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symbolic-numerical computation of the stability regions for Jameson's schemes

MATHEMATICS AND COMPUTERS IN SIMULATION 1996年第4-6期42卷 607-615页

作者： Ganzha, VG Vorozhtsov, EV RUSSIAN ACAD SCI INST THEORET & APPL MECHNOVOSIBIRSK 630090RUSSIA

We describe a symbolic-numerical method for the Fourier stability analyses of difference initial-value problems approximating the initial-value problems for hyperbolic or parabolic PDEs. The Fourier method is reduced to the algebra of the resultants. We further use the REDUCE computer algebra system for the symbolic computation of the resultant and for the generation of a FORTRAN function to compute the value of the resultant. Basing on this FORTRAN function we further construct a binary function to characterize the stability and instability points. Using this function we generate a bilevel digital picture, and the stability region boundaries are then detected in this picture with the aid of the efficient algorithm proposed previously by Pavlidis (1982). The above symbolic-numerical method has enabled us to obtain for the first time the stability regions of the considered Jameson's schemes as applied to the two-dimensional advection-diffusion equation. Analytical formulas are proposed for the approximation of the boundaries of the obtained stability regions. It is shown that these formulas underestimate insignificantly the actual sizes of the stability regions. Therefore, these formulas can efficiently be used in practical computations by Jameson's schemes.

关键词： partial differential equations difference schemes stability symbolic-numerical computation

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Parallel computation of real solving bivariate polynomial systems by zero-matching method

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APPLIED MATHEMATICS AND computation 2013年第14期219卷 7533-7541页

作者： Qin, Xiaolin Feng, Yong Chen, Jingwei Zhang, Jingzhong Chinese Acad Sci Chengdu Inst Comp Applicat Lab Automated Reasoning & Programming Chengdu 610041 Peoples R China Sichuan Univ Dept Math Chengdu 610064 Peoples R China Chinese Acad Sci Grad Sch Beijing 100049 Peoples R China

We present a new algorithm for solving the real roots of a bivariate polynomial system Sigma = {f(x;y);g(x;y)} with a finite number of solutions by using a zero-matching method. The method is based on a lower bound for the bivariate polynomial system when the system is non-zero. Moreover, the multiplicities of the roots of Sigma = 0 can be obtained by the associated quotient ring technique and a given neighborhood. From this approach, the parallelization of the method arises naturally. By using a multidimensional matching method this principle can be generalized to the multivariate equation systems. (C) 2013 Elsevier Inc. All rights reserved.

关键词： Bivariate polynomial system Zero-matching method Real roots symbolic-numerical computation Parallel computation

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OBTAINING EXACT INTERPOLATION MULTIVARIATE POLYNOMIAL BY APPROXIMATION

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Journal of Systems Science & Complexity 2011年第4期24卷 803-815页

作者： Yong FENG Xiaolin QIN Jingzhong ZHANG Xun YUAN Laboratory of Computer Reasoning and Trustworthy Computation University of Electronic b'cience and 'l~ech-nology of China Chengdu 611731 China. Laboratory for Automated Reasoning and Programming Chengdu Institute of Computer Applications ChineseAcademy of Sciences Chengdu 610041 China Graduate University of Chinese Academy of Sciences Beijing100049 China. Laborutory of Computer Reasoning and 7Yustworthy Computation University of Electronic Science and Tech-nology of China Chengdu 611731 China. Laboratory for Automated Reasoning and Programming Chengdu Institute of Computer Applications ChineseAcademy of Sciences Chengdu 610041 China.

In some fields such as Mathematics Mechanization, automated reasoning and Trustworthy Computing, etc., exact results are needed. symbolic computations are used to obtain the exact results. symbolic computations are of high complexity. In order to improve the situation, exact interpolating methods are often proposed for the exact results and approximate interpolating methods for the ap- proximate ones. In this paper, the authors study how to obtain exact interpolation polynomial with rational coefficients by approximate interpolating methods.

关键词： Continued fraction multivariate interpolation numerical approximate computation symbolic-numerical computation Vandermonde determinant.

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Obtaining exact value by approximate computations

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Science China Mathematics 2007年第9期50卷 1361-1368页

作者： Jing-zhong ZHANG Yong FENG Laboratory for Automated Reasoning and Programming Chengdu Institute of Computer ApplicationsChinese Academy of SciencesChengdu 610041China

numerical approximate computations can solve large and complex problems *** have the advantage of high *** they only give approximate results,whereas we need exact results in some *** is a gap between approximate computations and exact results. In this paper,we build a bridge by which exact results can be obtained by numerical approximate computations.

