Exact Solution of Linear Equations Using Residue Arithmetic (Remark on Algorithm 406)
Submittal of an algorithm for consideration for publication in Communications of the ACM implies unrestricted use of the algorithm within a computer is permissible. General permission to republish, but not for profit, all or part of this material is granted provided that ACM's copyright notice is given and that reference is made to the publication, to its date of issue, and to the fact that reprinting privileges were granted by permission of the Association for Computing Machinery. The procedure Coulomb can be used very weU to generate the Coulomb wave functions FL and GL and their derivatives, needed in elastic scattering calculations in nuclear physics. When the procedure is used many times for many values of rho and eta, it is not only very useful but also necessary to have in each instance an indication about the accuracy of the results. It is obvious to use the Wronskian relations FL'GL-FLGL' ~ l for the purpose of checking the results, as Fr6berg [1] states after formula (3.4). However, one has to be very careful in using these relations. The most significant check is given later on, but first it is shown what can go wrong. K61big pointed out already in the certification that Lutz and Karvelis [2] failed to notice discrepancies exceeding 100 units in the sixth significant digit in their tables although they state "when all the functions are generated we test to see how closely the Wron-skian relation FL'GL-FLGL' = 1 is obeyed." The way Lutz and Karvelis generate the functions goes as follows. First they calculate Go and Go'; then they use recurrence relations to get GL and GL' for L > 0; and lastly them use backward recurrence This investigation was part of the research program of the "Stichting voor Fundamental Onderzoek der Materie (F.O.M.)," which is financially supported by the "Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (Z.W.O.)". relations together with the relation FoG~-GoF1 = (n 2 + I)-~ to get FL and FL' for all L. This last relation is in fact a di:::rent form of the Wronskian relation, see e.g. Fr6berg [1] formula (3.5). The use of the Wronskian relations to check the results now gives information only about the stability in the use of the recurrence relations, not about the accuracy of the Coulomb wave functions. As an independent check on the function values, the following procedure can be used. It is easy to …