Matrix Fields, Regular and Irregular: a Complete Fundamental Characterization
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A GMRES Solver with ILU(K) Preconditioner for Large-Scale Sparse Linear Systems on Multiple Gpus
University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies The Vault: Electronic Theses and Dissertations 2015-09-28 A GMRES Solver with ILU(k) Preconditioner for Large-Scale Sparse Linear Systems on Multiple GPUs Yang, Bo Yang, B. (2015). A GMRES Solver with ILU(k) Preconditioner for Large-Scale Sparse Linear Systems on Multiple GPUs (Unpublished master's thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/24749 http://hdl.handle.net/11023/2512 master thesis University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY A GMRES Solver with ILU(k) Preconditioner for Large-Scale Sparse Linear Systems on Multiple GPUs by Bo Yang A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE GRADUATE PROGRAM IN CHEMICAL AND PETROLEUM ENGINEERING CALGARY, ALBERTA SEPTEMBER, 2015 c Bo Yang 2015 Abstract Most time of reservoir simulation is spent on the solution of large-scale sparse linear systems. The Krylov subspace solvers and the ILU preconditioners are the most commonly used methods for solving such systems. Based on excellent parallel computing performance, GPUs have been a promising hardware architecture. The work of developing preconditioned Krylov solvers on GPUs is necessary and challengeable. -
Product Irregularity Strength of Graphs with Small Clique Cover Number
Product irregularity strength of graphs with small clique cover number Daniil Baldouski University of Primorska, FAMNIT, Glagoljaˇska 8, 6000 Koper, Slovenia Abstract For a graph X without isolated vertices and without isolated edges, a product-irregular labelling ω : E(X) 1, 2,...,s , first defined by Anholcer in 2009, is a labelling of the edges of X such that→ for { any two} distinct vertices u and v of X the product of labels of the edges incident with u is different from the product of labels of the edges incident with v. The minimal s for which there exist a product irregular labeling is called the product irregularity strength of X and is denoted by ps(X). Clique cover number of a graph is the minimum number of cliques that partition its vertex-set. In this paper we prove that connected graphs with clique cover number 2 or 3 have the product-irregularity strength equal to 3, with some small exceptions. Keywords: product irregularity strength, clique-cover number. Math. Subj. Class.: 05C15, 05C70, 05C78 1 Introduction Throughout this paper let X be a simple graph, that is, a graph without loops or multiple edges, without isolated vertices and without isolated edges. Let V (X) and E(X) denote the vertex set and the edge set of X, respectively. Let ω : E(X) 1, 2,...,s be an integer → { } labelling of the edges of X. Then the product degree pdX (v) of a vertex v V (X) in the graph ∈ X with respect to the labelling ω is defined by pdX (v)= ω(e). -
UNIVERSALLY OPTIMAL MATRICES and FIELD INDEPENDENCE of the MINIMUM RANK of a GRAPH 1. Introduction. the Minimum Rank Problem Is
Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 18, pp. 403-419, July 2009 ELA http://math.technion.ac.il/iic/ela UNIVERSALLY OPTIMAL MATRICES AND FIELD INDEPENDENCE OF THE MINIMUM RANK OF A GRAPH∗ LUZ M. DEALBA†, JASON GROUT‡,LESLIEHOGBEN§, RANA MIKKELSON‡, AND KAELA RASMUSSEN‡ Abstract. The minimum rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose (i, j)th entry (for i = j) isnonzero whenever {i, j} isan edge in G and is zero otherwise. A universally optimal matrix is defined to be an integer matrix A such that every off-diagonal entry of A is0, 1, or −1, and for all fields F , the rank of A isthe minimum rank over F of its graph. Universally optimal matrices are used to establish field independence of minimum rank for numerousgraphs. Examplesare alsoprovided verifying lack of field independence for other graphs. Key words. Minimum rank, Universally optimal matrix, Field independent, Symmetric matrix, Rank, Graph, Matrix. AMS subject classifications. 