Re-Parametrization And High-Accuracy Computations Of Several Classes Of Structured Matrices

A matrix is called totally nonpositive if all its minors are nonpositive.A matrix is called almost strictly totally negative if all its nontrivial minors are negative.A matrix whose inverse is totally nonpositive is called inverse totally nonpositive.The several classes of matrices with special structures are widely used in areas of probability,stochastic,numerical algebra and so on.In numerical linear algebra,a ideal goal is to show numerical computations with high relative accuracy.Starting with the researches of Demmel and Kahan on computing singular values of bidiagonal matrices accurately,a great deal of work has been done on the high relative accuracy algorithms of eigenvalues and singular values of some structural matrices.One idea which plays a crucial role in high relative accuracy computations is the need of re-parameterizing matrices.In this paper,our contributions are that we research re-parametrization and develop new algorithms to compute several classes of structured matrices with high relative accuracy.The paper is organized as follows:In chapter one,we introduce some theoretical knowledge and recent works of totally nonpositive matrices,almost strictly totally negative matrices and inverse totally nonpositive matrices;present some basic symbols and definitions in this paper;describe the process of Neville elimination method and basic definitions of Bernstein-Vandermonde matrices.In chapter two,we provide a re-parametrization of the class of almost strictly totally negative matrices in term of Tn2 independent parameters.An optimal test is derived to check if a given matrix is almost strictly totally negative.Moreover,a necessary and sufficient condition is established for accurately computing these parameters.In chapter three,we first provide a re-parametrization of the class of Bernstein-Vandermonde-like totally nonpositive matrices and compute all Neville-type param-eters with high relative accuracy.Then,we present new O(n3)algorithms for solving linear systems and for computing all the singular values of the class of matrices to high relative accuracy.Finally,numerical experiment is provided to confirm the high relative accuracy of our algorithm.In chapter four,we present a re-parametrization of the class of Generalized-Bernstein-Vandermonde inverse totally nonpositive matrices and compute all parameters with high relative accuracy.Then,we develop new O(n3)algorithms for computing all the singular values and eigenvalues of the class of matrices with high relative accuracy.Finally,numerical experiment is provided to confirm the high relative accuracy of our algorithm.