Approximate polynomial GCD

In this correspondence, a new viewpoint is proposed for estimating an image from its distorted versions in presence of noise without the a priori knowledge of the distortion functions. In z-domain, the desired image can be regarded as the greatest common polynomial divisor among the distorted versions. With the assumption that the distortion filters are finite impulse response (FIR) and relatively coprime, in the absence of noise, this becomes a problem of taking the greatest common divisor (GCD) of two or more two-dimensional (2-D) polynomials. Exact GCD is not desirable because even extremely small variations due to quantization error or additive noise can destroy the integrity of the polynomial system and lead to a trivial solution. Our approach to this blind deconvolution approximation problem introduces a new robust interpolative 2-D GCD method based on a one-dimensional (1-D) Sylvester-type GCD algorithm. Experimental results with both synthetically blurred images and real motion-blurred pictures show that it is computationally efficient and moderately noise robust.

We present new methods for computing the greatest common right divisor of polynomial matrices. These methods involve the recently studied generalized Sylvester and generalized Bezoutian resultant matrices, which require no polynomial operations. They can provide a row proper greatest common right divisor, test for coprimeness and calculate dual dynamical indices. The generalized resultant matrices are developments of the scalar Sylvester and Bezoutian resultants and many of the familiar properties of these latter matrices are demonstrated to have analogs with the properties of the generalized resultant matrices for matrix polynomials.

Approximate square-free part and decomposition

The study of the Least Common Multiple (LCM) and the Greatest Common Divisor (GCD) explores the fundamental field of number theory. In addition to being essential to theoretical number theory, these mathematical ideas have extensive applications in computer science, algebra, and real-world settings like network routing and scheduling. Using techniques like the Euclidean algorithm for GCD calculation and prime factorization for LCM, the study starts by examining the fundamental concepts and notations of GCD and LCM. Then the author goes further into the detail about how these ideas are used to polynomial equations, highlighting how they may be used to solve challenging issues and simplify algebraic expressions. The studys practical applications, which demonstrate how GCD and LCM optimize scheduling tasks and boost network routing efficiency, are presented in the conclusion. The results of this study highlight the broad impact and adaptability of LCM and GCD in theoretical and applied mathematics, providing valuable insights for further research in these domains.

A matrix Euclidean algorithm induced by state space realization

We give a probabilistic algorithm for computing the greatest common divisor (GCD) of two polynomials over an algebraic number field. We can compute the GCD using O(llog5(l)) expected binary operations where l is the size of the GCD given by standard estimations. Since we require time Ω(l) just to write down the GCD, the algorithm is close to optimal.

SummaryThe implementation of cryptography algorithms is constantly observing threat from multiple techniques in breaking secure system. The current trend in breaking the secure system of cryptography is by power analysis technique. In order to break the secure key, the energy consumed in the digital circuit during computation is measured. This technique is commonly known as differential power analysis, explored by the hackers to break the secure systems. To circumvent these type of attacks, it is necessary to explore different designs, which dissipate less energy. Ideally, reversible circuits dissipate zero energy. We present a new reversible architecture for greatest common divisor (GCD) computation using modified Binary GCD algorithm. We present the generalized design methodology of reversible GCD computation unit. We compare the proposed GCD computation design with the existing design. The proposed reversible GCD architecture takes less number of iterations compared with the existing GCD architecture in the literature. The proposed design outperforms the existing GCD design in terms of Quantum Cost, Gate Count, and Ancilla Inputs. Copyright © 2016 John Wiley & Sons, Ltd.

We propose an algorithm to compute a greatest common divisor (GCD) of univariate polynomials with large integer coefficients on Graphics Processing Units (GPUs). At the highest level, our algorithm relies on modular techniques to decompose the problem into subproblems that can be solved separately. Next, we employ resultant-based or matrix algebra methods to compute a GCD of each modular image in parallel. Our approach exhibits block structure to distribute the computation of a single modular GCD over several thread blocks, and thus to remove any hardware limitations on the maximal size of polynomials that can be handled. To "combine" all modular results, we have adopted Mixed-Radix Conversion (MRC) algorithm running on the GPU. Our approach shows a significant speed-up over host-based GCD algorithm from Maple 13.

In this paper, the dielectric, piezoelectric, and elastic properties of the Ca3TaGa3Si2O14 (CTGS) single crystals were fully investigated over the temperature range of 21~800°C. Both the real and the complex coefficients were completely characterized. The real coefficients were determined through the measurements of the capacitance, impedance, or admittance of three kinds of plate-samples (X-cut, Y-cut, and Z-cut) and four types of bar-samples (XY-cut, (XYt)45°, (XYt)-30°, and (XYt)-85°), according to the IEEE dynamic methods. Of utmost importance are the complex coefficients, which were obtained by taking into account the electrical and mechanical dissipations of the specimens, and therefore they could describe the materials' performance more accurately. In the light of the characterization method developed by Du et al., three kinds of plate-samples (X-cut, Y-cut, and Z-cut) and two types of bar-samples (XY-cut and (XYt)-85°) were designed for the determination of the materials complex coefficients. Moreover, the temperature behaviors of all the independent dielectric, piezoelectric, and elastic coefficients (2, 2, and 6, respectively) were summarized: the real parts of the complex dielectric coefficients (equal to the real dielectric coefficients) rise gradually with the increase of the temperature, while the imaginary parts increase dramatically after 400 °C; both the real and complex piezoelectric coefficients stay relatively stable over the measurement temperature range; all the absolute values of the elastic compliance coefficients, to varying degrees, increase with the temperature, while the elastic stiffness coefficients decrease, and some of them exhibit excellent linearity.

