arXiv subjects

Yong Feng

Publications and source records attributed to Yong Feng.

31 records · Page 2Linked to original sources

arXiv2014

Structural index reduction algorithms for differential algebraic equations via fixed-point iteration

Motivated by Pryce's structural index reduction method for differential algebraic equations (DAEs), we show the complexity of the fixed-point iteration algorithm and propose a fixed-point iteration method with parameters. It leads to a block fixed-point iteration method which can be applied to large-scale DAEs with block upper triangular structure. Moreover, its complexity analysis is also given in this paper.

math.NA↗

arXiv2014

A Short Note on Zero-error Computation for Algebraic Numbers by IPSLQ

The PSLQ algorithm is one of the most popular algorithm for finding nontrivial integer relations for several real numbers. In the present work, we present an incremental version of PSLQ. For some applications needing to call PSLQ many times, such as finding the minimal polynomial of an algebraic number without knowing the degree, the incremental PSLQ algorithm is more efficient than PSLQ, both theoretically and practically.

cs.SC↗

arXiv2013

Numerical method for real root isolation of semi-algebraic system and its applications

In this paper, based on the homotopy continuation method and the interval Newton method, an efficient algorithm is introduced to isolate the real roots of semi-algebraic system. Tests on some random examples and a variety of problems including transcendental functions arising in many applications show that the new algorithm reduces the cost substantially compared with the traditional symbolic approaches.

math.NA↗

arXiv2012

Structural analysis of high-index DAE for process simulation

This paper deals with the structural analysis problem of dynamic lumped process high-index DAE models. We consider two methods for index reduction of such models by differentiation: Pryce's method and the symbolic differential elimination algorithm rifsimp. Discussion and comparison of these methods are given via a class of fundamental process simulation examples. In particular, the efficiency of the Pryce method is illustrated as a function of the number of tanks in process design.

eess.SY↗

arXiv2010

Detecting Simultaneous Integer Relations for Several Real Vectors

An algorithm which either finds an nonzero integer vector ${\mathbf m}$ for given $t$ real $n$-dimensional vectors ${\mathbf x}_1,...,{\mathbf x}_t$ such that ${\mathbf x}_i^T{\mathbf m}=0$ or proves that no such integer vector with norm less than a given bound exists is presented in this paper. The cost of the algorithm is at most ${\mathcal O}(n^4 + n^3 \log λ(X))$ exact arithmetic operations in dimension $n$ and the least Euclidean norm $λ(X)$ of such integer vectors. It matches the best complexity upper bound known for this problem. Experimental data show that the algorithm is better than an already existing algorithm in the literature. In application, the algorithm is used to get a complete method for finding the minimal polynomial of an unknown complex algebraic number from its approximation, which runs even faster than the corresponding \emph{Maple} built-in function.

cs.SC↗

arXiv2010

Exact Bivariate Polynomial Factorization in Q by Approximation of Roots

Factorization of polynomials is one of the foundations of symbolic computation. Its applications arise in numerous branches of mathematics and other sciences. However, the present advanced programming languages such as C++ and J++, do not support symbolic computation directly. Hence, it leads to difficulties in applying factorization in engineering fields. In this paper, we present an algorithm which use numerical method to obtain exact factors of a bivariate polynomial with rational coefficients. Our method can be directly implemented in efficient programming language such C++ together with the GNU Multiple-Precision Library. In addition, the numerical computation part often only requires double precision and is easily parallelizable.

math.AG↗

arXiv2010

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

We present a new algorithm for solving the real roots of a bivariate polynomial system $Σ=\{f(x,y),g(x,y)\}$ with a finite number of solutions by using a zero-matching method. The method is based on a lower bound for bivariate polynomial system when the system is non-zero. Moreover, the multiplicities of the roots of $Σ=0$ can be obtained by 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.

cs.SC↗

arXiv2010

A complete algorithm to find exact minimal polynomial by approximations

We present a complete algorithm for finding an exact minimal polynomial from its approximate value by using an improved parameterized integer relation construction method. Our result is superior to 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. we also discuss some applications to some transcendental numbers by approximations. 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.

cs.SC↗

arXiv2009

Finding Exact Minimal Polynomial by Approximations

We present a new algorithm for reconstructing an exact algebraic number from its approximate value 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 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 other methods, this method has the numerical computation advantage of high efficiency. The experimental results show that the method is more efficient than \emph{identify} in \emph{Maple} 11 for obtaining an exact algebraic number from its approximation. In this paper, we completely implement how to obtain exact results by numerical approximate computations.

cs.CC↗

arXiv2008

Obtaining Exact Interpolation Multivariate Polynomial by Approximation

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, exactly interpolating methods are often proposed for the exact results and approximate interpolating methods for the approximate ones. In this paper, we study how to obtain exact interpolation polynomial with rational coefficients by approximate interpolating methods.

cs.SC↗

arXiv2007

Multivariate Polynomial Factorization by Interpolation Method

Factorization of polynomials arises in numerous areas in symbolic computation. It is an important capability in many symbolic and algebraic computation. There are two type of factorization of polynomials. One is convention polynomial factorization, and the other approximate polynomial factorization. Conventional factorization algorithms use symbolic methods to get exact factors of a polynomial while approximate factorization algorithms use numerical methods to get approximate factors of a polynomial. Symbolic computation often confront intermediate expression swell problem, which lower the efficiency of factorization. The numerical computation is famous for its high efficiency, but it only gives approximate results. In this paper, we present an algorithm which use approximate method to get exact factors of a multivariate polynomial. Compared with other methods, this method has the numerical computation advantage of high efficiency for some class of polynomials with factors of lower degree. The experimental results show that the method is more efficient than {\it factor} in Maple 9.5 for polynomials with more variables and higher degree.

math.AG↗

arXiv2007

An Improved Algorithm for Recovering Exact Value from its Approximation

Numerical approximate computation can solve large and complex problems fast. It has the advantage of high efficiency. However it only gives approximate results, whereas we need exact results in many fields. There is a gap between approximate computation and exact results. A bridge overriding the gap was built by Zhang, in which an exact rational number is recovered from its approximation by continued fraction method when the error is less than $1/((2N+2)(N-1)N)$, where $N$ is a bound on absolute value of denominator of the rational number. In this paper, an improved algorithm is presented by which a exact rational number is recovered when the error is less than $1/(4(N-1)N)$.

math.AG↗

arXiv2006

Obtaining Exact Value by Approximate Computations

math.AG↗