  The solar dipole moment at cycle minimum is considered to be the most
successful precursor for the amplitude of the subsequent cycle. Numerical
simulations of the surface flux transport (SFT) model are widely used to
effectively predict the dipole moment at cycle minimum. Recently an algebraic
method has been proposed to quickly predict the contribution of an active
region (AR) to the axial dipole moment at cycle minimum instead of SFT
simulations. However, the method assumes a bipolar magnetic region (BMR)
configuration of ARs. Actually most ARs are asymmetric in configuration of
opposite polarities, or have more complex configurations. Such ARs evolve
significantly differently from that of BMR approximations. We propose a
generalized algebraic method to describe the axial dipole contribution of an AR
with an arbitrary configuration, and evaluate its effectiveness compared to the
BMR-based method. We employ mathematical deductions to obtain the generalized
method. We compare the results of the generalized method with SFT simulations
of observed ARs, artificially created BMRs, and ARs with more complex
configurations. We also compare the results with that from the BMR-based
method. The generalized method is equivalent to the SFT model, and precisely
predicts the ARs' contributions to the dipole moment. The method has a much
higher computational efficiency than SFT simulations. Although the BMR-based
method has similar computational efficiency as the generalized method, it is
only accurate for symmetric bipolar ARs. The BMR-based method systematically
overestimates the dipole contributions of asymmetric bipolar ARs, and randomly
miscalculate the contributions of more complex ARs. The generalized method
provides a quick and precise quantification of an AR's contribution to the
solar cycle evolution, which paves the way for the application into the
physics-based solar cycle prediction.
