The Eigenvalue Problem for an N - Sector Ring Keith

The problem of finding the eigenvalues and eigenvectors of a matrix referring to an N-sector ring can be reduced to a number of smaller eigenvalue problems in which the dimensions of the vector space are reduced by a factor N. Suppose we need to solve an eigenvalue problem for a matrix A associated with a storage ring, for example in analyzing orbit errors as in Ref.(l}. We number the N superperiods with an index t, running over the values 0, ±l, ±2, .. • ±(l-l), l, where l = N/2 = 20 for the APS. We assume N is even, as is virtually always the case; the case N odd can be handled without difficulty. It is convenient to let the index t run around the ring in both directions from sector 0 to the diametrically opposite sector l. Sums over t are understood to run over these values. All quantities are taken to be periodic in t, so that t =-l is equivalent to t = l, and t = l+l is equivalent to t =-(l-l). We wish to deal with a space of vectors x tm ' where the index m runs over a single superperiod. If, for example, x is the vertical magnet mis-alignments, then m runs over the magnets in a superperiod. If x is the horizontal orbit displacement from the reference orbit, then m may be a continuous index running from one end of the superperiod to the other, or may be a discrete index running over the places in which we are interested in the orbit. If x is the orbit error, then m may run over the beam position monitors. In the following discussion, we will take m to be an index running over the values 1, 2, • • • , M.