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A modular algorithm to compute the generalized Hermite normal form for Z[x]-lattices

JOURNAL OF SYMBOLIC COMPUTATION 2017年 81卷 97-118页

作者： Jing, Rui-Juan Yuan, Chun-Ming Chinese Acad Sci KLMM UCAS Acad Math & Syst Sci Beijing 100190 Peoples R China

In this paper, a modular algorithm is given to compute the generalized Hermite normal form of matrices over Z[x], or equivalently, the reduced Grobner basis of Z[x]-modules in Z[x](n). The main advantage of the algorithm is that the special structure of the Grobner basis of ideals in Z[x] is taken into consideration. The algorithm is deterministic and seems to be the most efficient available algorithm for inputs with relatively low degrees. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Generalized Hermite normal form Reduced Grober bases modular algorithm Hensel Lifting

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modular Resultant algorithm for Graphics Processors

Modular Resultant Algorithm for Graphics Processors

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10th International Conference on algorithms and Architectures for Parallel Processing

作者： Emeliyanenko, Pavel Max Planck Inst Informat Saarbrucken Germany

ISBN: (纸本)9783642131189

In this paper we report on the recent progress in computing bivariate polynomial resultants on Graphics Processing Units (GPU). Given two polynomials in Z[x, y], our algorithm first maps the polynomials to a prime field. Then, each modular image is processed individually. The GPU evaluates the polynomials at a number of points and computes univariate modular resultants in parallel. The remaining "combine" stage of the algorithm is executed sequentially on the host machine. Porting this stage to the graphics hardware is an object of ongoing research. Our algorithm is based on an efficient modular arithmetic from [1]. With the theory of displacement structure we have been able to parallelize the resultant algorithm up to a very fine scale suitable for realization on the GPU. Our benchmarks show a substantial speed-up over a host-based resultant algorithm [2] from CGAL (***).

关键词： polynomial resultants modular algorithm parallel computations graphics hardware GPU CUDA

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Inversion of residual gravity anomalies using neural network

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ARABIAN JOURNAL OF GEOSCIENCES 2013年第5期6卷 1509-1516页

作者： Al-Garni, Mansour A. King Abdulaziz Univ Fac Earth Sci Dept Geophys Jeddah 21441 Saudi Arabia

A new approach is presented in order to interpret residual gravity anomalies from simple geometrically shaped bodies such as horizontal cylinder, vertical cylinder, and sphere. This approach is mainly based on using modular neural network (MNN) inversion for estimating the shape factor, the depth, and the amplitude coefficient. The sigmoid function has been used as an activation function in the MNN inversion. The new approach has been tested first on synthetic data from different models using only one well-trained network. The results of this approach show that the parameter values estimated by the modular inversion are almost identical to the true parameters. Furthermore, the noise analysis has been examined where the results of the inversion produce satisfactory results up to 10% of white Gaussian noise. The reliability of this approach is demonstrated through two published real gravity field anomalies taken over a chromite deposit in Camaguey province, Cuba and over sulfide ore body, Nornada, Quebec, Canada. A comparable and acceptable agreement is obtained between the results derived by the MNN inversion method and those deduced by other interpretation methods. Furthermore, the depth obtained by the proposed technique is found to be very close to that obtained by drilling information.

关键词： Neural network inversion modular algorithm Gravity Simple-shaped bodies

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AN APPLICATION OF REGULAR CHAIN THEORY TO THE STUDY OF LIMIT CYCLES

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INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS 2013年第9期23卷 1350154-1350154页

作者： Chen, Changbo Corless, Robert M. Maza, Marc Moreno Yu, Pei Zhang, Yiming Univ Western Ontario Middlesex Coll Dept Comp Sci London ON N6A 5B7 Canada Univ Western Ontario Middlesex Coll Dept Appl Math London ON N6A 5B7 Canada

In this paper, the theory of regular chains and a triangular decomposition method relying on modular computations are presented in order to symbolically solve multivariate polynomial systems. Based on the focus values for dynamic systems obtained by using normal form theory, this method is applied to compute the limit cycles bifurcating from Hopf critical points. In particular, a quadratic planar polynomial system is used to demonstrate the solving process and to show how to obtain center conditions. The modular computations based on regular chains are applied to a cubic planar polynomial system to show the computation efficiency of this method, and to obtain all real solutions of nine limit cycles around a singular point. To the authors' best knowledge, this is the first article to simultaneously provide a complete, rigorous proof for the existence of nine limit cycles in a cubic system and all real solutions for these limit cycles.

