Efficient procedure for solving circuit algebraic-differential equations with modified sparse LU factorization improving fill-in suppression

In the paper, an efficient and reliable algorithm for solving the circuit algebraic-differential equations is characterized first, which is based on a sophisticated arrangement of the Newton interpolation polynomial. For enhancing the efficiency of repeated solutions of linear systems necessary in the Newton-Raphson method, a novel modification of the Markowitz criterion is suggested, which is compatible with the fast modes of the LU factorization. The modified criterion consists in an estimation of probabilities of the fill-in enlargement. The probabilities are determined for all columns of the system matrix before the LU factorization, where the column probability is calculated as the average value of the probabilities for all the column elements. Finally, the columns are reordered so that first and last should be those with the minimum and maximum probabilities, respectively. As a verification of the proposed algorithm, a comprehensive set of numerical tests has been performed.