Neural-Network-Based Approach for Extracting Eigenvectors and Eigenvalues of Real Normal Matrices and Some Extension to Real Matrices

Abstract

This paper introduces a novel neural-network-based approach for extracting some eigenpairs of real normal matrices of ordern. Based on the proposed algorithm, the eigenvalues that have the largest and smallest modulus, real parts, or absolute values of imaginary parts can be extracted, respectively, as well as the corresponding eigenvectors. Although the ordinary differential equation on which our proposed algorithm is built is onlyn-dimensional, it can succeed to extractn-dimensional complex eigenvectors that are indeed 2n-dimensional real vectors. Moreover, we show that extracting eigen-pairs of general real matrices can be reduced to those of real normal matrices by employing the norm-reducing skill. Numerical experiments verified the computational capability of the proposed algorithm.

Linear Simultaneous Equations’ Neural Solution and Its Application to Convex Quadratic Programming with Equality-Constraint

“…According to the traditional gradient-based algorithm [8,10,12], a scalar-valued norm-based energy function ( ) := ‖ − ‖ 2 2 /2 is firstly constructed, and then evolving along the descent direction resulting from such energy function, we could obtain the linear GNN model for the solution of linear algebraic (1); that is,̇=…”

“…Journal of Applied Mathematics where ∈ × is a positive definite Hessian matrix, coefficients ∈ and ∈ are vectors, and ∈ × is a full row-rank matrix. They are known as constant coefficients of the to be solved QP problem (8).…”

“…Mathematically, (8) can be written as minimize ( ) = (1/2) + . For analysis convenience, let * denote the theoretical solution of QP problem (8).…”

“…Therefore, we can obtain the solution ∈ of (8) by transforming QP problem (8) into matrix-vector equation (12). In other words, to get the solution ∈ of (8), QP problem (8) is firstly transformed into the matrix-vector equation 12, which is a linear matrix-vector equation similar to the linear simultaneous equations (1), and then, we thus can make full use of the neural solvers presented in Section 2 to solve the QP problem (8). Moreover, the first elements of solutioñ( ) of (12) compose the neural solution ( ) of (8), and the Lagrangian vector consists of the last elements.…”

“…In general, the minimal arithmetic operations of numerical algorithms are usually proportional to the cube of the dimension of the coefficient matrix, that is, ( 3 ) [7]. In order to be satisfied with the low complexity and real-time requirements, recently, numerous novel neural networks have been exploited based on the hardware implementation [2,4,5,[8][9][10][11][12][13]. For example, Tank and Hopfield solved the linear programming problems by using their proposed Hopfield neural networks (HNN) [9], which promoted the development of the neural networks in the optimization and other application problems.…”