Parameterization Sign Analyses And Accurate Computations Of Several Classes Of Structured Matrices

Structured matrices have important applications in many fields of modern life,such as economics,statistics,approximation theory,numerical computation and computer aided geometric design,etc.With the continuous development and progress of science and technology,the traditional numerical computations can not meet the needs of high relative accuracy in the theoretical and practical research of structured matrices.Therefore,it will be our research goal to achieve high relative accuracy for computations.Different from traditional numerical computations,high-accuracy computations can be effectively combined with high-efficiency computers to provide more accurate computation results under the premise of higher calculation speeds.In recent years,the study of high-accuracy computations is a research topic of great interest,and much work has been devoted to the study of high-accuracy algorithms for structured matrices.This paper focuses on studying parameterization sign analyses and accurate computations of several classes of structured matrices.The details are as follows:·A class of Cauchy-Polynomial-Vandermonde(CPV)matrices is introduced by the rational interpolation problem,and then the parameterization matrices of these matrices are accurately computed,so that all the eigenvalues of a product involving CPV matrices and other structured matrices are computed to high relative accuracy.Numerical experiments confirm the high relative accuracy.·By extending Cauchy-Vandermonde matrices,we introduce the class of quasi-Cauchy-Vandermonde(qCV)matrices belonging to the class of generalized sign regular matrix with signature(1,...,1,-1),which responds the problem pointed out by Koev and Dopico[85]:“In fact,we know of no efficient way to even generate sign regular matrices other than TN,TNJ or their negatives.Then,an algorithm is designed to accurately compute the parameterization matrix for qCV matrices,and all the eigenvalues of such matrices are computed to high relative accuracy.Error analysis and numerical experiments confirm the high relative accuracy of computed eigenvalues.·For the generalized Kronecker product(GKP)linear system that appears in the bivariate interpolation problem,we propose an error analysis for solving the GKP linear system associated with the consecutive-rank-descending(CRD)matrices,and obtain the "ideal"bound of componentwise forward errors.Moreover,the sign sequences of generalized Vandermonde(gV)matrices are proposed,which shows that the GKP linear system associated with gV matrices can be solved accurately with the desired componentwise forward error.Numerical experiments verify the error bounds and the high relative accuracy of the computed solutions.·We consider how to solve the least squares problem with linear equality constraints(LSE problem),which arises in many applications such as analysis of large scale structures,inequality constrained least squares problem,electromagnetic data processing,etc.Based on the parameterization matrices of CRD matrices and their sign regularities,we design a new algorithm to accurately solve the LSE problem associated with CRD matrices.Numerical experiments confirm the accuracy of computed solutions.