Perturbation Theory for the Solution of Systems of Linear Equations

Abstract

Abstract : We present expressions for absolute and relative errors in individual components of the solution to systems of linear equations. We consider three kinds of linear systems: non-singular, underdetermined of full row rank, and least squares of full column rank. No assumptions regarding the structure or distribution of the perturbations are required. Our expressions for component- wise relative errors allow the following conclusions: For any linear system there is at least one solution component whose sensitivity to perturbations is proportional to the condition number of the matrix; but - depending on the relation between right-hand side and matrix - there may exist components that are much better conditioned. For a least squares problem, the sensitivity of the components also depends on the right-hand side and may be as high as the square of the condition number. Least squares problems are therefore always more receptive to ill-conditioning than linear systems. In addition, we show that the component-wise relative errors for linear systems are reduced by column scaling only if column scaling manages to reduce the perturbations. Regarding underdetermined linear systems of full column rank, the problem of finding the minimal-norm solution can be formulated so that the same analysis as for least squares problems is applicable here as well. Finally, we define component-wise condition numbers that measure the sensitivity of the solution components to perturbations. They have simple geometric interpretations and can be command estimated as efficiently as the conventional condition numbers.