关键词： Regular chain modular algorithm triangularization Hilbert's 16th problem limit cycle focus value Maple

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Parallel Computation of Bivariate Polynomial Resultants on Graphics Processing Units

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10th Nordic International Conference on Applied Parallel Computing - State of the Art in Scientific and Parallel Computing (PARA)

作者： Stussak, Christian Schenzel, Peter Univ Halle Wittenberg Inst Comp Sci D-06120 Halle Saale Germany

ISBN: (纸本)9783642281440

Polynomial resultants are of fundamental importance in symbolic computations, especially in the field of quantifier elimination. In this paper we show how to compute the resultant res(y) (f, g) of two bivariate polynomials f, g is an element of Z[x, y] on a CUDA-capable graphics processing unit (GPU). We achieve parallelization by mapping the bivariate integer resultant onto a sufficiently large number of univariate resultants over finite fields, which are then lifted back to the original domain. We point out, that the commonly proposed special treatment for so called unlucky homomorphisms is unnecessary and how this simplifies the parallel resultant algorithm. All steps of the algorithm are executed entirely on the GPU. Data transfer is only used for the input polynomials and the resultant. Experimental results show the considerable speedup of our implementation compared to host-based algorithms.

关键词： polynomial resultants modular algorithm parallelization GPU CUDA graphics hardware symbolic computation

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Interpretation of spontaneous potential anomalies from some simple geometrically shaped bodies using neural network inversion

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ACTA GEOPHYSICA 2010年第1期58卷 143-162页

作者： Al-Garni, Mansour A. King Abdulaziz Univ Fac Earth Sci Dept Geophys Jeddah 21441 Saudi Arabia

A new approach is proposed in order to interpret spontaneous potential (self-potential) anomalies related to simple geometric-shaped models such as sphere, horizontal cylinder, and vertical cylinder. This approach is mainly based on using neural network inversion of SP anomalies, particularly modular algorithm, for estimating the parameters of different simple geometrical bodies. However, Hilbert transforms are involved to determine the origin location in order to reduce the parameters which minimize the ambiguity in the inverted models. The inversion has been tested first on synthetic data from different models, using only one well-trained network. The results of inversion show that the parameter values derived by the inversion are identical to the true values of parameters. Noise analysis has been also examined, where the results of the inversion produce acceptable results up to 10% of white Gaussian noise. The validity of the neural network inversion is demonstrated through published real field SP taken from southern Bavarian Woods, Germany. A comparable and acceptable agreement is shown between the results of inversion derived by the neural network and those from the real field data.

关键词： neural network inversion modular algorithm Hilbert transform spontaneous potential anomalies semi-infinite vertical cylinder

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Output-sensitive modular algorithms for polynomial matrix normal forms

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JOURNAL OF SYMBOLIC COMPUTATION 2007年第7期42卷 733-750页

作者： Cheng, Howard Labahn, George Univ Lethbridge Dept Math & Comp Sci Lethbridge AB T1K 3M4 Canada Univ Waterloo Symbol Computat Grp David R Cheriton Sch Comp Sci Waterloo ON N2L 3G1 Canada

We give modular algorithms to compute row-reduced forms, weak Popov forms, and Popov forms of polynomial matrices, as well as the corresponding unimodular transformation matrices. Our algorithms improve on existing fraction-firee algorithms. In each case, we define lucky homomorphisms, determine the appropriate normalization, as well as bound the number of homomorphic images required. The algorithms have the advantage that they are output-sensitive;that is, the number of homomorphic images required depends on the size of the output. Furthermore, there is no need to verify the result by trial division or multiplication. Our algorithms can be used to compute normalized one-sided greatest common divisors and least common multiples of polynomial matrices, along with irreducible matrix-fraction descriptions of matrix rational functions. When our algorithm is used to compute polynomial greatest common divisors, we obtain a new output-sensitive modular algorithm. (c) 2007 Elsevier Ltd. All rights reserved.

