2022-08-14T08:52:47Zhttps://ipsj.ixsq.nii.ac.jp/ej/?action=repository_oaipmhoai:ipsj.ixsq.nii.ac.jp:000597902017-03-31T05:36:57Z05471:05490:05494
Implicit Linear Multistep Methods with Nonnegative Coefficients for Solving Initial Value ProblemsImplicit Linear Multistep Methods with Nonnegative Coefficients for Solving Initial Value Problemsenghttp://id.nii.ac.jp/1001/00059790/Articlehttps://ipsj.ixsq.nii.ac.jp/ej/?action=repository_action_common_download&item_id=59790&item_no=1&attribute_id=1&file_no=1Copyright (c) 1989 by the Information Processing Society of JapanSendai National College of TechnologyKazufumi, OzawaConsider the linear multistep methods (LM methods)y_<n+k>+a_&k-1>y_<n;k-1>+...+a_0y_n=h(B_kf_<n+k>+...+B_0f_n) for solving initial value problems of ordinary differential equations. In many of the methods the signs of A's andB's are mixed. However the methods satisfying the conditions -A_j≧0 1 ... k-1 B_j0 j=0 1 ... k are preferable to others because these conditions prev.ent the cancellation of significant figures during the computations. In this paper we consider the existence of the implicit LM methods satisfying these conditions for each of the three types; these types consist of the Adams type the Milne type and the Radial type. It is found that the highest orders of such LM methods are 2 4 and 8 for the Adams for the Milne and for the Radial respectively. In particular the Adams type includes the A_o-and A-stable methods. For the Milne type it is found that the methods of order 3 4 are unstable and consequently only the one of order 2 is useful. For the Radial type of order from 3 to 5 the optimal parameters are obtained which minimize the round-off error propagations of the methods. The numerical example shows that these optimal methods are more accurate than the Adams-Moulton methods.Consider the linear multistep methods (LM methods)y_<n+k>+a_&k-1>y_<n;k-1>+...+a_0y_n=h(B_kf_<n+k>+...+B_0f_n) for solving initial value problems of ordinary differential equations. In many of the methods, the signs of A's andB's are mixed. However, the methods satisfying the conditions -A_j≧0,1,...,k-1,B_j0, j=0,1,...,k are preferable to others because these conditions prev.ent the cancellation of significant figures during the computations. In this paper, we consider the existence of the implicit LM methods satisfying these conditions, for each of the three types; these types consist of the Adams type, the Milne type, and the Radial type. It is found that the highest orders of such LM methods are 2, 4, and 8, for the Adams, for the Milne, and for the Radial, respectively. In particular, the Adams type includes the A_o-and A-stable methods. For the Milne type, it is found that the methods of order 3, 4 are unstable, and consequently only the one of order 2 is useful. For the Radial type of order from 3 to 5, the optimal parameters are obtained which minimize the round-off error propagations of the methods. The numerical example shows that these optimal methods are more accurate than the Adams-Moulton methods.AA00700121Journal of Information Processing 12142501989-03-311882-66522009-06-30