论文标题
统一符号分解和统一特征组件的迭代
Iterations for the Unitary Sign Decomposition and the Unitary Eigendecomposition
论文作者
论文摘要
我们构建快速，结构的迭代迭代，以计算统一矩阵$ a $的符号分解，而没有特征值等于$ \ pm i $。该分解将$ a $分解为引起矩阵的产物$ s = \ operatorAtorname {sign}（a）= a（a^2）^{ - 1/2} $ times a矩阵$ n =（a^2）^{1/2} $，带有频谱的频谱，均匀地符合复数平面的一半。我们的迭代依赖于最近发现的公式（从最小程度上讲）标量函数的最佳（最小值）单模型有理近似，$ \ permatatorName {sign}（z）= z/\ sqrt {z^2} $在单位圆的子集上。当$ a $的特征值接近$ \ pm i $时，迭代的收敛速度明显快于帕德迭代。数值证据表明迭代是向后稳定的，其向后误差通常比直接方法获得的误差小。这与其他迭代（例如牛顿迭代迭代量表）形成鲜明对比，如果$ a $具有$ \ pm i $ $ $ $的特征值，则具有数值不稳定性。作为应用程序，我们使用迭代来为单一特征组合构建稳定的光谱分裂和诱导算法。
We construct fast, structure-preserving iterations for computing the sign decomposition of a unitary matrix $A$ with no eigenvalues equal to $\pm i$. This decomposition factorizes $A$ as the product of an involutory matrix $S = \operatorname{sign}(A) = A(A^2)^{-1/2}$ times a matrix $N = (A^2)^{1/2}$ with spectrum contained in the open right half of the complex plane. Our iterations rely on a recently discovered formula for the best (in the minimax sense) unimodular rational approximant of the scalar function $\operatorname{sign}(z) = z/\sqrt{z^2}$ on subsets of the unit circle. When $A$ has eigenvalues near $\pm i$, the iterations converge significantly faster than Padé iterations. Numerical evidence indicates that the iterations are backward stable, with backward errors often smaller than those obtained with direct methods. This contrasts with other iterations like the scaled Newton iteration, which suffers from numerical instabilities if $A$ has eigenvalues near $\pm i$. As an application, we use our iterations to construct a stable spectral divide-and-conquer algorithm for the unitary eigendecomposition.
