Optimization algorithms on matrix manifolds

Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on...

Full description

Saved in:

Bibliographic Details
Main Authors	Absil, P.A, Mahony,R, Sepulchre,R
Format	eBook Book
Language	English
Published	Princeton Princeton University Press 2008
Edition	1
Subjects	Affine space Algorithm Algorithms Analytic function Complex projective space Computation Computational problem COMPUTERS / Computer Science Conjugate gradient method Constrained optimization Directional derivative Eigenvalue algorithm Eigenvalues and eigenvectors Equation Euclidean vector Finsler manifold Fixed-point theorem Gauss–Newton algorithm Geometry Global optimization Gradient descent Inclusion map Invariant subspace Invariant subspace problem Invertible matrix Iteration Iterative method Levenberg–Marquardt algorithm Line search Linear algebra Linear map Linear space (geometry) Linear subspace Local convergence Manifold Mathematical optimization MATHEMATICS MATHEMATICS / Applied Matrices Matrix (mathematics) Matrix decomposition Newton's method Normed vector space Numerical analysis Numerical integration Numerical linear algebra Optimization problem Principal component analysis Product topology Projection (linear algebra) Quadratic equation Quasi-Newton method Quotient space (topology) Rate of convergence Rayleigh quotient Rayleigh quotient iteration Real projective space Riemannian geometry Riemannian manifold Riemannian submanifold Row and column spaces Scale invariance Simultaneous equations Stiefel manifold Submanifold Submersion (mathematics) Subset Subspace topology Tangent space Tangent vector Taylor's theorem Technology Technology & Engineering / Engineering (General) Theorem Topological space Topology Vector space Subset Euclidean vector Iterative method Projection (linear algebra) Matrix (mathematics) Tangent vector Gradient descent Vector space Eigenvalue algorithm Eigenvalues and eigenvectors Invertible matrix Tangent space Finsler manifold Real projective space Numerical linear algebra Invariant subspace problem Topology Subspace topology Riemannian manifold Matrix decomposition Algorithm Complex projective space Gauss–Newton algorithm Line search Directional derivative Numerical analysis Quasi-Newton method Computation Topological space Submersion (mathematics) Linear algebra Inclusion map Normed vector space Mathematical optimization Rayleigh quotient Submanifold Invariant subspace Product topology Fixed-point theorem Iteration Quotient space (topology) Linear subspace Computational problem Riemannian geometry Taylor's theorem Manifold Linear map Rayleigh quotient iteration Stiefel manifold Theorem Row and column spaces Rate of convergence Local convergence Conjugate gradient method Scale invariance Numerical integration Linear space (geometry) Levenberg–Marquardt algorithm Equation Analytic function Affine space Quadratic equation Constrained optimization Geometry Simultaneous equations Global optimization Newton's method Optimization problem Riemannian submanifold Principal component analysis
Online Access	Get full text
ISBN	9780691132983 0691132984 1400830249 9781400830244
DOI	10.1515/9781400830244

Cover

Abstract	Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra.
AbstractList	Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists. Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. No detailed description available for "Optimization Algorithms on Matrix Manifolds". Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifoldsoffers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.
