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In the paper, an efficient and reliable algorithm for solving the circuit nonlinear algebraic-differential equations based on a sophisticated arrangement of Newton interpolation polynomial is characterized first. After that, a novel method is introduced for improving the convergence with four suggested criteria that are being compared. Unlike the similar algorithms focused on an operating point analysis only, the proposed method also works in a transient analysis. 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 this fundamental proposal of the paper, a comprehensive set of numerical tests has been carried out.