A method for the automatic correction of third-order aberrations in thick lenses has been tried on several lens problems. The procedure is to use the Newton method for solving the nonlinear equations involved. The derivatives are found by differentiating the third-order surface contribution formulas. The method has proven to be successful on a large variety of lens types.
R. E. HOPKINS, C. A. MCCARTHY, and R. WALTERS, "Automatic Correction of Third-Order Aberrations," J. Opt. Soc. Am. 45, 363-363 (1955)