numerical scheme is structurally simple: it evaluates the function (z^1/n) at a set of complex points distributed uniformly on a circle of radius (A), with ( (0, 1) ), and computes the arithmetic mean of their real parts. Despite its apparent simplicity, this construction embeds a nontrivial spectral mechanism. The periodic analyticity of the integrand ensures that the trapezoidal rule converges at an exponential rate, a phenomenon well established in the theory of spectral quadrature. Consequently, the approximation error decays as (O (N) ), where (N) is the number of quadrature nodes, thus achieving exponential convergence in the number of function evaluations. The theoretical analysis provided in the manuscript elucidates the origin of this convergence through a detailed expansion of the integrand in a Laurent series about the center (A). Upon substitution into the discrete summation, the orthogonality properties of complex exponentials—equivalently, the discrete Fourier basis—eliminate all Fourier modes except those whose frequencies are integer multiples of the sampling size (N). This filtering mechanism isolates the constant (zeroth) Fourier component while suppressing all lower-order perturbations, thereby explaining both the accuracy and the exponential decay of the error. The residual error is shown to be governed by higher-order coefficients in the expansion, which are exponentially damped by the factor (^mN). An explicit bound is derived in terms of generalized binomial coefficients, providing a rigorous and quantitative characterization of the convergence rate. An additional and conceptually significant aspect of the method is its exact algebraic equivalence to the extraction of the zeroth coefficient of the discrete Fourier transform (DFT) of the sampled function values. This observation situates the MICP within the broader framework of spectral methods and enables the use of fast Fourier transform (FFT) algorithms for efficient computation. Although the direct implementation has linear complexity in the number of nodes, the FFT-based formulation reduces this to (O (N N) ), while preserving the independence of the individual function evaluations. From a computational standpoint, the method exhibits a distinctive architectural property: all function evaluations are mutually independent, rendering the algorithm fully parallelizable without any inter-step dependency. This contrasts sharply with classical iterative schemes such as Newton’s or Halley’s methods, which, despite their rapid local convergence, are inherently sequential due to their recursive structure. In the MICP framework, the entire computation can, in principle, be executed in a single parallel round, with the total wall-clock time determined by the cost of a single function evaluation rather than by the number of iterations. This characteristic makes the method particularly well suited for modern parallel architectures, including GPUs and distributed computing systems, where massive concurrency can be exploited. The manuscript further develops practical aspects of the method, including parameter selection strategies for () and (N), supported by explicit error bounds and tabulated values for achieving prescribed precision levels. A series of numerical experiments corroborates the theoretical predictions, demonstrating consistent exponential convergence across a wide range of inputs, including large radicands, fractional exponents, and non-integer powers. These examples confirm that the asymptotic error behavior is observed in practice and that machine precision can be attained with a moderate number of quadrature points. Beyond scalar root extraction, the framework admits natural generalizations. Since the derivation relies solely on the holomorphicity of the function, the same methodology extends to arbitrary analytic functions, yielding exponentially convergent quadrature-based evaluations for a broad class of elementary and special functions. Moreover, by invoking the matrix-valued Cauchy integral formula, the approach generalizes to the computation of matrix functions, including matrix roots, where each term in the quadrature corresponds to the solution of an independent linear system. This extension preserves the parallel structure of the scalar case and aligns with established contour-integral methods in numerical linear algebra. In a broader conceptual sense, the MICP embodies a shift in computational paradigm. Rather than approximating a solution through successive local refinements, it reconstructs the desired value via a global averaging process over a complex domain. This perspective leverages the inherent rigidity of holomorphic functions, whereby local values are determined entirely by boundary data, and translates this property into an algorithmic advantage. The method thus integrates ideas from complex analysis, harmonic analysis, and numerical computation into a unified framework that is both theoretically rigorous and computationally distinctive. In summary, the manuscript offers a coherent synthesis of classical analytical tools and modern computational considerations, presenting a method that is mathematically sound, algorithmically transparent, and architecturally aligned with parallel computation. While it does not introduce fundamentally new analytical results, its contribution lies in the explicit formulation, rigorous error analysis, and computational interpretation of a contour-integral-based approach to root extraction and related problems.
Caio A. Rocha (Sat,) studied this question.