论文标题
收敛率分析并改善了数值半径计算的迭代
Convergence rate analysis and improved iterations for numerical radius computation
论文作者
论文摘要
计算数值半径的主要两种算法是Mengi和Overton的级别方法以及UHLIG的切削平面方法。通过新的分析，我们解释了为什么有时比级别设定的速度更快或慢得多，然后提出了一种在所有情况下均保持有效效率的新型混合算法。对于以值为中心的圆盘的矩阵，我们表明，Uhlig方法的成本相对于所需的相对准确性爆炸。更一般而言，我们还分析了值范围内最外部点的Uhlig切割程序的局部行为，这表明它通常具有快速的Q线性收敛速率，并且在拐角处是Q-Superlinear。最后，我们在级别和切割平面方法中识别并解决了效率低下，并提出了这些技术的精制版本。
The main two algorithms for computing the numerical radius are the level-set method of Mengi and Overton and the cutting-plane method of Uhlig. Via new analyses, we explain why the cutting-plane approach is sometimes much faster or much slower than the level-set one and then propose a new hybrid algorithm that remains efficient in all cases. For matrices whose fields of values are a circular disk centered at the origin, we show that the cost of Uhlig's method blows up with respect to the desired relative accuracy. More generally, we also analyze the local behavior of Uhlig's cutting procedure at outermost points in the field of values, showing that it often has a fast Q-linear rate of convergence and is Q-superlinear at corners. Finally, we identify and address inefficiencies in both the level-set and cutting-plane approaches and propose refined versions of these techniques.
