M. J. D. Powell Boston, MA Springer US Numerical and Computational Mathematics 3-28 The use of Band Matrices for Second Derivative Approximations in Trust Region Algorithms chapter chapters 2019-04-16T09:43 https://link.springer.com/10.1007%2F978-1-4613-3335-7_1 en https://scigraph.springernature.com/explorer/license/ 1998-01-01 1998 In many trust region algorithms for optimization calculations, each iteration seeks a vectord ∈ Rn that solves the linear system of equations (B +λI)d = -g, whereB is a symmetric estimate of a second derivative matrix, I is the unit matrix,g is a known gradient vector, and λ is a parameter that controls the length ofd. Several values of λ 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 Ω, is available such thatM =ΩTBΩ is an nxn matrix of bandwidth 2s+1, thenΩTd can be calculated in onlyO(ns2) operations for each new λ, by writing the system in the form (M+λI)(ΩTd) = — ΩTg. We find, unfortunately, that the construction ofM and Ω 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 Ω. 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. true Mathematical Sciences Yuan Ya-xiang Springer Nature - SN SciGraph project Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, CB3 9EW, Cambridge, England University of Cambridge 978-1-4613-3337-1 Advances in Nonlinear Programming 978-1-4613-3335-7 pub.1033777821 dimensions_id doi 10.1007/978-1-4613-3335-7_1 readcube_id a0e7c6594a1ed4d4b32a22506a804da845a846b4df37f8ef1fbb4bf6554ef115