Jingzhong Zhang

The parallel numerical method of mechanical theorem proving

Abstract In this paper, we present results of the work which allow us to prove geometry theorems by the parallel numerical method based on the multi-instance numerical verification of algebraic identity. The algebraic principle of the parallel numerical method is discussed and illustrated intuitively; the advantages of our method are given. It is acceptable on the complexity of both memory and time . It can be used to prove non-trivial geometric theorems by microcomputer, even by hand. We give some examples proved by parallel numerical method, including certain new unexpected results.

A review and prospect of readable machine proofs for geometry theorems

After half a century research, the mechanical theorem proving in geometries has become an active research topic in the automated reasoning field. This review involves three approaches on automated generating readable machine proofs for geometry theorems which include search methods, coordinate-free methods, and formal logic methods. Some critical issues about these approaches are also discussed. Furthermore, the authors propose three further research directions for the readable machine proofs for geometry theorems, including geometry inequalities, intelligent geometry softwares and machine learning.

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.

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.

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.

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 @S={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 @S=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.

Exact bivariate polynomial factorization over 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, the authors present an algorithm which use numerical method to obtain exact factors of a bivariate polynomial with rational coefficients. The proposed 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.

Computing the Dixon Derived Polynomial by Combination Resultant Method

Dixon resultant method is a fundamental elimination method for simultaneously eliminating several variables from polynomials. In this paper, we put forward the combination resultant method to construct any Dixon derived polynomial for any nonlinear polynomial system. Further we can obtain Dixon matrix. Through the method, we discover the mechanism of producing Dixon derived polynomials.