| At present,we are living in the information age of data development,ill conditioned algebraic systems are very common in mathematical research.The solution of ill-condition system exists but is not unique,and the trusted computing of its specific solution is a difficult problem.Therefore,the exploration and research of ill conditioned algebraic system is of great significance.In this paper,we combine symbolic numerical mixed computation with interval computation to establish a credible verification model for the solution of ill conditioned algebraic systems.Afterwards,we design an algorithm to solve the corresponding problem,which provides algorithmic guarantee for the application of trusted verification algorithm in ill-condition problems.The main research contents are as follows:(1)Given an underdetermined linear system with ill conditioned coefficient matrices,the algorithm is designed to output its slightly perturbed interval system.The algorithm ensures that there is an underdetermined linear system with accurate rank-deficient coefficient matrices in the interval system.The algorithm outputs the high-precision approximate solution of the minimum 2-norm solution and its corresponding credible error bound of the underdetermined linear system with accurate rank-deficient coefficient matrices.(2)Given an overdetermined linear system with ill conditioned coefficient matrices,the algorithm is designed to output its slightly perturbed interval system.The algorithm ensures that there is an overdetermined linear system with accurate rank-deficient coefficient matrices in the interval system.The algorithm outputs the high-precision approximate solution of the least square solution and its corresponding credible error bound of the overdetermined linear system with accurate rank-deficient coefficient matrices.(3)Given a square linear system with ill conditioned coefficient matrices Ax=b,we design an algorithm to calculate an interval linear system near it,so that the interval system contains a real linear system(?)x=(?),(?) is the nearest rank deficient matrix to A in the sense of Frobenius norm.At the same time,the algorithm outputs an interval vector,which contains the minimum 2-norm solution of the system(?)x=(?).(4)Given an underdetermined nonlinear system and an approximate solution,if the numerical rank of the Jacobian matrix of the nonlinear system at the given approximate solution is deficient,we design an algorithm to output a small perturbation interval nonlinear system and an interval vector of the nonlinear system,In that case,there is a real nonlinear system in the interval nonlinear system,and the exact solution of the real nonlinear system exists in the output interval vector. |
