The tableau approach to automated network design optimization via implicit, variable order, variable time-step integration, and adjoint sensitivity computation is described. In this approach, the only matrix operation required is that of repeatedly solving linear algebraic equations of fixed sparsity structure. Required partial derivatives and numerical integration is done at the branch level leading to a simple input language, complete generality and maximum sparsity of the characteristic coefficient matrix. The bulk of computation and program complexity is thus located in the sparse matrix routines; described herein are the routines OPTORD and 1-2-3 GNSO. These routines account for variability type of the matrix elements in producing a machine code for solution of Ax=b in nested iterations for which a weighted sum of total operations count and round-off error incurred in the optimization is minimized. © 1971, IEEE. All rights reserved.