Two quadrature-based algorithms for computing the matrix fractional power $A^\alpha$ are presented in this paper. These algorithms are based on the double exponential (DE) formula, which is well-known for its effectiveness in computing improper integrals as well as in treating nearly arbitrary endpoint singularities. The DE formula transforms a given integral into another integral that is suited for the trapezoidal rule; in this process, the integral interval is transformed to the infinite interval. Therefore, it is necessary to truncate the infinite interval into an appropriate finite interval. In this paper, a truncation method, which is based on a truncation error analysis specialized to the computation of $A^\alpha$, is proposed. Then, two algorithms are presented -- one computes $A^\alpha$ with a fixed number of abscissas, and the other computes $A^\alpha$ adaptively. Subsequently, the convergence rate of the DE formula for Hermitian positive definite matrices is analyzed. The convergence rate analysis shows that the DE formula converges faster than the Gaussian quadrature when $A$ is ill-conditioned and $\alpha$ is a non-unit fraction. Numerical results show that our algorithms achieved the required accuracy and were faster than other algorithms in several situations.
翻译:本文展示了两种基于二次方程式的算法, 用于计算矩阵分数功率 $A ⁇ alpha$。 这些算法基于双倍指数( DE) 公式, 因为它在计算不适当的整体元件和处理近乎任意的终点奇特方面是众所周知的。 DE 公式将给定的内分数转换成另一个内分数, 适合陷阱分裂规则; 在此过程中, 将整体间距转换为无限间隔。 因此, 有必要将无限的间距切换成一个适当的有限间隔。 在本文中, 提出了一种曲解法方法, 这种方法基于专门计算 $A ⁇ alpha$ 的调试错误分析。 然后, 提出了两种算法 -- 一种计算 $A ⁇ alpha$, 和 其它计算法则显示美元的不精确度。 当 $A 和 美元 等分数的计算结果显示时, DE 公式的趋近于高斯二次方程。