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A. Chakraborty, D.C.S. Allison, C.J. Ribbens, L.T. Watson, "The Parallel Complexity of Embedding Algorithms for the Solution of Systems of Nonlinear Equations," IEEE Transactions on Parallel and Distributed Systems, vol. 4, no. 4, pp. 458465, April, 1993.  
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@article{ 10.1109/71.219760, author = {A. Chakraborty and D.C.S. Allison and C.J. Ribbens and L.T. Watson}, title = {The Parallel Complexity of Embedding Algorithms for the Solution of Systems of Nonlinear Equations}, journal ={IEEE Transactions on Parallel and Distributed Systems}, volume = {4}, number = {4}, issn = {10459219}, year = {1993}, pages = {458465}, doi = {http://doi.ieeecomputersociety.org/10.1109/71.219760}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
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TY  JOUR JO  IEEE Transactions on Parallel and Distributed Systems TI  The Parallel Complexity of Embedding Algorithms for the Solution of Systems of Nonlinear Equations IS  4 SN  10459219 SP458 EP465 EPD  458465 A1  A. Chakraborty, A1  D.C.S. Allison, A1  C.J. Ribbens, A1  L.T. Watson, PY  1993 KW  Index Termsparallel complexity; embedding algorithms; nonlinear equations; Jacobian matrix;hypercube; static rectangular grid mapping; mathematical software package; dynamicassignment; computational complexity; nonlinear equations; parallel algorithms VL  4 JA  IEEE Transactions on Parallel and Distributed Systems ER   
Embedding algorithms used to solve nonlinear systems of equations do so by constructing a continuous family of systems and solving the given system by tracking the continuous curve of solutions to the family. Solving nonlinear equations by a globally convergent embedding algorithm requires the evaluation and factoring of a Jacobian matrix at many points along the embedding curve. Ways to optimize the Jacobian matrix on a hypercube are described. Several static and dynamical strategies for assigning components of the Jacobian to processors on the hypercube are investigated. It is found that a static rectangular grid mapping is the preferred choice for inclusion in a robust parallel mathematical software package. The static linear mapping is a viable alternative when there are many common subexpressions in the component evaluation, and the dynamic assignment strategy should only be considered when there is large variation in the evaluation times for the components, leading to a load imbalance on the processors.
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