05C50, 15A03. 1. Introduction. The minimum rank problem is, for a given graph G and field F , to determine the smallest possible rank among symmetric matrices over F whose off-diagonal pattern of zero-nonzero entries is described by G. Most work on minimum rank has been on the real minimum rank problem. See [10] for a survey of known results and discussion of the motivation for the minimum rank problem; an extensive bibliography is also provided there. Catalogs of minimum rank and other parameters for families of graphs [15] and small graphs [16] were developed at the AIM workshop “Spectra of families of matrices described by graphs, digraphs, and sign patterns” [2] and are available on-line; these catalogs are updated routinely. -
Visualizing Finite Field of Order P 2 Through Matrices
Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 16 Issue 1 Version 1.0 Year 2016 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249 -4626 & Print ISSN: 0975-5896 Visualizing Finite Field of Order p2 through Matrices By S. K. Pandey Sardar Patel University of Police, India Abstract- In this article we consider finite field of order p2 (p ≠ 2). We provide matrix representation of finite field of order at most p2 = 121. However this article provides a technique to yield matrix field of order p2 for every prime p > 2 . Keywords: finite field, matrix field, Galois field. GJSFR-F Classification : MSC 2010: 12E20 VisualizingFiniteFieldofOrderp2throughMatrices Strictly as per the compliance and regulations of : © 2016. S. K. Pandey. This is a research/review paper, distributed under the terms of the Creative Commons Attribution- Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. Ref Visualizing Finite Field of Order p2 through Matrices 2016 r ea Y S. K. Pandey 271 Abstract- In this article we consider finite field of order 2 (pp ≠ 2). We provide matrix representation of finite field of order at most p2 =121 . However this article provides a technique to yield matrix field of order p2 for every prime p > 2 . V I Keywords: finite field, matrix field, Galois field. ue ersion I s s MSCS2010: 12E20. I I. Intro duction XVI There is a finite field of order pn for every positive prime p and positive integer n . -
Phd Thesis Parallel and Scalable Sparse Basic Linear Algebra
UNIVERSITY OF COPENHAGEN FACULTY OF SCIENCE PhD Thesis Weifeng Liu Parallel and Scalable Sparse Basic Linear Algebra Subprograms [Undertitel på afhandling] Academic advisor: Brian Vinter Submitted: 26/10/2015 ii To Huamin and Renheng for enduring my absence while working on the PhD and the thesis To my parents and parents-in-law for always helping us to get through hard times iii iv Acknowledgments I am immensely grateful to my supervisor, Professor Brian Vinter, for his support, feedback and advice throughout my PhD years. This thesis would not have been possible without his encouragement and guidance. I would also like to thank Profes- sor Anders Logg for giving me the opportunity to work with him at the Department of Mathematical Sciences of Chalmers University of Technology and University of Gothenburg. I would like to thank my co-workers in the eScience Center who supported me in every respect during the completion of this thesis: James Avery, Jonas Bardino, Troels Blum, Klaus Birkelund Jensen, Mads Ruben Burgdorff Kristensen, Simon Andreas Frimann Lund, Martin Rehr, Kenneth Skovhede, and Yan Wang. Special thanks goes to my PhD buddy James Avery, Huamin Ren (Aalborg Uni- versity), and Wenliang Wang (Bank of China) for their insightful feedback on my manuscripts before being peer-reviewed. I would like to thank Jianbin Fang (Delft University of Technology and National University of Defense Technology), Joseph L. Greathouse (AMD), Shuai Che (AMD), Ruipeng Li (University of Minnesota) and Anders Logg for their insights on my papers before being published. I thank Jianbin Fang, Klaus Birkelund Jensen, Hans Henrik Happe (University of Copenhagen) and Rune Kildetoft (University of Copenhagen) for access to the Intel Xeon and Xeon Phi machines. -
