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.
Numerical bifurcation analysis of renewal equations via pseudospectral approximation
利用常微分方程给出了非线性更新方程的一个近似。我们考虑积分状态，它是绝对连续的，并满足延迟微分方程。将伪谱方法应用于微分方程的抽象形式，得到了一个近似的常微分方程组。我们提出了收敛性证明的平衡和相关的特征根，我们使用一些模型从生态学和流行病学来说明的好处的方法进行数值分岔分析的平衡和周期性的解决方案。数值模拟表明，新近似系统的实现比最初在[Breda et al，SIAM Journal on Applied动力系统，2016]中提出的系统效率高10倍，因为它避免了代数方程的数值求逆。
Complementary families of approximating polynomials with applications to finite element methods applied to differential equations of arbitrary even spatial order
