The Verification Of Eigenvalues Of Structured Matrix

Structure matrix is very important in matrix analysis and matrix computation.It not only exists in traditional mathematics,but also has been widely applied in chemistry,modern physics,economics,IT industry and other fields.It's very easy to simply solve the eigenvalues of a matrix.However,it is no longer so easy to apply the eigenvalues of matrix to other fields.The research on eigenvalues of structured matrices is motivated by applications,such as floating point error analysis,backward error analysis in linear algebra,numerical algorithm and other problems of eigenvalues.Therefore,it is of great significance to study the eigenvalues of structured matrix.With the rapid development of science and technology,there are higher requirements for the accuracy of computed results.Whether matrices or linear algebraic equations,the data are mostly obtained by observation or computation.Since the input is affected by uncertainty,the error always exists.In engineering experiments,there may be measurement errors,linearization or other simplifications,equipment may be worn or out of order,working conditions may change,etc.,which will lead to data errors in the mathematical model.Perturbation analysis of matrix eigenvalues is the study of how the small changes in the elements of a matrix affect the eigenvalues of a matrix.As an advanced method,the reliable verification is to prove the existence of the solution of mathematical problems in a certain interval by using computer.The issue to be solved in scientific computation is how to ensure that the errors in the computation process are controllable and the results are true and credible.This paper studies the influence of structured perturbation on nonlinear structured eigenvalues problems by using the Rump's interval algorithm and Kantorovich theorem.We design a credible verification algorithm for structured matrices eigenvalues.The main research contents are as follows:(1)Design algorithms for the verification of the spectra of the skew-symmetric matrix.Given a skew-symmetric matrix,we design two algorithms to compute its high-precision approximate spectra and verified error bound by Rump's interval Newton method and Kantorovich theorem.These algorithms guarantee that there exists a skew-symmetric matrix within computed error bound,whose exact spectra is the computed high-precision approximate spectra of the given matrix.The examples illustrate that the error bounds computed by these two algorithms are equal.(2)Design an algorithm for the verification of the eigenvalues of the other matrices.Given real symmetric,persymmetric and Hankel matrices,we firstly use eig code in Matlab to obtain their real eigenvalues on the basis of numerical computation.Then by Rump's interval method and Kantorovich theorem,we provide an algorithm to compute the verified error bounds of numerical eigenvalues,such that there exists a perturbation matrix of the same type within the computed error bound whose exact real eigenvalues are the computed real eigenvalues.