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A note on direct method for approximations of sparse Hessian matrices. (English) Zbl 0658.65058
This note is a brief review of methods available for the estimation of sparse Hessian matrices by finite-differences. Using the graph-coloring analogy developed by the reviewer and J. J. Moré [Math. Program. 28, 243-270 (1984; Zbl 0572.65029)], a new ordering scheme (coloring scheme) is presented. Numerical results are provided.
Reviewer: T.F.Coleman
MSC:
65K05 Numerical mathematical programming methods
65D25 Numerical differentiation
65H10 Numerical computation of solutions to systems of equations
90C30 Nonlinear programming
Software:
symrcm
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References:
[1] T. F. Coleman J. J. Moré: Estimation of Sparse Hessian Matrices and Graph Coloring Problems. Math. Prog. 28 (1984), 243-270. · Zbl 0572.65029
[2] G. C. Everstine: A Comparison of Three Resequencing Algorithms for the Reduction of Matrix Profile and Wavefront. International Journal on Numerical Methods in Engineering 14 (1979), 837-853. · Zbl 0401.73082
[3] A. George J. W. H. Liu: Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall, Inc. Englewood Cliffs. N. J. 07632, 1981. · Zbl 0516.65010
[4] P. Hanzálek J. Hřebíček J. Kučera: A Conversational Program System for Mathematical Optimization. Computer Physics Communications 41 (1986), 403 - 408.
[5] D. M. Matula L. L. Beck: Smallest-last Ordering and Clustering and Graph Coloring Algorithms. JACM 30 (1983), 417-427. · Zbl 0628.68054
[6] S. T. McCormick: Optimal Approximation of Sparse Hessians and its Equivalence to a Graph Coloring Problem. Math. Prog. 26 (1983), 153-171. · Zbl 0507.65027
[7] M. J. D. Powell, Ph. L. Toint: On the Estimation of Sparse Hessian Matrices. SIAM on Num. Anal. 16 (1979), 1060-1074. · Zbl 0426.65025
[8] M. N. Thapa: Optimization of Unconstrained Functions with Sparse Hessian Matrices: Newton-type Methods. Math. Prog. 19 (1984), 156-186. · Zbl 0538.49023
[9] O. C. Zienkiewicz: The Finite Element Method. McGraw Hill, London, 1977. · Zbl 0435.73072
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