Hyperbolic Pairs in the Method of Conjugate Gradients

Previous article Next article Hyperbolic Pairs in the Method of Conjugate GradientsDavid G. LuenbergerDavid G. Luenbergerhttps://doi.org/10.1137/0117118PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout1 Magnus R. Hestenes and , Eduard Stiefel, Methods of conjugate gradients for solving linear systems, J. Research Nat. Bur. Standards, 49 (1952), 409–436 (1953) MR0060307 (15,651a) 0048.09901 CrossrefISIGoogle Scholar2 Magnus R. Hestenes, The conjugate-gradient method for solving linear systems, Proceedings of Symposia in Applied Mathematics. Vol. VI. Numerical analysis, McGraw-Hill Book Company, Inc., New York, for the American Mathematical Society, Providence, R. I., 1956, 83–102 MR0084178 (18,824c) 0072.14102 Google Scholar3 F. S. Beckman, A. Ralston and , H. S. Wilf, The solution of linear equations by the conjugate gradient methodMathematical methods for digital computers, Wiley, New York, 1960, 62–72 MR0117910 (22:8684) Google Scholar4 David G. Luenberger, Optimization by vector space methods, John Wiley & Sons Inc., New York, 1969xvii+326 MR0238472 (38:6748) 0176.12701 Google Scholar5 Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965xvii+508 MR0197234 (33:5416) 0193.34701 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Krylov-Subspace Methods for Quadratic Hypersurfaces: A Grossone–based Perspective6 July 2022 Cross Ref Polarity and conjugacy for quadratic hypersurfaces: A unified framework with recent advancesJournal of Computational and Applied Mathematics, Vol. 390 Cross Ref Primal–Dual Methods20 August 2021 Cross Ref Conjugate Direction Methods20 August 2021 Cross Ref Planar methods and grossone for the Conjugate Gradient breakdown in nonlinear programming23 October 2017 | Computational Optimization and Applications, Vol. 71, No. 1 Cross Ref Conjugate Direction Methods and Polarity for Quadratic Hypersurfaces30 October 2017 | Journal of Optimization Theory and Applications, Vol. 175, No. 3 Cross Ref Primal-Dual Methods Cross Ref Basic Properties of Linear Programs Cross Ref Duality and Complementarity Cross Ref Conjugate Direction Methods Cross Ref Exploiting the Composite Step Strategy to the Biconjugate A -Orthogonal Residual Method for Non-Hermitian Linear SystemsJournal of Applied Mathematics, Vol. 2013 Cross Ref Lanczos Conjugate-Gradient Method and Pseudoinverse Computation on Indefinite and Singular Systems19 January 2007 | Journal of Optimization Theory and Applications, Vol. 132, No. 2 Cross Ref Planar Conjugate Gradient Algorithm for Large-Scale Unconstrained Optimization, Part 1: TheoryJournal of Optimization Theory and Applications, Vol. 125, No. 3 Cross Ref Planar Conjugate Gradient Algorithm for Large-Scale Unconstrained Optimization, Part 2: ApplicationJournal of Optimization Theory and Applications, Vol. 125, No. 3 Cross Ref Conjugate gradient (CG)-type method for the solution of Newton's equation within optimization frameworksOptimization Methods and Software, Vol. 19, No. 3-4 Cross Ref Planar-CG Methods and Matrix Tridiagonalization in Large Scale Unconstrained Optimization Cross Ref Iterative solution of linear systems in the 20th century Cross Ref Iterative solution of linear systems in the 20th centuryJournal of Computational and Applied Mathematics, Vol. 123, No. 1-2 Cross Ref A minimization method for the solution of large symmetriric eigenproblemsInternational Journal of Computer Mathematics, Vol. 70, No. 1 Cross Ref Lanczos-type solvers for nonsymmetric linear systems of equations7 November 2008 | Acta Numerica, Vol. 6 Cross Ref A Comparison of Three Basic Conjugate Direction MethodsNumerical Linear Algebra with Applications, Vol. 3, No. 6 Cross Ref A new taxonomy of conjugate gradient methodsComputers & Mathematics