Author	Mahony,R Sepulchre,R Absil, P.A
Author_xml	– sequence: 1 fullname: Absil, P.A – sequence: 2 fullname: Absil, P.A – sequence: 3 fullname: Mahony,R – sequence: 4 fullname: Sepulchre,R
BackLink	https://cir.nii.ac.jp/crid/1130282270670001664$$DView record in CiNii
BookMark	eNplkU1vEzEQho2giKbkyAkh9YBACAU8_vaRRv1AqtQegKvl3fVunOyug-1Qyq_H6UYghA8ej97H74zHM_RkDKND6AXgD8CBf9RSAcNYUUwYe4RmfxL9GM2LiIUGoEQreoRmBGOpieRUPUXHjCpJiQTxDM1TWuOyCHAp1DF6f7PNfvC_bPZhPLV9F6LPqyGdlmywOfqfJYy-DX2TnqOj1vbJzQ_xBH27OP-yvFpc31x-Xn66XljFtMILYikWSlak4UBqDU4q5qgAZ1vdAC2voHVNuRCtBFAt4ZZXjQNSaQuu1o6eoHeTsU0bd5dWoc_J_OhdFcImmX_m8Je9s312sXFd3N2XgxlsrP9j30zsNobvO5eyebCs3Zij7c352ZJxWXoq4MsD6GLvumCmypIyznRRX0_q6L2p_X4vdzBRhEgsZJkuCLGv9mrC1imHeLBYmzpnmYZhU_S3k-677a7qfVr5sTPb6Evn9-b26-3Z_qMoB_obUciSBQ
ContentType	eBook Book
Copyright	2008 Princeton University Press
Copyright_xml	– notice: 2008 Princeton University Press
DBID	E06 RYH YSPEL
DEWEY	518.1
DOI	10.1515/9781400830244
DatabaseName	Princeton University Press eBooks CiNii Complete Perlego
DatabaseTitleList
DeliveryMethod	fulltext_linktorsrc
Discipline	Engineering Mathematics
EISBN	1400830249 9781400830244
Edition	1
ExternalDocumentID	9781400830244 EBC457711 734549 BA84559669 j.ctt7smmk PUPB0000351
Genre	Electronic books
GroupedDBID	-VQ -VX 05P 089 20A 38. 6XM 92K A4I A4J AABBV AAGED AAJDW AAMYY AANYM AAUSU AAZEP ABARN ABCYV ABCYY ABHWV ABIAV ABIWA ABONK ABYBY ACEOT ACFYN ACHKM ACISH ACLGV ACQYL ADDXO ADVEM ADVQQ AEAED AECLD AEDVL AEEBL AERYV AEYCP AFHFQ AFRFP AGLEC AILDO AIOLA AIUUY AIXPE AJFER AKHYG ALMA_UNASSIGNED_HOLDINGS AMYDA APVFW ARPAB ARSQP ATDNW AZZ BBABE BFATZ BFRBX BJTYN CDLGT CZZ DGVNY DHNOV DUGUG E06 EBSCA EBZNK ECOWB GEOUK HELXT HF4 I4C IVK IWG J-X JJU JLPMJ KBOFU L7C MYL O7H PQQKQ QD8 SUPCW UE6 XI1 YSPEL ~I6 ABMRC ABQNV ABRSK ACBYE ACHUA AHWGJ AVGCG RYH
ID	FETCH-LOGICAL-a84980-2a30687b2d512c91e784e361eaf9d138143cc3566f7118f25a5bde12b9a1ec9e3
ISBN	9780691132983 0691132984 1400830249 9781400830244
IngestDate	Fri Nov 08 05:49:45 EST 2024 Sat Oct 25 00:45:46 EDT 2025 Wed Oct 29 00:48:23 EDT 2025 Tue Sep 30 17:15:04 EDT 2025 Thu Jun 26 21:52:05 EDT 2025 Tue Sep 16 03:33:23 EDT 2025 Wed Jul 17 03:28:08 EDT 2024
IsPeerReviewed	false
IsScholarly	false
Keywords	Subset Euclidean vector Iterative method Projection (linear algebra) Matrix (mathematics) Tangent vector Gradient descent Vector space Eigenvalue algorithm Eigenvalues and eigenvectors Invertible matrix Tangent space Finsler manifold Real projective space Numerical linear algebra Invariant subspace problem Topology Subspace topology Riemannian manifold Matrix decomposition Algorithm Complex projective space Gauss–Newton algorithm Line search Directional derivative Numerical analysis Quasi-Newton method Computation Topological space Submersion (mathematics) Linear algebra Inclusion map Normed vector space Mathematical optimization Rayleigh quotient Submanifold Invariant subspace Product topology Fixed-point theorem Iteration Quotient space (topology) Linear subspace Computational problem Riemannian geometry Taylor's theorem Manifold Linear map Rayleigh quotient iteration Stiefel manifold Theorem Row and column spaces Rate of convergence Local convergence Conjugate gradient method Scale invariance Numerical integration Linear space (geometry) Levenberg–Marquardt algorithm Equation Analytic function Affine space Quadratic equation Constrained optimization Geometry Simultaneous equations Global optimization Newton's method Optimization problem Riemannian submanifold Principal component analysis
