【文章內(nèi)容簡(jiǎn)介】 nodes any more, start with step b)。 b) Number the node so that no equivalent branches will be introduced when this node is eliminated. If more than one node meets this criterion, number the one with the smallest original number. If we can not start with step a) or step b), turn to step c)。 c) Number the node so that the fewest branches will be introduced when this node is eliminated. If not only node could introduce fewest branches, number the one with the largest degree. Once certain node is numbered in the step above, update the degree of relevant nodes and topological information. Until all the nodes are numbered, the process of node numbering ends up. TABLE I. REORDERED NODES USING SCHEME ONE Following the steps of scheme I, the sequence of the node numbered for the 6node work is given in table I. No fillin will be introduced during the procedure of solving the linear equation, so the table of factors and the Jacobian matrix will have pletely identical structure. So the memory requirement for the table of factors is 外文翻譯（原 文） 8 , which is the same with that for the Jacobian matrix. Normally, an acceptable solution can be obtained in four or five iterations by NewtonRaphson method. While, the number of iterations required for this example is thirtythree because of the illconditioned caused by the small impedance branch. 123 multiply operations will be performed during forward substitution and backward substitution for each iteration, and 7456 multiply operations will be performed throughout the whole process of solving. B. Puropse 2: Improving Accuracy Using Complete Pivoting Considering that plete pivoting is numerically preferable to partial pivoting, in this section plete pivoting is adopted to improve accuracy of the solution of the linear equations, aiming at reducing the number of iterations. Here nodes relate to large determinant of the diagonal submatrices intend to be arrange in front. To some extern, the modulus of the entries on the main diagonal of the admittance matrix could indicate the magnitude of the determinant of the submatrices on the main diagonal of the Jacobian matrix. For convenience, we make use of admittance matrix to determine the order of numbers. Scheme II a) Form the nodal admittance matrix。 b) Factorize the nodal admittance matrix with plete pivoting. Record the changes on the position of the nodes。 c) Determine the new number of the node according to the positong of node in the end of the factorization。 TABLE II. REORDERED NODES USING SCHEME TWO 外文翻譯（原 文） 9 Executing scheme II, plete pivoting might automatic performed without row and column exchanges. The module of entries on main diagonal corresponding to such node may bee larger by summing more branch parameter, as a result, the nodes, degree of which is larger, tend to be numbered first. So the results of such scheme may depart form the principle of node numbering guided by sparse matrix methods and many fillins might be introduced. The sequence of the node numbered for 6node work is list in table II. Six fillins will be produced, so more memory () and more operations (321 multiply operations) are spent in the procedure of forward and backward substitution during once iteration. The total number of iterations required reduces to thirteen, which suggests that the calculation accuracy for linear equations could be raised by plete pivoting. Finally, the number of multiply operations reduces to 5573 thanks to smaller number of iterations. C. Puropse 3: Improving Accuracy while preserving the sparsity Only one small impedance branch exists in the system, so only four entries (submatrices) corresponding to node 4 and node 6 are very large in admittance matrix (Jacobin matrix). During the process of forward substitution, once node 4 or node 6 is elimination, the submatrix prised of rest elements could keep good numerical stability and numbering of rest nodes would not make a difference to the accuracy of the solution. To take both accuracy and sparsity into account, we numbered node 4 first, then numbered other nodes following the method used for purpose 1. That is what we called scheme III for the 6node work. The sequence of the node numbered for the 6node work is given in table III. 外文翻譯（原 文） 10 Since only one small impedance branch exists in the system and it connects to node 4, the degree of which is one. Scheme III will meet the request of purpose 1. So the number of fillins, memory requirements and operations needed for factorization are all the same with scheme I. Only nine iterations will be needed to insure the convergence, result in a large save of calculation (only 2107 multiply operations). The reduction on the number of iterations indicates that more exact solutions for the linear equations could be got using scheme III. After analysis and parison, the reasons are as follows: ? The diagonal element related to node 4 is just a little smaller than the one related to node 6, so eliminate node 4 first will not decrease accuracy. The scheme could meet plete pivoting approximately. ? Fewer operations in scheme III reduce the rounding error of calculator floatingpoint numbers. Especially, if eliminate node 6 first, very small value might be added to diagonal element of node 2 and node 5, which would cause serious rounding error. While, if eliminate node 4 first, a sizable value will be added to diagonal element of node 6, producing a valu