Yong Feng

Obtaining exact value by approximate computations

Numerical approximate computations can solve large and complex problems fast. They have the advantage of high efficiency. However they only give approximate results, whereas we need exact results in some fields. There 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.

Finding exact minimal polynomial by approximations

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.

Computing real witness points of positive dimensional polynomial systems

Abstract We consider a critical point method for finding certain solution (witness) points on real solution components of real polynomial systems of equations. The method finds points that are critical points of the distance from a plane to the component with the requirement that certain regularity conditions are satisfied. In this paper we analyze the numerical stability and complexity of the method. We aim to find at least one well conditioned witness point on each connected component by using perturbation, path tracking and projection techniques. An optimal-direction strategy and an adaptive step size control strategy for path following on high dimensional components are given.

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: Pryces 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.

Exact polynomial factorization by approximate high degree algebraic numbers

For factoring polynomials in two variables with rational coefficients, an algorithm using transcendental evaluation was presented by Hulst and Lenstra. In their algorithm, transcendence measure was computed. However, a constant c is necessary to compute the transcendence measure. The size of c involved the transcendence measure can influence the efficiency of the algorithm greatly. In this paper, we overcome the problem arising in Hulst and Lenstras algorithm and propose a new polynomial time algorithm for factoring bivariate polynomials with rational coefficients. Using an approximate algebraic number of high degree instead of a variable of a bivariate polynomial, we can get a univariate one. A factor of the resulting univariate polynomial can then be obtained by a numerical root finder and the purely numerical LLL algorithm. The high degree of the algebraic number guarantees that this factor corresponds to a factor of the original bivariate polynomial. We prove that our algorithm saves a (log2(mn))2+ε factor in bit-complexity comparing with the algorithm presented by Hulst and Lenstra, where (n, m) represents the bi-degree of the polynomial to be factored. We also demonstrate on many significant experiments that our algorithm is practical. Moreover our algorithm can be generalized to polynomials with variables more than two.

A complete algorithm to find exact minimal polynomial by approximations

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.

Homomorphically encrypted arithmetic operations over the integer ring

Fully homomorphic encryption allows cloud servers to evaluate any computable functions for clients without revealing any information. It attracts much attention from both of the scientific community and the industry since Gentry’s seminal scheme. Currently, the Brakerski-Gentry-Vaikuntanathan scheme with its optimizations is one of the most potentially practical schemes and has been implemented in a homomorphic encryption C++ library HElib. HElib supplies friendly interfaces for arithmetic operations of polynomials over finite fields. Based on HElib, Chen and Guang (2015) implemented arithmetic over encrypted integers. In this paper, we revisit the HElib-based implementation of homomorphically arithmetic operations on encrypted integers. Due to several optimizations and more suitable arithmetic circuits for homomorphic encryption evaluation, our implementation is able to homomorphically evaluate 64-bit addition/subtraction and 16-bit multiplication for encrypted integers without bootstrapping. Experiments show that our implementation outperforms Chen and Guang’s significantly.

Two variants of HJLS-PSLQ with applications

The HJLS and PSLQ algorithms are the most popular algorithms for finding nontrivial integer relations for several real numbers. It has been already shown that PSLQ is essentially equivalent to HJLS under certain settings. We here call them HJLS-PSLQ. In the present work, we provide two variants of HJLS-PSLQ. The first one is a new modification of Bailey and Broadhursts multi-pair version. We prove the termination of our modification, while the original multi-pair version may not terminate. The second one is an incremental version of HJLS-PSLQ. For those applications requiring to call HJLS-PSLQ many times, such as finding the minimal polynomial of an algebraic number without knowing the degree, we show the incremental version is more efficient than HJLS-PSLQ, both theoretically and practically.

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

Computing the determinant of a matrix with polynomial entries by approximation

Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. This paper proposes an effective algorithm to compute the determinant of a matrix with polynomial entries using hybrid symbolic and numerical computation. The algorithm relies on the Newton’s interpolation method with error control for solving Vandermonde systems. The authors also present the degree matrix to estimate the degree of variables in a matrix with polynomial entries, and the degree homomorphism method for dimension reduction. Furthermore, the parallelization of the method arises naturally.