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。1995年STOC会议提出的分治算法在最优布尔时间（对数多项式因子内）支持该问题的求解，即其运行速度几乎等同于以输出精度所需精度访问p系数所需的时间。至今尚无其他算法能匹敌该复杂度记录，但1995年提出的求根器实现复杂且从未被实现。我们提出基于经典细分迭代新型变体的替代近最优求根器。与1995年的前身不同，我们需引入拉斯维加斯型随机化，从而能以可忽略的计算代价检测任何输出误差，但新求根器的实现复杂度远低于1995年的前身。针对仅包含部分新技术雏形的初步版本，使用标准测试多项式的广泛测试结果表明：新求根器与数十年间用户首选的MPSolve求根子程序包相比具有竞争力，且对大规模输入显著更优。与1995年的前身及所有已知用于多项式求根任务的快速算法不同，我们的新算法还可应用于由黑箱求值预言机（而非系数形式）给定的多项式。这使得求根器对可快速求值的多项式（如Mandelbrot多项式或少量平移单项式求和形式的多项式）特别高效。本算法可自然扩展至矩阵或矩阵多项式特征值的快速逼近。