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- W2011193026 abstract "where F : R” R” and F is differentiable with Jacobian matrix J(Z). The algorithms proposed in [21 are applicable to message-passing multiprocessor computers where each processor has local memory and there is no shared (global) memory; Coleman and Li ]2] discussed implementation details and provided results of numerical experiments obtained on an Intel hypercube computer (IPSC). The algorithms discussed in 12; are global algorithms based on the trust region/dogleg idea first proposed by Powell .6i and then refined and implemented in Minpack [5]. In this note we analyze the local behaviour of one of the methods proposed in ;2;: the multi-secant method. This method can be implemented on any multiprocessor topology but is most natural on a ring of processors. We assume that there are p processors, or nodes, labelled PO, PI, . . . . Pp-l such that Pi is connected to Pi+1 crn,,d p), for i = 0 : p 1. Further we assume that n 1 p and that each processor has enough local memory to store roughly n/p columns of the Jacobian approximation B. (Of course in practise we must be able to store a factorization of the matrix B but we ignore such details here see -21 .) We assume the columns of B have been partitioned amongst the p nodes: define 1(j) to be th e index set of columns of B stored on node j. Another major assumption behind the multi-secant method is that the evaluation of F(z) at any point z is not a distributed computation. Specifically, we assume that every node has a copy of the subroutine that evaluates F: this subroutine is sequential (Coleman and Li 121 also considered algorithms for the case when F(z) can be evaluated in a distributed parallel manner) and F( ) z can be evaluated by any node (given Z) without requiring further communication with other nodes. A high level description of the multi-secant algorithm is illustrated in Figure 1. The implementation (including globalization) is discussed in :2]; here we are concerned only with the asymptotic analysis concerning the multi-secant update. As mentioned in Figure 1, once s is determined F is evaluated at p points concurrently. Specifically, each node evaluates F at a different point. Node 0 evaluates" @default.
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- W2011193026 date "1988-01-01" @default.
- W2011193026 modified "2023-10-18" @default.
- W2011193026 title "Local convergence of the multi-secant method for the parallel solution of systems of nonlinear equations" @default.
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- W2011193026 doi "https://doi.org/10.1016/0893-9659(88)90059-6" @default.
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