High-accuracy Computations Of Product Singular Value Problem Of Two Classes Of Sign Regular Matrices

In numerical linear algebra,high accuracy numerical results are often required.Due to the limitations of MATLAB’s computing power and error,we have to design mathematical algorithms with high relative accuracy.In the last century,Demmel J.and Kahan W.performed high relative accuracy computations on tridiagonal symbolic regular matrices.Since then,more research on high relative accuracy algorithms of sign regular matrices have been studied and noticed.For the high relative accuracy computation of the singular value of the product of a totally nonnegative matrix(if all of its minors are nonnegative,simply TN matrix)and a totally nonpositive matrix(if all of its minors are nonpositive,simply TNP matrix),it is called the product singular value problem and it is also a kind of high relative accuracy algorithm problem of sign regular matrices.The general method of transforming a TN matrix into a TNP matrix is given first,and then it is theoretically proved that the singular values of the TN×TNP matrix can be obtained from the parameters of the nonsingular TN matrix and the nonsingular TNP matrix without subtraction.New algorithms achieve high relative accuracy for singular values of TN×TNP matrices.Finally,numerical examples illustrate the theoretical results.This paper is organized as follows:In chapter one,we introduce the research background and current situation of high relative accuracy computing of TNP matrices,and give definition of fundamental matrices and symbols,followed by some explanations of the theoretical method Neville elimination method and parametrization used in this paper.In chapter two,we can obtain the TNP matrix from the TN matrix by using the rank one perturbation.The vector structure used in this process is simple and general,and explores the parameter relationship between two matrices,and finally applies a specific example of the rank one perturbation.In chapter three,we theoretically give the feasibility of computing any nonsingular TN matrix and nonsingular TNP matrix to obtain all singular values of TN×TNP matrix.Then,according to the bidiagonal decomposition of TN matrix,the parameterization of TN×TNP matrix corresponding to a special nonsingular TN matrix is considered.Finally,a high relative accuracy algorithms for TN×TNP matrix is designed.In chapter four,we take the special TN×TNP matrix and random matrix as examples,and complete the numerical experiment of computing all singular values for them to verify the high relative accuracy of the algorithms.