LCCN	2007927538
LCCallNum	QA402.5
LCCallNum_Ident	QA402.5
Language	English
LinkModel	OpenURL
MergedId	FETCHMERGED-LOGICAL-a84980-2a30687b2d512c91e784e361eaf9d138143cc3566f7118f25a5bde12b9a1ec9e3
Notes	Includes bibliographical references and index
OCLC	438732716
PQID	EBC457711
PageCount	240
ParticipantIDs	askewsholts_vlebooks_9781400830244 walterdegruyter_marc_9781400830244 proquest_ebookcentral_EBC457711 perlego_books_734549 nii_cinii_1130282270670001664 jstor_books_j_ctt7smmk igpublishing_primary_PUPB0000351
ProviderPackageCode	J-X
PublicationCentury	2000
PublicationDate	[2008]
PublicationDateYYYYMMDD	2008-01-01
PublicationDate_xml	– year: 2008 text: [2008]
PublicationDecade	2000
PublicationPlace	Princeton
PublicationPlace_xml	– name: Princeton – name: Princeton, NJ
PublicationYear	2008
Publisher	Princeton University Press
Publisher_xml	– name: Princeton University Press
RestrictionsOnAccess	restricted access
SSID	ssj0000215768
Score	2.5112162
Snippet	Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure.... No detailed description available for "Optimization Algorithms on Matrix Manifolds".
SourceID	askewsholts walterdegruyter proquest perlego nii jstor igpublishing
SourceType	Aggregation Database Publisher
SubjectTerms	Affine space Algorithm Algorithms Analytic function Complex projective space Computation Computational problem COMPUTERS / Computer Science Conjugate gradient method Constrained optimization Directional derivative Eigenvalue algorithm Eigenvalues and eigenvectors Equation Euclidean vector Finsler manifold Fixed-point theorem Gauss–Newton algorithm Geometry Global optimization Gradient descent Inclusion map Invariant subspace Invariant subspace problem Invertible matrix Iteration Iterative method Levenberg–Marquardt algorithm Line search Linear algebra Linear map Linear space (geometry) Linear subspace Local convergence Manifold Mathematical optimization MATHEMATICS MATHEMATICS / Applied Matrices Matrix (mathematics) Matrix decomposition Newton's method Normed vector space Numerical analysis Numerical integration Numerical linear algebra Optimization problem Principal component analysis Product topology Projection (linear algebra) Quadratic equation Quasi-Newton method Quotient space (topology) Rate of convergence Rayleigh quotient Rayleigh quotient iteration Real projective space Riemannian geometry Riemannian manifold Riemannian submanifold Row and column spaces Scale invariance Simultaneous equations Stiefel manifold Submanifold Submersion (mathematics) Subset Subspace topology Tangent space Tangent vector Taylor's theorem Technology Technology & Engineering / Engineering (General) Theorem Topological space Topology Vector space
