Exploiting the sparsity in the solution of linear ordinary differential equations

AbstractLarge systems of linear ordinary differential equations appear in a natural way in many fields of science and engineering. The application of sparse matrix technique is a very useful option in a package for solving such systems numerically. Such an option, the code SPAR1, is described. Several criteria for comparing the efficiency of the code SPAR1 with the efficiency of the corresponding option, the code DENS1, in which the same integration method is used but the sparsity is not exploited, are formulated. Numerical examples indicate that in many cases it is better to combine the use of a sparse matrix technique with “dropping” small elements in the course of computations in order to preserve the sparsity of the Jacobian matrix. Moreover, it is necessary to develop a device by which all small elements are automatically dropped by the code. The consecutive phases in the implementation of the above ideas in code SPAR1 are discussed. The efficiency achieved at each phase is illustrated by the use of two representative test-examples arising in the modelling of physical phenomena in nuclear magnetic resonance spectroscopy. It is demonstrated that the dropping of small elements is the only option that could be used in practice when some classes of large and stringent problems are to be solved