  We demonstrate an efficient method to compute the
strong-gravitational-lensing deflection angle and magnification for any
elliptical surface-density profile. This method solves a numerical hurdle in
lens modelling that has lacked a general solution for nearly three decades. The
hurdle emerges because it is prohibitive to derive analytic expressions of the
lensing quantities for most elliptical mass profiles. In our method, we first
decompose an elliptical mass profile into Gaussian components. We introduce an
integral transform that provides us with a fast and accurate algorithm for the
Gaussian decomposition. We derive analytic expressions of the lensing
quantities for a Gaussian component. As a result, we can compute these
quantities for the total mass profile by adding up the contributions from the
individual components. This lensing analysis self-consistently completes the
kinematic description in terms of Gaussian components presented by Cappellari
(2008). Our method is general without extra computational burden unlike other
methods currently in use.
