The Sensitivity Analysis Of The Rank-deficient And Ill-Conditioned Least Squares Solution

The analysis of the sensitivity of a least squares problem is one of the most challenging problems in scientific computing.We consider the general solution of the least squares problem in an affine Grassmannian such that it owns the property of uniqueness even for the rank-deficient least squares problem.The deviation between the initial general solution and the perturbed general solution will be derived if and only if they are truncated in the same affine Grassmannian.Furthermore,we also proved that the numerical solution approximates one of the exact solutions when the coefficient matrix is rank-deficient.While the coefficient matrix is ill-conditioned but has well-determined numerical rank,any exact solution is an approximation to one of the perturbed solutions.