多项式零点与矩阵特征值的快速逼近
Fast Approximation of Polynomial Zeros and Matrix Eigenvalues
AI总结：
针对多项式零点与矩阵特征值逼近问题，提出基于细分迭代变体的Las Vegas型近最优求根算法，实现更简单、性能优于MPSolve，还支持黑盒求值多项式，可扩展至矩阵特征值求解。
摘要：
我们对d次单变量多项式p(x)的d个复零点，或是位于复平面上固定感兴趣区域（如圆盘或正方形）内的零点进行逼近。我们1995年发表于STOC的分治算法能以最优布尔时间（至多相差一个多对数因子）求解该问题，即其运行速度几乎与以支撑输出所需精度的必要精度读取多项式p的系数的速度一样快。这一复杂度纪录至今仍未被其他算法打破，但我们1995年提出的求根算法结构相当复杂，从未被实现过。\n我们基于经典细分迭代的新型变体，提出了若干替代的近最优求根算法。与1995年的前身算法不同，我们采用Las Vegas型随机化，能够以可忽略的计算开销检测任何输出误差，且新求根算法的实现难度远低于1995年的版本。针对仅融入了我们部分新技术的初步版本，我们使用标准测试多项式开展了大量测试，结果显示，新求根算法性能颇具竞争力，且对于一大类输入，其性能显著优于数十年来用户首选的求根子例程包MPSolve。\n与1995年的前身算法以及所有已知的用于上述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盒神谕（用于求值）而非系数给出的多项式。这使得我们的求根算法对于可快速求值的多项式p(x)（例如Mandelbrot多项式，或是由少量平移单项式之和给出的多项式）尤为高效。我们的算法可轻松扩展至矩阵或矩阵多项式特征值的快速逼近。
英文摘要：
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.
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