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＜電子ブック＞
Conjugate Direction Methods in Optimization / by M.R. Hestenes
(Stochastic Modelling and Applied Probability. ISSN:2197439X ; 12)

版	1st ed. 1980.
出版者	(New York, NY : Springer New York : Imprint: Springer)
出版年	1980
本文言語	英語
大きさ	X, 325 p : online resource
著者標目	*Hestenes, M.R author SpringerLink (Online service)
件　名	LCSH:System theory LCSH:Control theory LCSH:Mathematical optimization LCSH:Calculus of variations FREE:Systems Theory, Control FREE:Calculus of Variations and Optimization
一般注記	I Newton’s Method and the Gradient Method -- 1 Introduction -- 2 Fundamental Concepts -- 3 Iterative Methods for Solving g(x) = 0 -- 4 Convergence Theorems -- 5 Minimization of Functions by Newton’s Method -- 6 Gradient Methods—The Quadratic Case -- 7 General Descent Methods -- 8 Iterative Methods for Solving Linear Equations -- 9 Constrained Minima -- II Conjugate Direction Methods -- 1 Introduction -- 2 Quadratic Functions on En -- 3 Basic Properties of Quadratic Functions -- 4 Minimization of a Quadratic Function F on k-Planes -- 5 Method of Conjugate Directions (CD-Method) -- 6 Method of Conjugate Gradients (CG-Algorithm) -- 7 Gradient PARTAN -- 8 CG-Algorithms for Nonquadratic Functions -- 9 Numerical Examples -- 10 Least Square Solutions -- III Conjugate Gram-Schmidt Processes -- 1 Introduction -- 2 A Conjugate Gram-Schmidt Process -- 3 CGS-CG-Algorithms -- 4 A Connection of CGS-Algorithms with Gaussian Elimination -- 5 Method of Parallel Displacements -- 6 Methods of Parallel Planes (PARP) -- 7 Modifications of Parallel Displacements Algorithms -- 8 CGS-Algorithms for Nonquadratic Functions -- 9 CGS-CG-Routines for Nonquadratic Functions -- 10 Gauss-Seidel CGS-Routines -- 11 The Case of Nonnegative Components -- 12 General Linear Inequality Constraints -- IV Conjugate Gradient Algorithms -- 1 Introduction -- 2 Conjugate Gradient Algorithms -- 3 The Normalized CG-Algorithm -- 4 Termination -- 5 Clustered Eigenvalues -- 6 Nonnegative Hessians -- 7 A Planar CG-Algorithm -- 8 Justification of the Planar CG-Algorithm -- 9 Modifications of the CG-Algorithm -- 10 Two Examples -- 11 Connections between Generalized CG-Algorithms and Stadard CG- and CD-Algorithm -- 12 Least Square Solutions -- 13 Variable Metric Algorithms -- 14 A Planar CG-Algorithm for Nonquadratic Functions -- References Shortly after the end of World War II high-speed digital computing machines were being developed. It was clear that the mathematical aspects of com­ putation needed to be reexamined in order to make efficient use of high-speed digital computers for mathematical computations. Accordingly, under the leadership of Min a Rees, John Curtiss, and others, an Institute for Numerical Analysis was set up at the University of California at Los Angeles under the sponsorship of the National Bureau of Standards. A similar institute was formed at the National Bureau of Standards in Washington, D. C. In 1949 J. Barkeley Rosser became Director of the group at UCLA for a period of two years. During this period we organized a seminar on the study of solu­ tions of simultaneous linear equations and on the determination of eigen­ values. G. Forsythe, W. Karush, C. Lanczos, T. Motzkin, L. J. Paige, and others attended this seminar. We discovered, for example, that even Gaus­ sian elimination was not well understood from a machine point of view and that no effective machine oriented elimination algorithm had been developed. During this period Lanczos developed his three-term relationship and I had the good fortune of suggesting the method of conjugate gradients. We dis­ covered afterward that the basic ideas underlying the two procedures are essentially the same. The concept of conjugacy was not new to me. In a joint paper with G. D HTTP:URL=https://doi.org/10.1007/978-1-4612-6048-6

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