A recurrent neural network computing the largest imaginary or real part of eigenvalues of real matrices
A recurrent neural network computing the largest imaginary or real part of eigenvalues of real matrices
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DOI：
10.1016/j.camwa.2006.09.004
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发表时间：
2007
期刊：
影响因子：
--
通讯作者：
Yiguang Liu;Zhisheng You;Liping Cao
中科院分区：
文献类型：
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作者：
Yiguang Liu;Zhisheng You;Liping Cao
由于矩阵尤其是一般实数矩阵的特征对的高效计算在工程上具有重要意义，而神经网络异步运行且计算性能较高，因此本文引入循环神经网络（RNN）来提取某些特征对。 RNN 的连接权值取决于矩阵，可以转化为变量 z(t) 为复向量的复微分系统。通过|z(t)|2的解析表达式，详细分析了RNN的收敛特性。使用一般的非零初始复向量，RNN 获得所有特征值的最大虚部。通过重新排列连接矩阵，获得最大实部。 7×7矩阵的实践表明了该方法的有效性。采用维数分别为 50 和 100 的两个矩阵来测试维数变大时该方法的效率。结果表明网络进入平衡状态的迭代次数对维数不敏感。该RNN可用于估计特征值的最大模等。与其他为类似目的而设计的神经网络相比，该RNN适用于一般的实矩阵。
As the efficient calculation of eigenpairs of a matrix, especially, a general real matrix, is significant in engineering, and neural networks run asynchronously and can achieve high performance in calculation, this paper introduces a recurrent neural network (RNN) to extract some eigenpair. The RNN, whose connection weights are dependent upon the matrix, can be transformed into a complex differential system whose variable z(t) is a complex vector. By the analytic expression of |z(t)|2, the convergence properties of the RNN are analyzed in detail. With general nonzero initial complex vector, the RNN obtains the largest imaginary part of all eigenvalues. By a rearrangement of connection matrix, the largest real part is obtained. A practice of a 7×7 matrix indicates the validity of this method. Two matrices, whose dimensionalities are 50 and 100, respectively, are employed to test the efficiency of this approach when dimension number becomes large. The results imply that the iteration number at which the network enters into equilibrium state is not sensitive with dimensionality. This RNN can be used to estimate the largest modulus of eigenvalues, etc. Compared with other neural networks designed for the similar aims, this RNN is applicable to general real matrices.