2 research outputs found
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.Comment: 1
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. In this paper, an effective algorithm is presented for
computing 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. It is
also based on a novel approach for estimating the degree of variables, and the
degree homomorphism method for dimension reduction. Furthermore, the
parallelization of the method arises naturally.Comment: 17 pages, 2 figure