Contraction-Anti Power Algorithm for Computing the Second Largest Eigenvalue of Dissipate Coupled Matrix
LU Jun-ana · Fuza xitong yu fuzaxing kexue · 2007
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 eigenvector,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.