关键词： numerical approximate computation symbolic-numerical computation continued fraction 33F10

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Finding Exact Minimal Polynomial by Approximations

Finding Exact Minimal Polynomial by Approximations

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International Workshop on symbolic-Numeric computation

作者： Qin, Xiaolin Feng, Yong Chen, Jingwei Zhang, Jingzhong CAS Chengdu Inst Comp Applicat Lab Automated Reasoning & Programming Chengdu 610041 Peoples R China

ISBN: (纸本)9781605586649

We present a new algorithm for reconstructing an exact algebraic number from its approximate value by using an improved parameterized integer relation construction method. Our result is consistent with the existence of error controlling on obtaining an exact rational number from its approximation. The algorithm is applicable for finding exact minimal polynomial of an algebraic number by its approximate root. This also enables us to provide an efficient method of converting the rational approximation representation to the minimal polynomial representation, and devise a simple algorithm to factor multivariate polynomials with rational coefficients. Compared with the subsistent methods, our method combines advantage of high efficiency in numerical computation, and exact, stable results in symbolic computation. The experimental results show that the method is more efficient than identify in Maple for obtaining an exact algebraic number from its approximation. Moreover, the Digits of our algorithm is far less than the LLL-lattice basis reduction technique in theory. In this paper, we completely implement how to obtain exact results by numerical approximate computations.

关键词： algebraic number numerical approximate computation symbolic-numerical computation integer relation algorithm minimal polynomial

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symbolic-numerical OBJECT-ORIENTED FINITE ELEMENT PROGRAMMING

SYMBOLIC-NUMERICAL OBJECT-ORIENTED FINITE ELEMENT PROGRAMMIN...

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作者： PISITH SAM THAMMASAT UNIVERSITY

学位级别：硕士

There are times when closed-form analysis of solids and structures is needed. When only a structural member is considered, it is certainly possible to perform this type of analysis by hand. However, when a structure with several members is to be considered, manual calculation becomes practically impossible even when closed-form solutions are theoretically obtainable. A finite element (FE) program that can symbolically analyze elasticity problems of solids and structures can therefore be quite useful. Unfortunately, symbolic computations are usually difficult to perform when problems are complex. In these cases, numerical computations are still necessary. This study presents an object-oriented FE program that can perform both symbolic and numerical computations. The program is implemented in MATLAB. The object-oriented programming (OOP) technique is used to enhance the maintainability, extendibility, and reusability of the program. The proposed program is capable of performing symbolic and numerical finite element analysis depending on the input it receives. The obtained program is tested using some engineering problems in order to demonstrate its usefulness. The results obtained from the program are found to be satisfactory.

关键词： Finite element programming Closed-form analysis Linear elasticity symbolic computation numerical computation symbolic-numerical computation Object-oriented programming

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Direct transformation from Cartesian into geodetic coordinates on a triaxial ellipsoid

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COMPUTERS & GEOSCIENCES 2020年 142卷

作者： Maria Diaz-Toca, Gema Marin, Leandro Necula, Ioana Univ Murcia Dept Ingn & Tecnol Comp Murcia Spain Univ Seville Dept Matemat Aplicada 1 Seville Spain

This paper1 presents two new direct symbolic-numerical algorithms for the transformation of Cartesian coordinates into geodetic coordinates considering the general case of a triaxial reference ellipsoid. The problem in both algorithms is reduced to finding a real positive root of a sixth degree polynomial. The first approach consists of algebraic manipulations of the equations describing the geometry of the problem and the second one uses Groeurobner bases. In order to perform numerical tests and accurately compare efficiency and reliability, our algorithms together with the iterative methods presented by M. Ligas (2012) and J. Feltens (2009) have been implemented in C++. The numerical tests have been accomplished by considering 10 celestial bodies, referenced in the available literature. The obtained results show that our algorithms improve the aforementioned, up-to-date reference, iterative methods, in terms of both efficiency and accuracy.

关键词： Coordinate transformation Cartesian coordinates Geodetic coordinates Triaxial ellipsoid symbolic-numerical computation

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numerical-symbolic Methods for Searching Relative Equilibria in the Restricted Problem of Four Bodies

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MATHEMATICAL MODELLING AND ANALYSIS 2018年第3期23卷 507-525页

作者： Prokopenya, Alexander N. Warsaw Univ Life Sci SGGW Nowoursynowska Str 159 PL-02776 Warsaw Poland

We discuss here the problem of solving the system of two nonlinear algebraic equations determining the relative equilibrium positions in the planar circular restricted four-body problem formulated on the basis of the Euler collinear solution of the three-body problem. The system contains two parameters mu(1), mu(2) and all its solutions coincide with the corresponding solutions in the three-body problem if one of the parameters equals to zero. For small values of one parameter the solutions are found in the form of power series in terms of this parameter, and they are used for separation of different solutions and choosing the starting point in the numerical procedure for the search of equilibria. Combining symbolic and numerical computation, we found all the equilibrium positions and proved that there are 18 different equilibrium configurations of the system for any reasonable values of the two system parameters mu(1), mu(2). All relevant symbolic and numerical calculations are performed with the aid of the computer algebra system Wolfram Mathematica.

关键词： restricted four-body problem relative equilibria symbolic-numerical computation Wolfram Mathematica

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A complete algorithm to find exact minimal polynomial by approximations

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2012年第17期89卷 2333-2344页

Based on an improved parameterized integer relation construction method, a complete algorithm is proposed for finding an exact minimal polynomial from its approximate root. It relies on a study of the error controlling for its approximation. We provide a sufficient condition on the precision of the approximation, depending only on the degree and the height of its minimal polynomial. Our result is superior to the existent error controlling on obtaining an exact rational or algebraic number from its approximation. Moreover, some applications are presented and compared with the subsistent methods.

关键词： error controllable algorithm symbolic-numerical computation integer relation construction minimal polynomial

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