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Manuscript Title: A program for closed orbit minimization by analytic technique.
Authors: J.V. Trotman
Program title: COMBAT
Catalogue identifier: ABCF_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 5(1973)56
Programming language: Fortran.
Computer: IBM 360/195.
Operating system: OS (IBM 360).
RAM: 40K words
Word size: 32
Keywords: Nuclear physics, High energy, General purpose, Closed orbit, Minimization, Synchrotron, Apparatus design.
Classification: 4.9, 17.1.

Nature of problem:
Closed orbit errors resulting from guide field imperfections in a synchrotron, increase the magnet aperture required. It is impractical to eliminate all imperfections but it is possible to minimise the closed errors by a suitable arrangement of correcting elements and beam sensors. For minimisation one must calculate the correlation between the distorted closed orbit and the correcting elements.

Solution method:
The correlation between the distorted closed orbit and the correcting elements may be expressed as a set of linear simultaneous equations: c= [A] [C]**-1 y. c= necessary correction, y= closed orbit distortion vector. The matrix [A] is tri-diagonal, the coefficients being functions of the (2X2) linear transfer matrices of the machine components. The matrix [C] is bi-diagonal with leading diagonal terms equal to unity. [C]**-1 is readily obtained algebraically. In general the sub-diagonal terms are much less than unity and [C]**-1 may therefore be considered a correcting matrix. The solution of the matrix equation may be obtained to any order of accuracy desired: from the full analytic solution to the zeroth order approximation. The zeroth order approximation considers the matrix [C]**-1 = [I]. For the nth order approximation only nth order products are retained in [C]**-1.