关键词： matrices row-reduced form weak Popov form Popov form modular algorithm

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modular dynamic simulation for integrated particulate processes by means of tool integration

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CHEMICAL ENGINEERING SCIENCE 2005年第7期60卷 2069-2083页

作者： Kulikov, V Briesen, H Grosch, R Yang, AD von Wedel, L Marquardt, W Rhein Westfal TH Aachen Lehrstuhl Prozesstechn D-52064 Aachen Germany AixCAPE D-52072 Aachen Germany

In this contribution a sequential modular strategy for the dynamic simulation of particulate process flowsheets is presented and the efficiency of the approach is demonstrated by means of an example process for the crystallization of pentaerythritol. The flowsheet of the process consists of a number of different unit operations, e.g. evaporator, crystallizer, hydrocyclone, and mixer, which are described by mathematical models of largely varying complexity and structure. A key advantage of the presented sequential modular strategy is that specialized tools can be selected for the modelling and solution of each unit operation in the flowsheet. The tools are then coupled together by means of the tool integration framework CHEOPS in order to capture the overall structure of the flowsheet. In a case study, a startup of a crystallization flowsheet is carried out. As a result, detailed information about the dynamic plantwide process behavior is obtained. The practical relevance of the approach is demonstrated by means of a scenario where potential blocking of the filters following the crystallizer has been analyzed. (c) 2004 Elsevier Ltd. All rights reserved.

关键词： crystallization sequential modular approach dynamic simulation tool integration modular algorithm population balance

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A modular algorithm for linear, periodic train-track models

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ARCHIVE OF APPLIED MECHANICS 2001年第6-7期71卷 473-486页

作者： Kruse, H Popp, K Univ Hannover Inst Mech Fac Mech Engn D-30167 Hannover Germany

An algorithm is presented which can be used for the investigation of a large variety of train-track models. These models only have to fulfil the requirements of linearity and periodicity with respect to the track length direction. A steady-state solution is obtained for a vehicle moving on a tangent track with constant velocity. The algorithm itself can be split into three modules: one for the whole train-track system, one for the track, and one for a single rail support. These modules and their interfaces are described in detail. The article demonstrates the applicability of the algorithm by means of four examples. The first example shows the influence of the sleeper elasticity on the sleeper motion. The second one illustrates the effect of an advanced subsoil model on the wheel/rail contact force. Subsequently, as a further example, the compliance frequency-response functions of a ballasted track and a rigid track are compared. The last example deals with the sleeper passing excitation. Here, it is shown that even in the case of resonance, the wheel/rail contact-force fluctuations remain below ten percent of the static value.

关键词： track model vehicle modular algorithm integral transformation periodic system

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modular MATRIX COMPUTATIONS ON MULTILINEAR VLSI ARRAYS

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PARALLEL COMPUTING 1995年第8期21卷 1269-1284页

作者： BARADA, HR Electrical Engineering and Computer Science Department Packard Lab Lehigh University Bethlehem PA 18015 USA

This paper describes the synthesis of modularly extensible matrix algorithms on multi-linear VLSI arrays with application to rectangular matrix multiplication. Multi-linear arrays differ from two-dimensional (2D) arrays in that one of the dimensions in the arrays is independent of the size of the problem and depends only on the I/O bandwidth of the host computer. They feature bounded clock skew (unlike 2D arrays) and higher throughput and communication bandwidth than linear arrays. A significant feature of the algorithms discussed in this paper is that they exhibit an interdependence between the number of processing elements (PEs) per row in the multi-linear array, the storage inside each PE, and the size of the problem. This interrelationship permits changes in one of these measures to be accommodated by altering one or more of the other measures. One particular benefit of this feature is that it enables the design of modularly extensible algorithms which can accommodate changes in the problem size by only changing the number of PEs per row in the array. Therefore, for a given host I/O bandwidth, larger problem sizes can be handled by cascading smaller multi-linear arrays, all of which have the same number of rows and are comprised of PEs with a fixed amount of local storage. A second feature of our algorithms is that their performance is also a function of the number of rows of PEs in the array which is in turn a function of the I/O bandwidth of the host. A third feature of our design is that the algorithms are controlled by tag bits inserted with the data and no control memory is required inside the PEs.

关键词： LINEAR ALGEBRA MATRIX MULTIPLICATION VLSI TECHNOLOGY MULTILINEAR ARRAY modular algorithm

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