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Abstract : |
Abstract. Existing definitions of backward error and condition number for linear systems do not cater for structure in the coefficient matrix, except possibly for sparsity. We extend the definitions so that when the coefficient matrix has structure the perturbed matrix has this structure too. We show that when the structure comprises linear dependence on a set of parameters the structured componentwise backward error is given by the solution of minimal 1-norm to an underdetermined linear system; we also obtain an explicit expression for the condition number in this linear case. Applications to symmetric matrices, Toeplitz matrices and the least squares problem are discussed and illustrated through numerical examples., |