Computing the greatest common divisor of a set of polynomials is a problem which plays an important role in different fields, such as linear system, control, and network theory. In practice, the polynomials are obtained through measurements and computations, so that their coefficients are inexact. This poses the problem of computing an approximate common factor. We propose an improvement and a generalization of the method recently proposed in Guglielmi et al. (SIAM J. Numer. Anal. 55, 1456–1482, 2017), which restates the problem as a (structured) distance to singularity of the Sylvester matrix. We generalize the algorithm in order to work with more than 2 polynomials and to compute an Approximate GCD (Greatest Common Divisor) of degree k ≥ 1; moreover, we show that the algorithm becomes faster by replacing the eigenvalues by the singular values.

This paper describes the implementation of recursive algorithms in unique factorization domains, namely multivariate polynomial greatest common divisors (gcd) and factorization into irreducible parts in the Java computer algebra library (JAS). The implementation of gcds, resultants and factorization is part of the essential building blocks for any computation in algebraic geometry, in particular in automated deduction in geometry. There are various implementations of these algorithms in procedural programming languages. Our aim is an implementation in a modern object oriented programming language with generic data types, as it is provided by Java programming language. We exemplify that the type design and implementation of JAS is suitable for the implementation of several greatest common divisor algorithms and factorization of multivariate polynomials. Due to the design we can employ this package in very general settings not commonly seen in other computer algebra systems. As for example, in the coefficient arithmetic for advanced Grobner basis computations like in polynomial rings over rational function fields or (finite, commutative) regular rings. The new package provides factory methods for the selection of one of the several implementations for non experts. Further we introduce a parallel proxy for gcd implementations which runs different implementations concurrently.

The objective of this is to show the application of Euclid's Lemma to calculate the Greatest Common Divisor in Elementary Education and evaluate the receptivity that the students had the method. We studied and wrote about the division in integers, divisibility and properties involved, the theorem euclidian division, the greatest common divisor, Euclid's Lemma, the euclidean algorithm, the Fundamental Theorem of Arithmetic and also on the Diophantine Equations they are an interesting application of the Greatest Common Divisor. We use a short course on Greatest Common Divisor in two elementary school classes and present some results obtained.

This paper presents a preliminary report on a new algorithm for computing the Greatest Common Divisor (GCD) of two multivariate polynomials over the integers. The algorithm is strongly influenced by the method used for factoring multivariate polynomials over the integers. It uses an extension of the Hensel lemma approach originally suggested by Zassenhaus for factoring univariate polynomials over the integers. We point out that the cost of the Modular GCD algorithm applied to sparse multivariate polynomials grows at least exponentially in the number of variables appearing in the GCD. This growth is largely independent of the number of terms in the GCD. The new algorithm, called the EZ (Extended Zassenhaus) GCD Algorithm, appears to have a computing bound which in most cases is a polynomial function of the number of terms in the original polynomials and the sum of the degrees of the variables in them. Especially difficult cases for the EZ GCD Algorithm are described. Applications of the algorithm to the computation of contents and square-free decompositions of polynomials are indicated.

THE purpose of this paper is to describe a short method for solving arbitrary systems of linear algebraic equations, and evaluating determinants, the quantities involved being either real or complex. For proofs and for treatment of complex systems whose symmetrical coefficients are conjugate, see appendix. The cases considered are: 1. Arbitrary systems with real coefficients, which occur in obtaining stresses in structures, in solving systems of linear differential equations with constant coefficients (transient problems), etc. 2. Symmetrical systems with real coefficients, which occur with d-c networks, undamped vibration, deflections in structures, least square processes, Ritz's method, etc. 3. Symmetrical systems with complex coefficients, which occur with a-c networks, and forced vibration with dissipation. 4. Arbitrary systems with complex coefficients, which occur in certain vibration problems involving gyroscopic action. 5. Systems involving two sets of variables, which occur when the currents in a network are to be found for a variety of impressed voltages, also in the approximate solution of integral equations arising in electric field problems.

A new numerical method for the computation of the greatest common divisor (GCD) of an m-set of polynomials of R [s], Pm,d of maximal degree d, is presented. This method is based on a recently developed theoretical algorithm (Karcanias 1987) that uses elementary transformations and shifting operations; the present algorithm takes into account the non-generic nature of GCD and thus uses steps, which minimize the introduction of additional errors and defines the GCD in an approximate sense. For a given set Pm,d with a basis matrix Pm, the method defines first, the most orthogonal uncorrupted base Pr from the rows of Pm, where r = rank(Pm) ≤ m. By applying successively gaussian transformations and shifting, on the basis matrix Pr ε Rr×(d+1) we produce each time a new basis matrix Pz with z = rank(Pz) < r. The method terminates when the rank of Pz is approximately equal to 1; the coefficient vector of the GCD is then defined as a row of the unit rank matrix Pz. The method defines the exact degree of the GCD, successfully evaluates an approximate solution and works satisfactorily with large numbers of polynomials of any fixed degree.