SubjectTermsDisplay	Mathematical optimization
TableOfContents	Optimization algorithms on matrix manifolds -- Contents -- List of Algorithms -- Foreword -- Notation Conventions -- Chapter One: Introduction -- Chapter Two: Motivation and Applications -- Chapter Three: Matrix Manifolds: First-Order Geometry -- Chapter Four: Line-Search Algorithms on Manifolds -- Chapter Five: Matrix Manifolds: Second-Order Geometry -- Chapter Six: Newton's Method -- Chapter Seven: Trust-Region Methods -- Chapter Eight: A Constellation of Superlinear Algorithms -- Appendix A: Elements of Linear Algebra, Topology, and Calculus -- Bibliography -- Index Front Matter Table of Contents List of Algorithms Foreword Notation Conventions Chapter One: Introduction Chapter Two: Motivation and Applications Chapter Three: Matrix Manifolds: Chapter Four: Line-Search Algorithms on Manifolds Chapter Five: Matrix Manifolds: Chapter Six: Newton’s Method Chapter Seven: Trust-Region Methods Chapter Eight: A Constellation of Superlinear Algorithms Appendix A Bibliography Index Intro -- Contents -- List of Algorithms -- Foreword -- Notation Conventions -- 1. Introduction -- 2. Motivation and Applications -- 2.1 A case study: the eigenvalue problem -- 2.1.1 The eigenvalue problem as an optimization problem -- 2.1.2 Some benefits of an optimization framework -- 2.2 Research problems -- 2.2.1 Singular value problem -- 2.2.2 Matrix approximations -- 2.2.3 Independent component analysis -- 2.2.4 Pose estimation and motion recovery -- 2.3 Notes and references -- 3. Matrix Manifolds: First-Order Geometry -- 3.1 Manifolds -- 3.1.1 Definitions: charts, atlases, manifolds -- 3.1.2 The topology of a manifold* -- 3.1.3 How to recognize a manifold -- 3.1.4 Vector spaces as manifolds -- 3.1.5 The manifolds Rnxp and Rnxp -- 3.1.6 Product manifolds -- 3.2 Differentiable functions -- 3.2.1 Immersions and submersions -- 3.3 Embedded submanifolds -- 3.3.1 General theory -- 3.3.2 The Stiefel manifold -- 3.4 Quotient manifolds -- 3.4.1 Theory of quotient manifolds -- 3.4.2 Functions on quotient manifolds -- 3.4.3 The real projective space RPn-1 -- 3.4.4 The Grassmann manifold Grass(p, n) -- 3.5 Tangent vectors and differential maps -- 3.5.1 Tangent vectors -- 3.5.2 Tangent vectors to a vector space -- 3.5.3 Tangent bundle -- 3.5.4 Vector fields -- 3.5.5 Tangent vectors as derivations -- 3.5.6 Differential of a mapping -- 3.5.7 Tangent vectors to embedded submanifolds -- 3.5.8 Tangent vectors to quotient manifolds -- 3.6 Riemannian metric, distance, and gradients -- 3.6.1 Riemannian submanifolds -- 3.6.2 Riemannian quotient manifolds -- 3.7 Notes and references -- 4. Line-Search Algorithms on Manifolds -- 4.1 Retractions -- 4.1.1 Retractions on embedded submanifolds -- 4.1.2 Retractions on quotient manifolds -- 4.1.3 Retractions and local coordinates* -- 4.2 Line-search methods -- 4.3 Convergence analysis -- 4.3.1 Convergence on manifolds 6.5 Analysis of Rayleigh quotient algorithms -- 6.5.1 Convergence analysis -- 6.5.2 Numerical implementation -- 6.6 Notes and references -- 7. Trust-Region