Maximum Eigenvalue Algorithm And Application Of Symmetric Nonnegative Constrained Matrices

The issues of the eigenvalue of matrices are among the most important problems in numerical mathematics.Eigenvalues are widely used in the economic,engineering,military and other fields.Most of the actual problems often come down to the largest eigenvalue of matrices.Therefore,computing the largest eigenvalue of matrices becomes especially important.Many scholars designed efficient algorithms of nonnegative irreducible matrices.However,in the calculation of the actual problem,it is very costly to test the reducibility for high dimension matrices.So,we want to find an algorithm to obtain the largest eigenvalue of nonnegative reducible matrices.According to the research of the largest eigenvalue of nonnegative irreducible matrices,we generalize the conclusions and algorithms to the symmetric nonnegative reducible matrices and give the algorithm for the largest eigenvalue of the symmetric nonnegative reducible matrices.Further,we apply the algorithm to test whether a matrix is an H-matrix and test the positive definiteness of Z-matrices.In chapter 1,we introduce the basic knowledge of reducible matrices and irreducible matrices and give some methods to solving the largest eigenvalue of nonnegative irreducible matrices.In chapter 2,we propose diagonal transformation algorithm to obtain the largest eigenvalue of symmetric nonnegative reducible matrices based on the nonnegative irreducible matrices.The new algorithm does not need test the reducibility of matrices and decompose matrices.It is proved that the algorithm is convergent under any conditions.Numerical results are reported to demonstrate the effectiveness of the proposed algorithm.Lastly,we apply the diagonal transformation algorithm to test whether a matrix is an H-matrix.In chapter 3,we propose an efficient algorithm for computing the largest eigenvalue of symmetric nonnegative reducible matrices based on the nonnegative irreducible matrices.In the selection of the initial vector,the new algorithm requires that each component is strictly greater than zero.After each iteration,the vector need normalization.The algorithm for any symmetric nonnegative reducible matrices is convergent.Numerical results are reported to demonstrate the effectiveness of the proposed algorithm.As applications,we present an algorithm for testing the positive definiteness of Z-matrices.Finally,we summarize the dissertation and present some questions for future researches.