Multi-Color Low-Rank Preconditioner for General Sparse Linear Systems ∗
MULTI-COLOR LOW-RANK PRECONDITIONER FOR GENERAL SPARSE LINEAR SYSTEMS ∗ QINGQING ZHENG y , YUANZHE XI z , AND YOUSEF SAAD x Abstract. This paper presents a multilevel parallel preconditioning technique for solving general large sparse linear systems of equations. Subdomain coloring is invoked to reorder the coefficient matrix by multicoloring the quotient graph of the adjacency graph of the subdomains, resulting in a two-level block diagonal structure. A full binary tree structure T is then built to facilitate the construction of the preconditioner. We show that the difference between the inverse of a general block 2-by-2 SPD matrix and that of its block diagonal part can be well approximated by a low-rank matrix. This property and the block diagonal structure of the reordered matrix are exploited to develop a Multi-Color Low-Rank (MCLR) preconditioner. The construction procedure of the MCLR preconditioner follows a bottom-up traversal of the tree T . All irregular matrix computations, such as ILU factorizations and related triangular solves, are restricted to leaf nodes where these operations can be performed independently. Computations in non-leaf nodes only involve easy-to-optimize dense matrix operations. In order to further reduce the number of iteration of the Preconditioned Krylov subspace procedure, we combine MCLR with a few classical block-relaxation techniques. Numerical experiments on various test problems are proposed to illustrate the robustness and efficiency of the proposed approach for solving large sparse symmetric and nonsymmetric linear systems. Key words. Low-rank approximation; parallel preconditioners; domain decomposition; recur- sive multilevel methods; Krylov subspace methods. AMS subject classifications. -
Fast Large-Integer Matrix Multiplication
T.J. Smeding Fast Large-Integer Matrix Multiplication Bachelor thesis 11 July 2018 Thesis supervisors: dr. P.J. Bruin dr. K.F.D. Rietveld Leiden University Mathematical Institute Leiden Institute of Advanced Computer Science Abstract This thesis provides two perspectives on fast large-integer matrix multiplication. First, we will cover the complexity theory underlying recent developments in fast matrix multiplication algorithms by developing the theory behind, and proving, Schönhage’s τ-theorem. The theorems will be proved for matrix multiplication over commutative rings. Afterwards, we will discuss two newly developed programs for large-integer matrix multiplication using Strassen’s algorithm, one on the CPU and one on the GPU. Code from the GMP library is used on both of these platforms. We will discuss these implementations and evaluate their performance, and find that multi-core CPU platforms remain the most suited for large-integer matrix multiplication. 2 Contents 1 Introduction 4 2 Complexity Theory 5 2.1 Matrix Multiplication . .5 2.2 Computation . .5 2.3 Tensors . .6 2.3.1 Reduction of Abstraction . .6 2.3.2 Cost . .7 2.3.3 Rank . .8 2.4 Divisions & Rank . 10 2.5 Tensors & Rank . 13 2.5.1 Tensor Properties . 15 2.5.2 Rank Properties . 16 2.6 Matrix Multiplication Exponent . 17 2.7 Border Rank . 21 2.7.1 Schönhage’s τ-Theorem . 23 2.7.2 A Simple Application . 26 3 Performance 27 3.1 Introduction . 27 3.1.1 Input Data Set . 27 3.2 Large-Integer Arithmetic . 28 3.2.1 Multiplication Algorithms . 28 3.3 Implementations . -
Sparse Linear System Solvers on Gpus: Parallel Preconditioning, Workload Balancing, and Communication Reduction
