For a square matrix $A$ with no eigenvalues on the closed real axis, its principal $p$th root, $A^{1∕p}$, is uniquely defined. When all eigenvalues of $A$ are in a suitable region in the complex plane, Newton’s method can be used to approximate $A^{1∕p}$, starting from the identity matrix. Such a region is called a convergence region for Newton’s method. For $p=2$, the convergence region is the whole region for which $A^{1∕2}$ is defined. For $p\ge 3$, however, the convergence region will be much smaller. In this paper, we obtain for $p\ge 3$ explicit $p$-dependent convergence regions that significantly expand existing explicit convergence regions.

中文翻译：

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用于矩阵 p th 根的牛顿法的显式 p 相关收敛区域

对于方阵 $\mathrm{一种}$ 在闭合实轴上没有特征值，它的主 $磷$根， ${\mathrm{一种}}^{1∕磷}$, 是唯一定义的。当所有特征值$\mathrm{一种}$ 在复平面的合适区域内，牛顿法可用于近似 ${\mathrm{一种}}^{1∕磷}$，从单位矩阵开始。这样的区域被称为牛顿方法的收敛区域。为了$磷=2$, 收敛区域是整个区域 ${\mathrm{一种}}^{1∕2}$被定义为。为了$磷\ge 3$，然而，收敛区域会小得多。在本文中，我们得到$磷\ge 3$ 明确的 $磷$显着扩展现有显式收敛区域的依赖收敛区域。