We approximate the d complex zeros of a univariate polynomial p(x) of a degree d or those zeros that lie in a fixed region of interest on the complex plane such as a disc or a square. Our divide and conquer algorithm of STOC 1995 supports solution of this problem in optimal Boolean time (up to a poly-logarithmic factor), that is, runs nearly as fast as one can access the coefficients of p with the precision necessary to support required accuracy of the output. That record complexity has not been matched by any other algorithm yet, but our root-finder of 1995 is quite involved and has never been implemented. We present alternative nearly optimal root-finders based on our novel variants of the classical subdivision iterations. Unlike our predecessor of 1995, we require randomization of Las Vegas type, allowing us to detect any output error at a dominated computational cost, but our new root-finders are much simpler to implement than their predecessor of 1995. According to the results of extensive test with standard test polynomials for their preliminary version, which incorporates only a part of our novel techniques, the new root-finders compete and for a large class of inputs significantly supersedes the package of root-finding subroutines MPSolve, which for decades has been user's choice package. Unlike our predecessor of 1995 and all known fast algorithms for the cited tasks of polynomial root-finding, our new algorithms can be also applied to a polynomial given by a black box oracle for its evaluation rather than by its coefficients. This makes our root-finders particularly efficient for polynomials p(x) that can be evaluated fast such as the Mandelbrot polynomials or those given by the sum of a small number of shifted monomials. Our algorithm can be readily extended to fast approximation of the eigenvalues of a matrix or a matrix polynomial.

翻译：我们逼近单变量多项式p(x)的d个复零点（次数为d），或逼近其位于复平面固定关注区域（如圆盘或正方形）内的零点。基于1995年STOC会议论文的分治算法，该问题可在最优布尔时间复杂度内求解（仅相差多对数因子），即运行速度几乎等同于以支持输出所需精度而访问p系数所需的时间。这一创纪录的复杂度至今未被其他算法超越，但1995年提出的求根算法相当复杂且从未被实现过。我们基于经典细分迭代的新变体，提出替代性近最优求根器。与1995年的前身不同，我们采用拉斯维加斯型随机化方法，使得我们能够在可忽略的计算成本下检测任何输出误差，但新求根算法的实现难度远低于1995年的前身。根据对仅包含部分新技术初步版本的标准多项式测试结果，新求根器在竞争中表现优异，且对大量输入显著超越了长期作为用户选择包的MPSolve求根子程序库。与1995年的前身及所有已知的快速多项式求根算法不同，我们的新算法还可应用于通过黑箱预言机（而非系数）进行求值的多项式。这使得求根器对可快速求值的多项式（如Mandelbrot多项式或由少量移位单项式之和表示的多项式）尤为高效。我们的算法可便捷扩展至矩阵或矩阵多项式特征值的快速逼近。