UNIVERSIDAD JAUME I DE CASTELLÓN E. S. DE TECNOLOGÍA Y CIENCIAS EXPERIMENTALES SPARSE LINEAR SYSTEM SOLVERS ON GPUS: PARALLEL PRECONDITIONING, WORKLOAD BALANCING, AND COMMUNICATION REDUCTION CASTELLÓN DE LA PLANA,MARCH 2019 TESIS DOCTORAL PRESENTADA POR:GORAN FLEGAR DIRIGIDA POR:ENRIQUE S. QUINTANA-ORTÍ HARTWIG ANZT UNIVERSIDAD JAUME I DE CASTELLÓN E. S. DE TECNOLOGÍA Y CIENCIAS EXPERIMENTALES SPARSE LINEAR SYSTEM SOLVERS ON GPUS: PARALLEL PRECONDITIONING, WORKLOAD BALANCING, AND COMMUNICATION REDUCTION GORAN FLEGAR Abstract With the breakdown of Dennard scaling during the mid-2000s, and the end of Moore’s law on the horizon, hard- ware vendors, datacenters, and the high performance computing community are turning their attention towards unconventional hardware in hope of continuing the exponential performance growth of computational capacity. Among the available hardware options, a new generation of graphics processing units (GPUs), designed to support a wide variety of workloads in addition to graphics processing, is achieving the widest adoption. These processors are employed by the majority of today’s most powerful supercomputers to solve the world’s most complex problems in physics simulations, weather forecasting, data analytics, social network analysis, and ma- chine learning, among others. The potential of GPUs for these problems can only be unleashed by developing appropriate software, specifically tuned for the GPU architectures. Fortunately, many algorithms that appear in these applications are constructed out of the same basic building blocks. One example of a heavily-used building block is the solution of large, sparse linear systems, a challenge that is addressed in this thesis. After a quick overview of the current state-of-the-art methods for the solution of linear systems, this dis- sertation pays detailed attention to the class of Krylov iterative methods. -
A Study of Linear Error Correcting Codes
University of Plymouth PEARL https://pearl.plymouth.ac.uk 04 University of Plymouth Research Theses 01 Research Theses Main Collection 2007 A STUDY OF LINEAR ERROR CORRECTING CODES TJHAI, CEN JUNG http://hdl.handle.net/10026.1/1624 University of Plymouth All content in PEARL is protected by copyright law. Author manuscripts are made available in accordance with publisher policies. Please cite only the published version using the details provided on the item record or document. In the absence of an open licence (e.g. Creative Commons), permissions for further reuse of content should be sought from the publisher or author. A STUDY OF LINEAR ERROR CORRECTING CODES C. J., Tjhai Ph.D. September 2007 - Un-.vers.ty^o^H.y...ou-m Copyright © 2007 Cen Jung Tjhai This copy of the thesis has been supplied on condition that anyone who consults it is understood to recognise that its copyright rests with its author and that no quotation from the thesis and no information derived from it may be pubHshed without author's prior consent. A STUDY OF LINEAR ERROR CORRECTING CODES A thesis submitted to the University of Plymouth in partial fulfillment of the requirements for the degree of Doctor of Philosophy Cen Jung Tjhai September 2007 School of Computing, Communications and Electronics " *^ * Faculty of Technology University of Plymouth, UK A Study of Linear Error Correcting Codes Cen Jung Tjhai Abstract Since Shannon's ground-breaking work in 1948, there have been two main development streams of channel coding in approaching the limit of communication channels, namely classical coding theory which aims at designing codes with large minimum Hamming distance and probabilistic coding which places the emphasis on low complexity probabilistic decoding using long codes built from simple constituent codes. -
A Matricial Perpective of Wedderburn's Little Theorem
