Efficient Optimal Image Reconstruction for the Nancy Grace Roman Space Telescope and Beyond: II. Implementation of {\sc Effortless}

Weak gravitational lensing is a promising but technically demanding cosmological probe. For space missions like the forthcoming Nancy Grace Roman Space Telescope, a major challenge is that native images are undersampled and need to be reconstructed to enable accurate measurements. {\sc Effortless} (EFFicient Optimal image ReconsTruction using LESS memory; previously known as Fast {\sc Imcom}) is a new algorithm designed for that purpose. My companion paper has exhibited promising first results to demonstrate that {\sc Effortless} can make point spread functions (PSFs) uniform and regular across reconstructed images more efficiently than its predecessor {\sc Imcom} and has the potential to outperform {\sc Imcom} in terms of control over systematic errors. In this paper, I present the mathematical formalism, software implementation, and practical issues in detail. Foremost, while the Nyquist--Shannon sampling theorem remains true, the conditions of the theorem are subtly (and importantly) different from the problem in survey data processing, and finite sampling effects can be substantially reduced via a simple post-measurement calibration. Imperfections caused by numerical artifacts, finiteness of input pixel windows, and unavailability of some input pixels are understood and under control. The {\sc Effortless} application programming interface is general and can support use cases beyond weak lensing cosmology and beyond the Roman Space Telescope.