论文标题
改进了Aberth-Ehrlich的根找到算法及其进一步的二元微透镜应用
Improved Aberth-Ehrlich root-finding algorithm and its further application for Binary Microlensing
论文作者
论文摘要
在重力微透析形式主义和建模二进制光曲线中，关键步骤是求解二进制镜头方程。当前，在建模二进制光曲线时，使用了Skowron \＆Gould（SG）最初引入的Newton和Laguerre方法的组合。在本文中，我们首先引入了一种基于Aberth-Ehrlich（AE）方法的单变量多项式的快速结发现算法，该方法于1967年首次开发，以改进牛顿的方法。事实证明，AE算法比牛顿，拉瓜尔和杜兰德·内凯尔的方法快得多，与其他钓鱼算法不同，它能够同时产生所有根。在改进了基本AE算法并讨论其属性之后，我们将优化它用于求解二进制镜头方程，该方程是具有复杂系数的第五度多项式。我们的方法约为$ 1.8 $至$ 2.0的$ 2.0 $ $倍，比SG算法快。由于为了计算点状或有限源星的放大因子，因此有必要求解二进制镜头方程，并首先在图像平面中找到产生的图像的位置，因此这种新方法将提高二进制微透镜建模的速度和准确性。
In gravitational microlensing formalism and for modeling binary light curves, the key step is solving the binary lens equation. Currently, a combination of the Newton's and Laguerre's methods which was first introduced by Skowron \& Gould (SG) is used while modeling binary light curves. In this paper, we first introduce a fast root-finding algorithm for univariate polynomials based on the Aberth-Ehrlich (AE) method which was first developed in 1967 as an improvement over the Newton's method. AE algorithm has proven to be much faster than Newton's, Laguerre's and Durand-Kerner methods and unlike other root-finding algorithms, it is able to produce all the roots simultaneously. After improving the basic AE algorithm and discussing its properties, we will optimize it for solving binary lens equations, which are fifth degree polynomials with complex coefficients. Our method is about $1.8$ to $2.0$ times faster than the SG algorithm. Since, for calculating magnification factors for point-like or finite source stars, it is necessary to solve the binary lens equation and find the positions of the produced images in the image plane first, this new method will improve the speed and accuracy of binary microlensing modeling.
