New Power Method Based On Interval Arithmetic For Interval Matrices

The interval computation has received much attention by many mathematicians and engineers since the interval analysis has introduced in the beginning of the sixties in the last century. Now it is widely used in science and engineering fields. Since every element is uncertain in an interval matrix, the interval computation is usually regarded as a very complex and hard problem. It is more difficult to apply the interval analysis in some methods with interval computation for solving some linear algebra problems, such as solving the eigenvalue problems with interval matrices. Up to date, no literature has been found in this field. Due to its hardness, existed computational methods we have found involve generalized Disk Theorem, some Perturbation Methods, Spectral Radius Method and so on, which combine certain theoretical conclusions with some computation for quantitative matrices, and then derive some inequalities for getting eigenvalue bounds of interval matrices. In fact, these obtained eigenvalue bounds are usually outer bounds for the true eigenvalue ranges. On the other hands, these methods also have many restrictions for interval matrices which limit their applications.Instead of these existed methods, in this paper, we apply the ordinary power method to interval matrices by introducing interval arithmetic, and take the different unit divisors in the separate computations for upper and lower bounds to the maximum interval eigenvalue of an interval matrix. Furthermore, we introduce some new definitions for the subtraction rule of the interval arithmetic, and apply these definitions into the power method. Many numerical examples show that these improved steps in the interval power method can not only enlarge the restrictions for interval matrices but also produce much better numerical results.