On the absolute stability regions corresponding to partial sums of the exponential function
On the absolute stability regions corresponding to partial sums of the exponential function
复制标题
关于指数函数部分和对应的绝对稳定区域
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
Lajos L'oczi
中科院分区：
文献类型：
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作者：
D. Ketcheson;T. A. Kocsis;Lajos L'oczi
初值问题的某些数值方法以指数函数的第n次部分和为稳定性函数。我们研究了稳定区域，即复平面上第n次部分和至多有单位模的集合。已知在左半平面稳定区域的一部分的渐近形状是一个半圆盘。我们通过提供包围或被稳定区域或其左半平面部分包围的圆盘来量化这一点。对1 n 2 0，确定了以包含稳定区域的原点(或其在左半平面上的那一部分)为中心的最小圆盘的半径.证明了n 2在这个半径上的界；证明了在极限n+1内，稳定域及其补集是最优的。我们证明了稳定域及其补域由长度趋于n！1的交替区间组成。最后，我们证明了左半平面上垂直边界为虚轴、中心在原点的半圆包含在稳定域中当且仅当n0 mod 4或n3 mod 4。对1 n 2 0，精确地确定了这种半圆的最大半径。
Certain numerical methods for initial value problems have as stability function the n th partial sum of the exponential function. We study the stability region, i.e., the set in the complex plane over which the n th partial sum has at most unit modulus. It is known that the asymptotic shape of the part of the stability region in the left half-plane is a semi-disk. We quantify this by providing disks that enclose or are enclosed by the stability region or its left half-plane part. The radius of the smallest disk centered at the origin that contains the stability region (or its portion in the left half-plane) is determined for 1 n 20. Bounds on such radii are proved for n 2; these bounds are shown to be optimal in the limit n ! +1. We prove that the stability region and its complement, restricted to the imaginary axis, consist of alternating intervals of length tending to , as n!1. Finally, we prove that a semi-disk in the left half-plane with vertical boundary being the imaginary axis and centered at the origin is included in the stability region if and only if n 0 mod 4 or n 3 mod 4. The maximal radii of such semi-disks are exactly determined for 1 n 20.