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：我们近似计算d次单变量多项式p(x)的d个复零点，或位于复平面固定感兴趣区域（如圆盘或正方形）内的零点。基于1995年STOC会议的分治算法，该问题可在最优布尔时间（至多相差一个多对数因子）内求解，即运行速度与支持输出精度所需的高精度系数存取速度几乎相当。这一创纪录的复杂度至今未被任何其他算法超越，但1995年的求根器实现复杂且从未被实际部署。我们提出基于经典细分迭代新变体的近似最优求根器。与1995年算法不同，我们引入拉斯维加斯型随机化，允许以可控计算开销检测任意输出误差，但新求根器的实现复杂度远低于1995年算法。根据对标准测试多项式初版（仅集成部分新技术）的大量测试结果，新求根器在大量输入类别上可与业界数十年首选求根子程序包MPSolve竞争，并显著超越后者。与1995年算法及所有已知多项式求根快速算法不同，我们的新算法还可应用于通过黑箱求值预言（而非系数）给定的多项式。这使得求根器对曼德博多项式或少量移位单项式求和的快速求值多项式p(x)尤为高效。本算法可便捷扩展至矩阵或矩阵多项式特征值的快速逼近。