A guaranteed-convergence algorithm for coupled leaf photosynthesis–transpiration–stomatal conductance models

The photosynthesis-transpiration-stomatal conductance (An-E-gs) model framework is widely used for estimating photosynthesis, transpiration, and stomatal conductance in plants. The model equations are solved by numerical iteration, and the converged model values are deemed the solution. However, there has been no general guarantee that the iterative procedure converges to a solution or that the procedure leads to convergence. Building on the recent proof of the existence of a unique set of solutions, we herewith propose a numerical algorithm that is guaranteed to converge to the solution for the An-E-gs model framework. We first analytically prove that the proposed algorithm necessarily converges to a solution. We then demonstrate the convergence across contrasting combinations of leaf temperature, relative humidity, light, atmospheric CO2, and wind speed. We further demonstrate rapid convergence with the algorithm: no more than ca. 10 iterations for approximately 10-3 mol CO2 m-2 s-1 precision in net photosynthesis and no more than ca. 20 iterations for 10-7 mol CO2 m-2 s-1 precision. By guaranteeing convergence to the solution, this algorithm eliminates concerns about nonconvergence in leaf gas-exchange calculations and is expected to serve as a robust foundation for a range of studies from leaf-level gas exchange to global-scale carbon and water cycle dynamics.