Backward Error and Condition of Structured Linear Systems

Existing definitions of backward error and condition number for linear systems do not cater to structure in the coefficient matrix, except possibly for sparsity. The definitions are extended so that when the coefficient matrix has structure the perturbed matrix has this structure too. It is shown that when the structure comprises linear dependence on a set of parameters, the structured componentwise backward error is given by the solution of minimal $\infty $ -norm to an underdetermined linear system; an explicit expression for the condition number in this linear case is also obtained. Applications to symmetric matrices, Toeplitz matrices and the least squares problem are discussed and illustrated through numerical examples. Existing definitions of backward error and condition number for linear systems do not cater to structure in the coefficient matrix, except possibly for sparsity. The definitions are extended so that when the coefficient matrix has structure the perturbed matrix has this structure too. It is shown that when the structure comprises linear dependence on a set of parameters, the structured componentwise backward error is given by the solution of minimal $\infty $ -norm to an underdetermined linear system; an explicit expression for the condition number in this linear case is also obtained. Applications to symmetric matrices, Toeplitz matrices and the least squares problem are discussed and illustrated through numerical examples.