| The indefinite least squares(ILS) problem comes from the total least squares problem and optimization problem areas. Under the premise that the ILS problem has a unique solution, many experts and scholars have given many algorithms for solving ILS problem.Backward error analysis can determine whether the approximate solution of the ILS problem is backward stable or not, so we study the estimate of the backward error bounds for the ILS problem.Firstly, we have launched a series of theorems. Using these theorems, we get a subsetΘILS+of backward perturbation matrix set for the ILS problem. Besides, the set ΘILS+contains a positive condition, so the smallest norm is difficult to calculate, so we can consider the set Θ, which does not contain the positive condition. If the optimal matrix E* which makes the smallest norm η(y) in Θ meets the positive condition, then η(y) can be seen as a backward error bound of the ILS problem. When the problem degenerates into the least squares problem backward error, the error and the corresponding optimal perturbation matrix will degenerate into the least squares problem backward error and the corresponding optimal perturbation matrix. Finally, based on the singular value decomposition of matrix J A, we find a upper bound of η(y). We can see that the computation of η(y) is less than η(y)’. Backward error bounds can be used to determine whether the method of calculation for solving the ILS problem is backward stable or not, but also can be used as termination criterion which solves the ILS problem. |
