To put the world of linear algebra to advanced use, it is not enough to merely understand the theory; there is a significant gap between the theory of linear algebra and its myriad expressions in nearly every computational domain. To bridge this gap, it is essential to process the theory by solving many exercises, thus obtaining a firmer grasp of its diverse applications. Similarly, from a theoretical perspective, diving into the literature on advanced linear algebra often reveals more and more topics that are deferred to exercises instead of being treated in the main text. As exercises grow more complex and numerous, it becomes increasingly important to provide supporting material and guidelines on how to solve them, supporting students' learning process. This book provides precisely this type of supporting material for the textbook "Numerical Linear Algebra and Matrix Factorizations," published as Vol. 22 of Springer's Texts in Computational Science and Engineering series. Instead of omitting details or merely providing rough outlines, this book offers detailed proofs, and connects the solutions to the corresponding results in the textbook. For the algorithmic exercises the utmost level of detail is provided in the form of MATLAB implementations. Both the textbook and solutions are self-contained. This book and the textbook are of similar length, demonstrating that solutions should not be considered a minor aspect when learning at advanced levels.

After reading this book, students should be able to analyze computational problems in linear algebra such as linear systems, least squares- and eigenvalue problems, and to develop their own algorithms for solving them. Since these problems can be large and difficult to handle, much can be gained by understanding and taking advantage of special structures. This in turn requires a good grasp of basic numerical linear algebra and matrix factorizations. Factoring a matrix into a product of simpler matrices is a crucial tool in numerical linear algebra, because it allows us to tackle complex problems by solving a sequence of easier ones. The main characteristics of this book are as follows: It is self-contained, only assuming that readers have completed first-year calculus and an introductory course on linear algebra, and that they have some experience with solving mathematical problems on a computer. The book provides detailed proofs of virtually all results. Further, its respective parts can be used independently, making it suitable for self-study. The book consists of 15 chapters, divided into five thematically oriented parts. The chapters are designed for a one-week-per-chapter, one-semester course. To facilitate self-study, an introductory chapter includes a brief review of linear algebra.

Designed to provide tools for independent study, this book contains student-tested mathematical exercises joined with MATLAB programming exercises. Most chapters open with a review followed by theoretical and programming exercises, with detailed solutions provided for all problems including programs. Many of the MATLAB exercises are presented as Russian dolls: each question improves and completes the previous program and results are provided to validate the intermediate programs. The book offers useful MATLAB commands, advice on tables, vectors, matrices and basic commands for plotting. It contains material on eigenvalues and eigenvectors and important norms of vectors and matrices including perturbation theory; iterative methods for solving nonlinear and linear equations; polynomial and piecewise polynomial interpolation; Bezier curves; approximations of functions and integrals and more. The last two chapters considers ordinary differential equations including two point boundary value problems, and deal with finite difference methods for some partial differential equations. The format is designed to assist students working alone, with concise Review paragraphs, Math Hint footnotes on the mathematical aspects of a problem and MATLAB Hint footnotes with tips on programming.

This volume constitutes the thoroughly refereed post-conference proceedings of the 8th International Conference on Mathematical Methods for Curves and Surfaces, MMCS 2012, held in Oslo, Norway, in June/July 2012. The 28 revised full papers presented were carefully reviewed and selected from 135 submissions. The topics range from mathematical analysis of various methods to practical implementation on modern graphics processing units. The papers reflect the newest developments in these fields and also point to the latest literature.

This volume constitutes the thoroughly refereed post-conference proceedings of the 7th International Conference on Curves and Surfaces, held in Avignon, in June 2010. The conference had the overall theme: "Representation and Approximation of Curves and Surfaces and Applications." The 39 revised full papers presented together with 9 invited talks were carefully reviewed and selected from 114 talks presented at the conference. The topics addressed by the papers range from mathematical foundations to practical implementation on modern graphics processing units and address a wide area of topics such as computer-aided geometric design, computer graphics and visualisation, computational geometry and topology, geometry processing, image and signal processing, interpolation and smoothing, scattered data processing and learning theory and subdivision, wavelets and multi-resolution methods.

