In practice, Equation ( 48 ) is solved in a least squares sense: a number of angles, R, are chosen at which the Yc and Ys terms are evaluated from Equations ( 50 ) and ( 51 ). The number of multipoles, M, is chosen such that M < R, and the f2m terms are evaluated, according to Equation ( 53 ), for each of the multipoles at each of the angles. Thus there are more linear equations than unknowns. It may be shown that the least squares solution to this system of equations may be expressed, in matrix form, as in Equation ( 50 ). This system may then easily be solved by Gauss elimination or any other matrix solving method, such as Gauss Seidel or SOR (successive over-relaxation).
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ATAx = AT b |
( 55 ) |