The Modified Splitting Strategy for the Parallel Multisection Algorithm

Based on Cuppen’s divide-and-conquer method [4] and bisection and multisection with the Sturm sequence evaluation coupled with inverse iteration [12, 8, 1], numerous parallel algorithms for solving the eigenvalue problem of real symmetric tridiagonal matrices Tn are presented in the literature [5, 3, 10, 7]. While Cuppen’s divide-and-conquer method achieves the better accuracy results in terms of residual ‖TnX−ΛX‖ and orthogonality ‖X X−I‖ [7], bisection and multisection achieve the better absolute error results for the eigenvalue evaluation according to the numerical results in [9]. When they are implemented on the distributed-memory hypercube multiprocessor, bisection and multisection coupled with inverse iteration are significantly faster than Cuppen’s method in finding all eigenvalues and eigenvectors of Tn [7]. Furthermore, with bisection and multisection, eigenvalues of Tn can be computed without evaluating eigenvectors. Therefore, when only eigenvalues are required to be computed for Tn, bisection and multisection are the methods of choice. The multisection method is the generalization of the bisection method adopted by some parallel algorithms for the eigenvalue problem of symmetric tridiagonal matrices Tn. For the convenience of exposition, we use Pm and Pb to denote the parallel multisection algorithm and the parallel bisection algorithm respectively in the discussion below. The execution of Pm consists of the two phases, the isolation phase and the extraction phase. In the isolation phase, all processors work together to split the initial interval that contains all eigenvalues of Tn into subintervals such that each subinterval contains only one eigenvalue or one cluster of close eigenvalues of Tn. In the extraction phase, all processors extract eigenvalues from their isolating intervals by recursively performing bisection on the intervals or performing some multisecting operation on the intervals depending on whether those intervals contain one eigenvalue or one cluster of close eigenvalues. When Pm is implemented on the distributed-memory multiprocessor, as shown in [7], a static task scheduling algorithm will be used to allocate subintervals produced by the isolation phase to all processors to achieve the load balance among processors. As shown by the numerical results in [7], when implemented on the distributed-memory hypercube multiprocessor, Pm is slower than Pb in solving eigenvalue problem for the symmetric tridiagonal matrices with the eigenvalue spectra of uniform distribution, especially when the number of processors is large. This fact is attributed to the communicational and computational overhead of Pm’s implementation. However, compared with Pb, Pm has the following advantages. First,