Liu Yiguang, You Zhisheng, Cao Liping
Institute of Image and Graphics, School of Computer Science and Engineering, Sichuan University, No. 29 Wangjiang Road, Chengdu 610064, Sichuan Province, People's Republic of China.
Neural Netw. 2005 Dec;18(10):1293-300. doi: 10.1016/j.neunet.2005.04.008. Epub 2005 Sep 8.
How to quickly compute eigenvalues and eigenvectors of a matrix, especially, a general real matrix, is significant in engineering. Since neural network runs in asynchronous and concurrent manner, and can achieve high rapidity, this paper designs a concise functional neural network (FNN) to extract some eigenvalues and eigenvectors of a special real matrix. After equivalent transforming the FNN into a complex differential equation and obtaining the analytic solution, the convergence properties of the FNN are analyzed. If the eigenvalue whose imaginary part is nonzero and the largest of all eigenvalues is unique, the FNN will converge to the eigenvector corresponding to this special eigenvalue with general nonzero initial vector. If all eigenvalues are real numbers or there are more than one eigenvalue whose imaginary part equals the largest, the FNN will converge to zero point or fall into a cycle procedure. Comparing with other neural networks designed for the same domain, the restriction to matrix is very slack. At last, three examples are employed to illustrate the performance of the FNN.
如何快速计算矩阵，特别是一般实矩阵的特征值和特征向量，在工程领域具有重要意义。由于神经网络以异步和并发方式运行，并且能够实现高速度，本文设计了一种简洁的函数神经网络（FNN）来提取特殊实矩阵的一些特征值和特征向量。在将FNN等效转化为复微分方程并得到解析解之后，分析了FNN的收敛特性。如果虚部非零且为所有特征值中最大的特征值是唯一的，那么对于一般的非零初始向量，FNN将收敛到对应于这个特殊特征值的特征向量。如果所有特征值都是实数或者存在多个虚部等于最大值的特征值，FNN将收敛到零点或陷入循环过程。与为同一领域设计的其他神经网络相比，对矩阵的限制非常宽松。最后，通过三个例子来说明FNN的性能。
