Numerical Algorithm For The Maximum Eigenvalue Of Nonnegative Matrix And Its Application

The estimation and calculation of the maximum eigenvalue of nonnegative matrix is the classical content of nonnegative matrix theory,which is of great significance in numerical algebra.In this paper,two new numerical algorithms for the maximum eigenvalue of nonnegative matrix are presented.Firstly,by using the Perron-Frobenius theorem of spectral radius of nonnegative matrix and the diagonal similarity transformation of matrix,through analysis and theoretical proof,a diagonal similar iterative algorithm for the maximum eigenvalue and the corresponding eigenvector of nonnegative matrix is constructed.At each step of the iteration,the algorithm uses the row sum of the last iteration matrix and a variable parameter to form a positive diagonal matrix to do diagonal similarity transformation,which has great flexibility.Then,by using the properties of irreducible nonnegative matrix and Collatz-Wielandt function,based on translation transformation,a new matrix form B?28??A?10??I?^n-1,is constructed,in which A is an irreducible nonnegative matrix,??29?0.Thus,an improved Collatz-Wielandt algorithm for the maximum eigenvalue and the corresponding eigenvector of nonnegative matrix is proposed.The iterative scheme is defined and the convergence of the algorithm is proved theoretically.The algorithm has a good convergence rate when the translation parameters are properly selected.Finally,a numerical example is given to illustrate the feasibility of two algorithms and the influence of parameters on the convergence rate.As an application,an algorithm of minimum eigenvalue of M-matrix and an iterative discriminant method of M-matrix?generalized strictly diagonally dominant matrix?are given,which is of great significance to the application of Jacobi iterative method and Gauss-Seidel iterative method for solving linear equations.