Research On The Accurate Computations Of Totally Negative Matrices

A matrix with all minors negative is called totally negative matrix,which is usually an ill-conditioned matrix.In numerical algebra,we want to obtain numerical results with high relative accuracy.But,the numerical results we obtained by using traditional algorithms may haven't accuracy.Therefore,the research about algorithms of ill-conditioned matrices attracts the attention of scholars.In this paper,we consider accurate computations of rectangular totally negative matrices.The paper is structured as following:In chapter one,we introduce research background and status about totally negative matrix,the explanation of symbols and Neville elimination which is a key tool of our research.In chapter two,we obtain the bidiagonal decomposition,and then provide a new necessary and sufficient condition of rectangular totally negative matrices by using of Neville elimination to re-parameterize them.In chapter three,we use the parameters to perform QR decomposition.By this process,we get the conclusion that the upper triangular matrix is computed with high relative accuracy when all parameters are accurate.In chapter four,we design algorithms to compute QR decomposition,singular values,the Moore-Penrose inverse and the solution of least squares problem by using of parameters.Numerical experiments show that we can accurately compute each element of upper triangular matrix,each singular values,the Moore-Penrose inverse in the sense of matrix 2-norm and the solution of least squares problem in the sense of vector 2-norm.