TheSeventhInternationalConferenceonMathematicalMethodsforCurvesand SurfacestookplaceJune26-July 1,2008, inTonsberg, Norway. Theearlier conferences in the series took place in Oslo (1988), Biri (1991), Ulvik (1994), Lillehammer(1997), Oslo(2000), andTromso(2004). Theconferencegathered 165participants fromalmost30countries who presenteda total of129talks. Thisincludesnineinvitedtalksandsevenmini-symposia. Thisbookcontains28originalarticlesbasedontalkspresentedattheconf- ence. Thetopicsrangefrommathematicalanalysisofvariousmethodstoprac- calimplementationonmoderngraphicsprocessingunits. Thepapersre?ectthe newestdevelopmentsinthese?eldsandalsopointtothelatestliterature. The papershavebeensubjecttotheusualpeerreviewprocess, andwethankboth theauthorsandthereviewersfortheirhardworkandhelpfulcollaboration. Wewishtothankthosewhohavesupportedandhelpedorganizetheconf- ence. Firstandforemostitisapleasuretoacknowledgethegenerous?nancial support from the Department of Informatics and the Centre of Mathematics forApplications(CMA)attheUniversityofOslo, andtheResearchCouncilof Norway. WewouldalsoliketothankAndrewMcMurryforhishelpwithwith technicalmatters, andSaraMorkenforhelpwiththeregistration. November2009 Theeditors Organization Organizing Commitee and Editors MortenDaehlen UniversityofOslo, Norway MichaelFloater UniversityofOslo, Norway TomLyche UniversityofOslo, Norway Jean-LouisMerrien INSAdeRennes, France KnutMorken UniversityofOslo, Norway LarryL. Schumaker VanderbiltUniversity, USA Invited Speakers Jean-DanielBoissonnat, SophiaAntipolis, France MassimoFornasier, Linz, Austria TomHughes, Austin, USA JorgPeters, Gainesville, USA RagniPiene, Oslo, Norway RobertSchaback, Gottingen, Germany PeterSchroder, Caltech, USA JonathanShewchuk, Berkeley, USA JoachimWeickert, Saarland, Germany Mini-Symposia Organizers OlegDavydov, Glasgow, UK TorDokken, Oslo, Norway BinHan, Edmonton, Canada ChuckHansen, SaltLakeCity, USA RimvydasKrasauskas, Vilnius, Lithuania TrondKvamsdal, Trondheim, Norway CarlaManni, Rome, Italy Sponsoring Institutions DepartmentofInformatics, UniversityofOslo CentreofMathematicsforApplications, UniversityofOslo ResearchCouncilofNorway Table of Contents MMCS 2008 Partial Di?erential Equations for Interpolation and Compression of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Egil Bae and Joachim Weickert Construction of Rational Curves with Rational Rotation-Minimizing Frames via Mob ] ius Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Michael Barton, ? Bert Juttl ] er, and Wenping Wang Fat Arcs for Implicitly De?ned Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Szilvia B ela and Bert Juttl ] er Geometric Properties of the Adaptive Delaunay Tessellation. . . . . . . . . . . 41 Tom Bobach, Alexandru Constantiniu, Paul Steinmann, and Georg Umlauf Quadrangular Parameterization for Reverse Engineering . . . . . . . . . . . . . . 55 David Bommes, Tobias Vossemer, and Leif Kobbelt A Comparison of Three Commodity-Level Parallel Architectures: Multi-core CPU, Cell BE and GPU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Andr e Rigland Brodtkorb and Trond Runar Hagen Mean Distance from a Curve to Its Control Polygon. . . . . . . . . . . . . . . . . . 81 Jesu s Carnicer and Jorge Delgado Compactly Supported Splines with Tension Properties on a Three-Direction Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . .

To put the world of linear algebra to advanced use, it is not enough to merely understand the theory; there is a significant gap between the theory of linear algebra and its myriad expressions in nearly every computational domain. To bridge this gap, it is essential to process the theory by solving many exercises, thus obtaining a firmer grasp of its diverse applications. Similarly, from a theoretical perspective, diving into the literature on advanced linear algebra often reveals more and more topics that are deferred to exercises instead of being treated in the main text. As exercises grow more complex and numerous, it becomes increasingly important to provide supporting material and guidelines on how to solve them, supporting students' learning process. This book provides precisely this type of supporting material for the textbook "Numerical Linear Algebra and Matrix Factorizations," published as Vol. 22 of Springer's Texts in Computational Science and Engineering series. Instead of omitting details or merely providing rough outlines, this book offers detailed proofs, and connects the solutions to the corresponding results in the textbook. For the algorithmic exercises the utmost level of detail is provided in the form of MATLAB implementations. Both the textbook and solutions are self-contained. This book and the textbook are of similar length, demonstrating that solutions should not be considered a minor aspect when learning at advanced levels.

