Accurate Computations Of Said-Ball-Vandermonde-like Totally Nonpositive Matrices And Inverse Totally Nonpositive Matrices

Sign regular structure matrices are widely used in the field of statistics,sociology,economics and computer assistance,so they have attracted many attentions of scholars and the research of sign regular structure matrix has also emerged a lot of results,while the results of the research about totally nonpositive matrices and inverse totally nonpositive matrices are not numerous.In this text,we mainly do the research for highly accurate computing eigenvalues and singular values of two kinds of special structure matrices which are respectively called Said-Ball-Vandermonde-like totally nonpositive matrices and Inverse Said-Ball-Vandermonde-like totally nonpositive matrices.This article presents a class of the matrices for the first time with distinctive structural features and characteristics.In this text,we solve the eigenvalues problems and linear systems through the parameters of the matrix by programming high relative accuracy algorithms.In the end,some numerical experiments are displayed to show the excellent results of the algorithms and our outcome..In chapter one,we introduce some research background of the totally nonpositive matrices,and some nations and definitions are given to help understand the passage.In addition,we present some methods and theories that will be used later at the end of chapter.·In chapter two,we use the Neville Elimination to get the bi-diagonal decomposition of the Said-Ball-Vandermonde-like matrix A.Then the sub-diagonal non-zero elements of each bi-diagonal matrices in the decomposition form the parameters who are determined by the quotient of some minors of A,resulting no cancellation between two calculated values.The quotient is determined so as to avoid the situation where the calculated values are destructive.We then use these high-precision parameters to prove that A is a totally nonpositive matrix and design a high relative accuracy algorithms to compute the matrix's eigenvalue problems and the linear system problems.Finally,some numerical experiments are presented to verify the effectiveness of the algorithms and the results.·In chapter three,we give the inverse totally nonpositive matrix B and its bi-diagonal decomposition.After reparameterization of B,we use the parameters to prove the matrix is an inverse totally nonpositive matrix.Noticed that the parameters of B are the quotients in which there is no cancellation.so the parameters who will be used to program high-precision algorithms for solving our problems,are computed to high relative accuracy.In the end of this chapter,there are some numerical examples to illustrate the high accuracy of the algorithms and the results.