Methods -- 7.1 Models -- 7.1.1 Models in Rn -- 7.1.2 Models in general Euclidean spaces -- 7.1.3 Models on Riemannian manifolds -- 7.2 Trust-region methods -- 7.2.1 Trust-region methods in Rn -- 7.2.2 Trust-region methods on Riemannian manifolds -- 7.3 Computing a trust-region step -- 7.3.1 Computing a nearly exact solution -- 7.3.2 Improving on the Cauchy point -- 7.4 Convergence analysis -- 7.4.1 Global convergence -- 7.4.2 Local convergence -- 7.4.3 Discussion -- 7.5 Applications -- 7.5.1 Checklist -- 7.5.2 Symmetric eigenvalue decomposition -- 7.5.3 Computing an extreme eigenspace -- 7.6 Notes and references -- 8. A Constellation of Superlinear Algorithms -- 8.1 Vector transport -- 8.1.1 Vector transport and affine connections -- 8.1.2 Vector transport by differentiated retraction -- 8.1.3 Vector transport on Riemannian submanifolds -- 8.1.4 Vector transport on quotient manifolds -- 8.2 Approximate Newton methods -- 8.2.1 Finite difference approximations -- 8.2.2 Secant methods -- 8.3 Conjugate gradients -- 8.3.1 Application: Rayleigh quotient minimization -- 8.4 Least-square methods -- 8.4.1 Gauss-Newton methods -- 8.4.2 Levenberg-Marquardt methods -- 8.5 Notes and references -- A. Elements of Linear Algebra, Topology, and Calculus -- A.1 Linear algebra -- A.2 Topology -- A.3 Functions -- A.4 Asymptotic notation -- A.5 Derivatives -- A.6 Taylor's formula -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Z 4.3.2 A topological curiosity* -- 4.3.3 Convergence of line-search methods -- 4.4 Stability of fixed points -- 4.5 Speed of convergence -- 4.5.1 Order of convergence -- 4.5.2 Rate of convergence of line-search methods* -- 4.6 Rayleigh quotient minimization on the sphere -- 4.6.1 Cost function and gradient calculation -- 4.6.2 Critical points of the Rayleigh quotient -- 4.6.3 Armijo line search -- 4.6.4 Exact line search -- 4.6.5 Accelerated line search: locally optimal conjugate gradient -- 4.6.6 Links with the power method and inverse iteration -- 4.7 Refining eigenvector estimates -- 4.8 Brockett cost function on the Stiefel manifold -- 4.8.1 Cost function and search direction -- 4.8.2 Critical points -- 4.9 Rayleigh quotient minimization on the Grassmann manifold -- 4.9.1 Cost function and gradient calculation -- 4.9.2 Line-search algorithm -- 4.10 Notes and references -- 5. Matrix Manifolds: Second-Order Geometry -- 5.1 Newton's method in Rn -- 5.2 Affine connections -- 5.3 Riemannian connection -- 5.3.1 Symmetric connections -- 5.3.2 Definition of the Riemannian connection -- 5.3.3 Riemannian connection on Riemannian submanifolds -- 5.3.4 Riemannian connection on quotient manifolds -- 5.4 Geodesics, exponential mapping, and parallel translation -- 5.5 Riemannian Hessian operator -- 5.6 Second covariant derivative* -- 5.7 Notes and references -- 6. Newton's Method -- 6.1 Newton's method on manifolds -- 6.2 Riemannian Newton method for real-valued functions -- 6.3 Local convergence -- 6.3.1 Calculus approach to local convergence analysis -- 6.4 Rayleigh quotient algorithms -- 6.4.1 Rayleigh quotient on the sphere -- 6.4.2 Rayleigh