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By M. Bocher

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Extra info for An Introduction to the Study of Integral Equations

Example text

6). Thus the factorization is complete in O(«/2) stages. At the kih stage, n — 2k 2 x 2 systems need to be solved to determine the w 's at that stage, and these 2 x 2 systems can be solved in parallel. Also, an n—2kXn-2k submatrix of A needs to be updated and these calculations can also be done in parallel. Hence, the degree of parallelism at the kth stage is O(n-2k) and the overall average degree of parallelism is O(n/2). To complete the solution of A\ = b, we then need to solve the systems Wy = b and Zx = y.

This synchronization can be carried out in a number of ways but, in essence, it requires that each processor wait after completion of its part of the computation until all processors have completed their work and the next iteration can begin. This adds two forms of overhead to the computation: one is the work required to verify that every processor is ready for the next iteration, the other is the idle time that some processors may experience while waiting for all processors to complete their tasks.

10) may be evaluated using vector operations on a computer like the Cray or the Cyber 200. 10) is completed, the resulting system under a reordering is again tridiagonal but only half as large. Thus the process may be continued for k steps until, in the case that n = 2* — 1, only one equation remains; then all of the unknowns are recovered in a back substitution process. The details of these observations are given in Lambiotte and Voigt [1975], where it is also shown that cyclic reduction requires O{n) operations and is thus consistent.

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