High-accuracy Computations Of Eigenvalues And Linear Systems Of Several Classes Of Vandermonde Matrices

Matrix theory is widely used in many fields such as approximation theory,differential or integral equations,combinatorics,economics,or biology,computer aided geometric design.Structured matrix is one of the focuses of many scholars,there are many research results about them.Bernstein-Vandermonde,q-BernsteinVandermonde and Cauchy-Vandermonde matrices are very classical examples in structured matrices,which are determined by the nodes of the basis.Previous studies established by scholars designed algorithms to calculate the singular values,eigenvalues and linear systems on the basis that the matrix is totally nonnegative.This paper will study these types of structured matrices and discuss the situation when their interpolation nodes change and are no longer totally nonnegative matrices.we discuss parameter matrices symbol characteristics,when the interpolation nodes are changed,design algorithm to accurately calculate eigenvalues of the matrix.Then we introduces the problem of bivariate interpolation,provide an algorithm to solve the linear system of bivariate interpolation based on Bernstein-Vandermondeclass basis and q-Bernstein-Vandermonde basis.Finally,some numerical examples illustrating the high relative accuracy are included.