After reading this book, students should be able to analyze computational problems in linear algebra such as linear systems, least squares- and eigenvalue problems, and to develop their own algorithms for solving them. Since these problems can be large and difficult to handle, much can be gained by understanding and taking advantage of special structures. This in turn requires a good grasp of basic numerical linear algebra and matrix factorizations. Factoring a matrix into a product of simpler matrices is a crucial tool in numerical linear algebra, because it allows us to tackle complex problems by solving a sequence of easier ones. The main characteristics of this book are as follows: It is self-contained, only assuming that readers have completed first-year calculus and an introductory course on linear algebra, and that they have some experience with solving mathematical problems on a computer. The book provides detailed proofs of virtually all results. Further, its respective parts can be used independently, making it suitable for self-study. The book consists of 15 chapters, divided into five thematically oriented parts. The chapters are designed for a one-week-per-chapter, one-semester course. To facilitate self-study, an introductory chapter includes a brief review of linear algebra.

This book takes readers on a multi-perspective tour through state-of-the-art mathematical developments related to the numerical treatment of PDEs based on splines, and in particular isogeometric methods. A wide variety of research topics are covered, ranging from approximation theory to structured numerical linear algebra. More precisely, the book provides (i) a self-contained introduction to B-splines, with special focus on approximation and hierarchical refinement, (ii) a broad survey of numerical schemes for control problems based on B-splines and B-spline-type wavelets, (iii) an exhaustive description of methods for computing and analyzing the spectral distribution of discretization matrices, and (iv) a detailed overview of the mathematical and implementational aspects of isogeometric analysis. The text is the outcome of a C.I.M.E. summer school held in Cetraro (Italy), July 2017, featuring four prominent lecturers with different theoretical and application perspectives. The book may serve both as a reference and an entry point into further research.

This volume constitutes the thoroughly refereed post-conference proceedings of the 9th International Conference on Mathematical Methods for Curves and Surfaces, MMCS 2016, held in Tonsberg, Norway, in June 2016. The 17 revised full papers presented were carefully reviewed and selected from 115 submissions. The topics range from mathematical theory to industrial applications.

This volume constitutes the thoroughly refereed post-conference proceedings of the 8th International Conference on Curves and Surfaces, held in Paris, France, in June 2014. The conference had the overall theme: "Representation and Approximation of Curves and Surfaces and Applications". The 32 revised full papers presented were carefully reviewed and selected from 39 submissions. The scope of the conference was on following topics: approximation theory, computer-aided geometric design, computer graphics and visualization, computational geometry and topology, geometry processing, image and signal processing, interpolation and smoothing, mesh generation, finite elements and splines, scattered data processing and learning theory, sparse and high-dimensional approximation, subdivision, wavelets and multi-resolution method.

After Ole-Johan's retirement at the beginning of the new millennium, some of us had thought and talked about making a "Festschrift" in his honor. When Donald Knuth took the initiative by sending us the ?rst contribution, the p- cess began to roll! In early 2002 an editing group was formed, including Kristen Nygaard, who had known Ole-Johan since their student days, and with whom he had developed the Simula language. Then we invited a number of prominent researchers familiar with Ole-Johan to submit contributions for a book hon- ing Ole-Johan on the occasion of his 70th birthday. Invitees included several members of the IFIP 2. 3 working group, a forum that Ole-Johan treasured and enjoyedparticipating in throughouthis career. In spite of the shortdeadline, the response to the invitations was overwhelmingly positive. The original idea was to complete the book rather quickly to make it a gift he could read and enjoy, because by then he had had cancer for three years, and his health was gradually deteriorating. Kristen had been regularly visiting Ole-Johan,who was in the hospitalat that time, and they were workingon their Turing award speech. Ole-Johan was grati?ed to hear about the contributions to this book, but modestly expressed the feeling that there was no special need to undertake a book project on his behalf. Peacefully accepting his destiny, Ole- Johan died on June 29, 2002.

These volumes contain carefully edited selections of papers that were presented at the Symposium on Trends in Approximation Theory, held in May 2000, and at the Oslo Conference on Mathematical Methods for Curves and Surfaces, held in July 2000. Both contain several invited surveys written by leading experts in the field, along with contributed research papers on the most current developments in approximation theory and in the theory and application of curves and surfaces. These books will be of great interest to mathematicians, engineers, and computer scientists.

These volumes contain carefully edited selections of papers that were presented at the Symposium on Trends in Approximation Theory, held in May 2000, and at the Oslo Conference on Mathematical Methods for Curves and Surfaces, held in July 2000. Both contain several invited surveys written by leading experts in the field, along with contributed research papers on the most current developments in approximation theory and in the theory and application of curves and surfaces. These books will be of great interest to mathematicians, engineers, and computer scientists.

Contains more than fifty carefully refereed and edited full-length papers on the theory and applications of mathematical methods arising out of the Fourth International Conference on Mathematical Methods in Computer Aided Geometric Design, held in Lillehammer, Norway, in July 1997.