Complexity issues on designing tridiagonal solvers on 2-dimensional mesh interconnection networks

We consider the problem of designing optimal and efficient algorithms for solving tridiagonal linear systems with multiple right-hand side vectors on two-dimensional mesh interconnection networks. We derive asymptotic upper and lower bounds for these solvers using odd-even cyclic reduction. We present various important lower bounds on execution time for solving these systems including general lower bounds which are independent of initial data assignment, and lower bounds based on classifications of initial data assignments which classify assignments via the proportion of initial data assigned amongst processors. Finally, different algorithms are designed in order to achieve running times that are within a small constant factor of the lower bounds provided.