with Applications, Vol. 31, No. 4-5 Cross Ref A breakdown of the block CG methodOptimization Methods and Software, Vol. 7, No. 1 Cross Ref Phase unwrapping of MR phase images using Poisson equationIEEE Transactions on Image Processing, Vol. 4, No. 5 Cross Ref A composite step bi-conjugate gradient algorithm for nonsymmetric linear systemsNumerical Algorithms, Vol. 7, No. 1 Cross Ref A composite step conjugate gradients squared algorithm for solving nonsymmetric linear systemsNumerical Algorithms, Vol. 7, No. 1 Cross Ref Rational interpolation via orthogonal polynomialsComputers & Mathematics with Applications, Vol. 26, No. 5 Cross Ref Block conjugate gradient methods2 November 2010 | Optimization Methods and Software, Vol. 2, No. 1 Cross Ref Conjugate gradient-type algorithms for a finite-element discretization of the Stokes equationsJournal of Computational and Applied Mathematics, Vol. 39, No. 1 Cross Ref Iterative solution of linear systems7 November 2008 | Acta Numerica, Vol. 1 Cross Ref Generating conjugate directions for arbitrary matrices by matrix equations I.Computers & Mathematics with Applications, Vol. 21, No. 1 Cross Ref Use of indefinite pencils for computing damped natural modesLinear Algebra and its Applications, Vol. 140 Cross Ref Efficient direct and iterative electrodynamic analysis of geometrically complex MIC and MMIC structuresInternational Journal of Numerical Modelling: Electronic Networks, Devices and Fields, Vol. 2, No. 3 Cross Ref Some History of the Conjugate Gradient and Lanczos Algorithms: 1948–1976Gene H. Golub and Dianne P. O’Leary2 August 2006 | SIAM Review, Vol. 31, No. 1AbstractPDF (6558 KB)Orthogonal Error MethodsVance Faber and Thomas A. Manteuffel14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 24, No. 1AbstractPDF (1824 KB)Planar quasi-Newton algorithms for unconstrained saddlepoint problemsJournal of Optimization Theory and Applications, Vol. 43, No. 2 Cross Ref On preconditioned iterative methods for solving (A−λB)x=0Computing, Vol. 32, No. 2 Cross Ref On the simplification of generalized conjugate-gradient methods for nonsymmetrizable linear systemsLinear Algebra and its Applications, Vol. 52-53 Cross Ref Truncated-newtono algorithms for large-scale unconstrained optimizationMathematical Programming, Vol. 26, No. 2 Cross Ref The numerical solution of equality constrained quadratic programming problems1 January 1983 | Mathematics of Computation, Vol. 41, No. 163 Cross Ref Computational methods of linear algebraJournal of Soviet Mathematics, Vol. 15, No. 5 Cross Ref Plasma Collection by High-Voltage Spacecraft at Low Earth OrbitJournal of Spacecraft and Rockets, Vol. 18, No. 1 Cross Ref Plasma collection by high voltage spacecraft at low earth orbit17 August 2012 Cross Ref Conjugate gradient methods for indefinite systems21 September 2006 Cross Ref Solution of Sparse Indefinite Systems of Linear EquationsC. C. Paige and M. A. Saunders14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 12, No. 4AbstractPDF (1332 KB)The method of conjugate gradients used in inverse iterationBIT, Vol. 12, No. 4 Cross Ref On the Convergence of the Conjugate Gradient Method for Singular Linear Operator EquationsW. J. Kammerer and M. Z. Nashed1 August 2006 | SIAM Journal on Numerical Analysis, Vol. 9, No. 1AbstractPDF (1764 KB)The Conjugate Residual Method for Constrained Minimization ProblemsDavid G. Luenberger14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 7, No. 3AbstractPDF (845 KB) Volume 17, Issue 6| 1969SIAM Journal on Applied Mathematics History Submitted:26 August 1968Published online:17 February 2012 InformationCopyright © 1969 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0117118Article page range:pp. 1263-1267ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics