Comparison of the simulated final dipoles of the ARs and those predicted using the algebraic methods. Each diamond represents an AR during CRs 2145-2159. The horizontal x axis represents the final dipole produced by SFT simulations. The vertical y axis represents the final dipole predicted by algebraic methods. Panel a: results of the generalized method that we introduce; panel b: results of the BMR-based method. The diagonal line represents y = x. The gray shaded area represents the region where the final dipole produced by the algebraic methods shows no greater than 0.009G deviation from the result produced by SFT simulations.