论文标题
多个精度求解器加速较低的代数方程，精度为eigensolver
Acceleration of multiple precision solver for ill-conditioned algebraic equations with lower precision eigensolver
论文作者
论文摘要
有一些类型的不良条件代数方程在获得准确的根和系数上很难以多个精确的浮点数表示。当它们所有的根都很简单时，如果相应的伴随矩阵的状况较小，则通过Eigensolver（特征值法）解决的问题就会得到很好的条件。但是，由于其根部分布的密度增加，应将使用牛顿或同时迭代方法（简称直接迭代方法）直接解决它们。尽管在直接迭代方法中，比特征值方法需要更多的浮点算术摩氏菌，但显然无法确定总计算成本。在这项研究中，我们针对威尔金森的榜样和Chebyshev正交问题，作为不良条件代数方程的例子，并证明了一些具体的数值结果，以证明直接迭代方法可以比标准的eigensolver更好地执行。
There are some types of ill-conditioned algebraic equations that have difficulty in obtaining accurate roots and coefficients that must be expressed with a multiple precision floating-point number. When all their roots are simple, the problem solved via eigensolver (eigenvalue method) is well-conditioned if the corresponding companion matrix has its small condition number. However, directly solving them with Newton or simultaneous iteration methods (direct iterative method for short) should be considered as ill-conditioned because of increasing density of its root distribution. Although a greater number of mantissa of floating-point arithmetic is necessary in the direct iterative method than eigenvalue method, the total computational costs cannot obviously be determined. In this study, we target Wilkinson's example and Chebyshev quadrature problem as examples of ill-conditioned algebraic equations, and demonstrate some concrete numerical results to prove that the direct iterative method can perform better than standard eigensolver.
