The LC method described in this work seeks to approximate the roots of polynomial equations in one variable. This book allows you to explore the LC method, which uses geometric structures of Lines L and Circumferences C in the plane of complex numbers, based on polynomial coefficients. These structures depend on the inclination angle of a line with fixed point that seeks to contain one of the roots; they are associated with an error measure that indicates the degree of proximity to that root, without knowing a priori its location. Using a computer with parallel processing capabilities, it is feasible to construct several of these geometric structures at the same time, varying the inclination angle of the lines with fixed point, in order to obtain an error measure map, with which it is possible to identify, approximately, the location of all polynomial roots. To show how the LC method works, this book includes numerical examples for quadratic, cubic, and quartic polynomials, and also for polynomials of degree greater than or equal to 5; this book also includes R programs that allow you to reproduce the results of the examples on a typical personal computer; these R programs use vectorization of operations instead of loops, which can be seen as a basic and accessible form of parallel processing. This book, in the end, invites us to explore beyond the basic ideas and concepts described here, motivating the development of a more efficient and complete computational implementation of the LC method.

翻译：本文描述的LC方法旨在求解单变量多项式方程的根。本书探讨了LC方法——该方法基于多项式系数，在复数平面上利用直线L和圆周C的几何结构。这些结构取决于过定点直线的倾角，该直线旨在包含其中一个根；它们与一种误差度量相关联，该度量可指示与根接近的程度，而无需先验已知根的位置。借助具备并行处理能力的计算机，可同时构建多个此类几何结构（通过改变过定点直线的倾角），从而获得一张误差度量图，据此能够近似识别所有多项式根的位置。为展示LC方法的运作原理，本书提供了二次、三次、四次多项式以及次数≥5的多项式的数值示例；本书还附有R语言程序，可用于在普通个人计算机上复现示例结果。这些R程序采用操作向量化而非循环的方式——这可视作一种基本且易于实现的并行处理形式。最终，本书邀请读者超越此处所述的基本思想与概念展开探索，推动LC方法更高效、更完整的计算实现。