Parallel solutions for large-scale general sparse nonlinear systems of equations

Abstract

In solving application problems, many large-scale nonlinear systems of equations result in sparse Jacobian matrices. Such nonlinear systems are called sparse nonlinear systems. The irregularity of the locations of nonzero elements of a general sparse matrix makes it very difficult to generally map sparse matrix computations to multiprocessors for parallel processing in a well balanced manner. To overcome this difficulty, we define a new storage scheme for general sparse matrices in this paper. With the new storage scheme, we develop parallel algorithms to solve large-scale general sparse systems of equations by interval Newton/Generalized bisection methods which reliably find all numerical solutions within a given domain.

In Section 1, we provide an introduction to the addressed problem and the interval Newton’s methods. In Section 2, some currently used storage schemes for sparse systems are reviewed. In Section 3, new index schemes to store general sparse matrices are reported. In Section 4, we present a parallel algorithm to evaluate a general sparse Jacobian matrix. In Section 5, we present a parallel algorithm to solve the corresponding interval linear system by the all-row preconditioned scheme. Conclusions and future work are discussed in Section 6.