A general iterative sparse linear solver and its parallelization for interval Newton methods

Interval Newton/Generalized Bisection methods reliably find all numerical solutions within a given domain. Both computational complexity analysis and numerical experiment s have shown that solving the corresponding interval linear system generated by interval Newton's methods can be computationally expensive (especially when the nonlinear system is large). In applications, many large-scale nonlinear systems of equations result in sparse interval Jacobian matrices. In this paper, we first profx)se a general indexed storage scheme to store sparse interval matrices We then present an iterative interval linear solver that utilizes the proposed index storage scheme It is expected that the newly proposed general interval iterative sparse linear solver will improve the overall performance for interval Newton/Generalized bisection methods when the Jacobian matrices are sparse. In Section 1, we briefly review interval Newton's methods. In Section 2, we review some currently used storage schemes for sparse systems. In Section 3, we introduce a new index scheme to store general sparse matrices. In Section 4, we present both sequential and parallel algorithms to evaluate a general sparse Jacobian matrix. In Section 5, we present both .~xluential and parallel algorithms to solve the corresponding interval linear system by the all-row preconditioned scheme. Condusions and future work are discussed in Section 6.