Applications

Abstract

In calculating an eigenvalue numerically, it is desirable to find definite upper and lower bounds for it rather than only an approximate value. There are various methods for calculating such bounds, most of which operate somewhat as follows: A formula or process is given which when applied to any trial vector yields two numbers which enclose at least one eigenvalue between them. The interval between the numbers comes out larger or smaller depending on how close the trial vector happens to be to an eigenvector. The trick, of course, is to choose the trial vector so that the interval is as small as possible. Supposing this is done, one still does not know which eigenvalue (according to its relative position in the spectrum) has been enclosed. This information can be very important. In the problem of finding the critical speeds of a rotating shaft, for example, the neighborhoods of the eigenvalues are danger regions. Here, it is not so important to know where the eigenvalues are as to know where they aren’t! It is therefore important to know which ones have been found and which, if any, have been missed.