quotient on the Grassmann manifold -- 6.4.3 Generalized eigenvalue problem -- 6.4.4 The nonsymmetric eigenvalue problem -- 6.4.5 Newton with subspace acceleration: Jacobi-Davidson Chapter Six. Newton’s Method List of Algorithms Notation Conventions Index - / Chapter Four. Line-Search Algorithms On Manifolds Contents Foreword Chapter One. Introduction Chapter Three. Matrix Manifolds: First-Order Geometry Chapter Eight. A Constellation Of Superlinear Algorithms A. Elements Of Linear Algebra, Topology, And Calculus Chapter Two. Motivation and Applications Frontmatter -- Chapter Seven. Trust-Region Methods Chapter Five. Matrix Manifolds: Second-Order Geometry Bibliography
Title	Optimization algorithms on matrix manifolds
URI	http://portal.igpublish.com/iglibrary/search/PUPB0000351.html https://www.jstor.org/stable/j.ctt7smmk https://cir.nii.ac.jp/crid/1130282270670001664 https://www.perlego.com/book/734549/optimization-algorithms-on-matrix-manifolds-pdf https://ebookcentral.proquest.com/lib/[SITE_ID]/detail.action?docID=457711 https://www.degruyterbrill.com/isbn/9781400830244 https://www.vlebooks.com/vleweb/product/openreader?id=none&isbn=9781400830244&uid=none
hasFullText	1
inHoldings	1
isFullTextHit
isPrint
link	http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV3Pb9MwFH6C9kIPCBgTAQYV4jZl5IdTx8dtKpqQAhVsaDcrcV7abmkbNSm__nqekzRtMg7Axaqj2HX81fH3nt_7CvBWIW2S3FGmr2JlMqUErTkUpos22rEnEsvRCc7Bx9HFFftw7V3vtJ7L7JIiOlG__phX8j-o0jXCVWfJ_gOyTad0gT4TvlQSwlR2yG9TrcD9RAt9UWdQHp-m0xWZ-LNF6fkPtOb-Dx3WMk9WadxQ5tMor9y9kxOzQS0IdWR6K27wC2abVM0q1_RnMlrTbIYt34Df8Q1MtL8etTxHN-2wZUWSjWVpGbBKiPHOO9Ur5Sfu3NfWru7sKU2kX6vZfeg7jChaD_q06Y6DxhmmiQfZPbUGKn3hu1a7AQzC_JZe-rQhFLkWlZ1mjZNuG0hK5GA5n5MZk-E6xemqZTI8_F4GH8Q4XW9-FtvD7pJDXD6CPurEksdwD5dPYLAnBEm1oFHPzQ_geB_e4Q7eIdUqeIcNvE_h6_vx5fmFWf-phRn6TPiW6YRkpfk8cmLiWkrYyH2G7sjGMBGxTQSKuUq5xLITTsZf4nihF8VoO5EIbVQC3UPoLVdLfAZDRlw45J6InFHMPB6HFktYxDy0qCPOEgPe7M2b_JaWB_C5bE2uAcP96ZRZpXIiJ1eTM6s6aTbgsJxiWTW_kaooeL5Y3BpwRHMu1VyXtj4DJ77Jy9QvMiVG1PdBjUbdlLvMY8KA11toZDmkOhBZjs_O6TnouWnkHcSklnZpj_z539z0Ah7sVsZL6BXrDR4RjyyiV_Wv8DfCG22u
linkProvider	ProQuest Ebooks
openUrl	ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.title=Optimization+Algorithms+on+Matrix+Manifolds&rft.au=Absil%2C+P.-A&rft.au=Mahony%2C+R&rft.au=Sepulchre%2C+Rodolphe&rft.date=2008-01-01&rft.pub=Princeton+University+Press&rft.isbn=9781400830244&rft_id=info:doi/10.1515%2F9781400830244&rft.externalDBID=n%2Fa&rft.externalDocID=9781400830244
thumbnail_l	http://utb.summon.serialssolutions.com/2.0.0/image/custom?url=https%3A%2F%2Fwww.perlego.com%2Fbooks%2FRM_Books%2Fprinceton_university_press_rydfxi%2F9781400830244.jpg
thumbnail_m	http://utb.summon.serialssolutions.com/2.0.0/image/custom?url=https%3A%2F%2Fwww.degruyterbrill.com%2Fdocument%2Fcover%2Fisbn%2F9781400830244%2Foriginal http://utb.summon.serialssolutions.com/2.0.0/image/custom?url=https%3A%2F%2Fvle.dmmserver.com%2Fmedia%2F640%2F97814008%2F9781400830244.jpg
thumbnail_s	http://utb.summon.serialssolutions.com/2.0.0/image/custom?url=http%3A%2F%2Fportal.igpublish.com%2Figlibrary%2Famazonbuffer%2FPUPB0000351_null_0_320.png