耗散耦合矩阵第二大特征值的先收缩后反幂算法

Contraction-Anti Power Algorithm for Computing the Second Largest Eigenvalue of Dissipate Coupled Matrix

无权无向耦合动态网络所对应的耗散耦合矩阵的第二大特征值是判断网络同步稳定性的重要指标。试图改善目前一般直接调用Matlab中函数eig计算1 000阶以上耗散耦合矩阵第二大特征值复杂度高、时间长的问题,利用耗散耦合矩阵具有一个零特征值及其对应的特征向量为[1,1,…,1]T的特点,提出了一种先收缩后反幂算法,证明了收缩矩阵的特征值与原矩阵的非零特征值误差为零,其对应特征向量相等,并导出对于1 000阶以上矩阵,先收缩后反幂法所需乘法次数比Matlab中调用函数eig所用的QR算法大幅度减少。数值计算验证了理论分析的正确性。

The second largest eigenvalue of dissipate coupling matrix is an important index to evaluate the complex dynamic network synchronous stability. In this paper, we try to improve the issue that the general method, which is used to compute the second largest eigenvalue of dissipate coupling matrix,is calling the function eig in Matlab directly, however, it will cost very high computational complexity and rather long time, when the matrix order is over 1 000. We make use of a special property of dissipate coupling matrices i. e. they have an eigenvalue zero and a corresponding eigenveetor[ 1,1,..., 1 ]^T ,and propose another algorithm Contraction-Anti-Power-Method. We prove that using Contraction Method to reduce the order of dissipate coupling matrices will not change the non-zero eigenvalue and their corresponding eigenvectors, and show that the needed multiplication times can be reduced by a wide margin,if we use Contraction-Anti-Power-Method,rather than QR algorithm used in the function eig of Matlab, especially for over 1 000 order matrices, The numerical calculations have confirmed in the correctness of this theoretical analysis.