_:N600fe81952b44c77a1d0e134f07e4645 .
_:N600fe81952b44c77a1d0e134f07e4645 "Springer US" .
_:N44b536706ca4410ca1c569dda3b95d07 .
_:N4b90a2a15ef34c95a7ba3c256ad69923 "Advances in Nonlinear Programming" .
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_:N4b90a2a15ef34c95a7ba3c256ad69923 .
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"2019-04-16T09:43" .
"1998" .
_:N44b536706ca4410ca1c569dda3b95d07 _:N1020c6fcb2074534a6cf737ea273ab9b .
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_:N4402f57ab5834f19a915f9979366e90c "10.1007/978-1-4613-3335-7_1" .
_:N4b90a2a15ef34c95a7ba3c256ad69923 "978-1-4613-3337-1" .
_:N4402f57ab5834f19a915f9979366e90c .
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"Powell" .
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.
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_:N600fe81952b44c77a1d0e134f07e4645 .
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"Numerical and Computational Mathematics" .
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_:N1020c6fcb2074534a6cf737ea273ab9b "Ya-xiang" .
_:N19c51eecbe0747f8aedf3697afff4b98 "Springer Nature - SN SciGraph project" .
_:N1020c6fcb2074534a6cf737ea273ab9b .
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_:N136916345815433daf507210b08febac .
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_:Nebd31c32484a4da2bc2eaff594cfe82f .
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_:Nebd31c32484a4da2bc2eaff594cfe82f "pub.1033777821" .
_:N80fa61c41d0d43e19696e11c9b95f3c8 .
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"M. J. D." .
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_:N1020c6fcb2074534a6cf737ea273ab9b "Yuan" .
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"The use of Band Matrices for Second Derivative Approximations in Trust Region Algorithms" .
_:N600fe81952b44c77a1d0e134f07e4645 "Boston, MA" .
_:N19c51eecbe0747f8aedf3697afff4b98 .
"3-28" .
_:N4b90a2a15ef34c95a7ba3c256ad69923 "978-1-4613-3335-7" .
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"Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, CB3 9EW, Cambridge, England" .
"en" .
_:N80fa61c41d0d43e19696e11c9b95f3c8 .
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"Mathematical Sciences" .
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"https://scigraph.springernature.com/explorer/license/" .
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"In many trust region algorithms for optimization calculations, each iteration seeks a vectord \u2208 Rn that solves the linear system of equations (B +\u03BBI)d = -g, whereB is a symmetric estimate of a second derivative matrix, I is the unit matrix,g is a known gradient vector, and \u03BB is a parameter that controls the length ofd. Several values of \u03BB may be tried on each iteration, and, when there is no helpful sparsity inB, it is usual for each solution to require O(n3) operations. However, if an orthogonal matrix \u03A9, is available such thatM =\u03A9TB\u03A9 is an nxn matrix of bandwidth 2s+1, then\u03A9Td can be calculated in onlyO(ns2) operations for each new \u03BB, by writing the system in the form (M+\u03BBI)(\u03A9Td) = \u2014 \u03A9Tg. We find, unfortunately, that the construction ofM and \u03A9 fromB is usually more expensive than the solution of the original system, but in variable metric and quasi-Newton algorithms for unconstrained optimization, each iteration changesB by a matrix whose rank is at most two, and then updating techniques can be applied to \u03A9. Thus it is possible to reduce the average work per iteration fromO(n3) to O(n7/3) operations. Here the elements of each orthogonal matrix are calculated explicitly, but instead one can express the orthogonal matrix updates as products of Givens rotations, which allows the average work per iteration to be only O(n11/5) operations. Details of procedures that achieve these savings are described, and the O(n7/3) complexity is confirmed by numerical results." .
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_:N19c51eecbe0747f8aedf3697afff4b98 .
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"https://link.springer.com/10.1007%2F978-1-4613-3335-7_1" .