International Journal of Algebra, Vol. 5, 2011, no. 27, 1305 - 1311 A Matricial Perpective of Wedderburn’s Little Theorem Celino Miguel Instituto de Telecomunuca¸c˜oes P´olo de Coimbra Laborat´orio da Covilh˜a [email protected] Abstract Wedderburn’s classic Little Theorem states that all finite division rings are commutative. This beautiful result was been proved by many people using a variant of different ideias. In this paper we give a simple proof based in Matrix Theory. Mathematics Subject Classification: 12E15, 12E20 , 15A21 Keywords: Division Ring, Finite Field, Eigenvalue 1 Some Historical Notes. Up until the XIX century, the emergence of number systems was a natural process, mostly fostered by the solving of polynomial equations. For instance, the field of complex numbers C results from the mathematicians attempts to establish a process for solving cubic equations (for the history of the complex numbers origin see [10] or [2]). The mathematicians showed then no major concern for the precise definition of the structure of these new numbers which were making their entry into the world of mathematics. In 1843 Hamilton first discovers the Quaternions and shortly thereafter the Octonions are discovered independently by John T. Graves and Arthur Cayley. These new Hypercomplex Systems are no longer fields. The commutative law of multiplication is no longer applicable to quaternions and with octonions the associative law of multiplication must also be abandoned. But every non- zero element still has an inverse. Moreover, division by a non-zero element is still possible, a property which was regarded by the founders of the theory as crucial. -
Optimizing the Sparse Matrix-Vector Multiplication Kernel for Modern Multicore Computer Architectures
¨ thesis March 11, 2013 15:54 Page 1 © National Technical University of Athens School of Electrical and Computer Engineering Division of Computer Science Optimizing the Sparse Matrix-Vector Multiplication Kernel for Modern Multicore Computer Architectures ¨ ¨ © © Ph.D. esis Vasileios K. Karakasis Electrical and Computer Engineer, Dipl.-Ing. Athens, Greece December, 2012 ¨ © ¨ thesis March 11, 2013 15:54 Page 2 © ¨ ¨ © © ¨ © ¨ thesis March 11, 2013 15:54 Page 3 © National Technical University of Athens School of Electrical and Computer Engineering Division of Computer Science Optimizing the Sparse Matrix-Vector Multiplication Kernel for Modern Multicore Computer Architectures Ph.D. esis Vasileios K. Karakasis Electrical and Computer Engineer, Dipl.-Ing. Advisory Committee: Nectarios Koziris Panayiotis Tsanakas ¨ Andreas Stafylopatis ¨ © © Approved by the examining committee on December 19, 2012. Nectarios Koziris Panayiotis Tsanakas Andreas Stafylopatis Associate Prof., NTUA Prof., NTUA Prof., NTUA Andreas Boudouvis Giorgos Stamou Dimitrios Soudris Prof., NTUA Lecturer, NTUA Assistant Prof., NTUA Ioannis Cotronis Assistant Prof., UOA Athens, Greece December, 2012 ¨ © ¨ thesis March 11, 2013 15:54 Page 4 © Vasileios K. Karakasis Ph.D., National Technical University of Athens, Greece. ¨ ¨ © © Copyright © Vasileios K. Karakasis, 2012. Με επιφύλαξη παντός δικαιώματος. All rights reserved Απαγορεύεται η αντιγραφή, αποθήκευση και διανομή της παρούσας εργασίας, εξ ολοκλήρου ή τμήματος αυτής, για εμπορικό σκοπό. Επιτρέπεται η ανατύπωση, αποθήκευση και διανομή για σκοπό μη κερδοσκοπικό, εκπαιδευτικής ή ερευνητικής φύσης, υπό την προϋπόθεση να αναφέρε- ται η πηγή προέλευσης και να διατηρείται το παρόν μήνυμα. Ερωτήματα που αφορούν τη χρήση της εργασίας για κερδοσκοπικό σκοπό πρέπει να απευθύνονται προς τον συγγραφέα. Η έγκριση της διδακτορικής διατριβής από την Ανώτατη Σχολή των Ηλεκτρολόγων Μηχανικών και Μηχανικών Υπολογιστών του Ε. -
NATURAL PRODUCT ×N on MATRICES
NATURAL PRODUCT n ON MATRICES W. B. Vasantha Kandasamy Florentin Smarandache 2012 NATURAL PRODUCT n ON MATRICES 2 CONTENTS Preface 5 Chapter One INTRODUCTION 7 Chapter Two POLYNOMIALS WITH MATRIX COEFFICIENTS 11 Chapter Three ALGEBRAIC COEFFICIENTS USING MATRIX COEFFICIENT POLYNOMIALS 39 Chapter Four NATURAL PRODUCT ON MATRICES 91 Chapter Five NATURAL PRODUCT ON SUPERMATRICES 163 3 Chapter Six SUPERMATRIX LINEAR ALGEBRAS 225 Chapter Seven APPLICATIONS OF THESE ALGEBRAIC STRUCTURES WITH NATURAL PRODUCT 295 Chapter Eight SUGGESTED PROBLEMS 297 FURTHER READING 337 INDEX 339 ABOUT THE AUTHORS 342 4 PREFACE In this book the authors introduce a new product on matrices called the natural product. We see when two row matrices of 1 × n order are multiplied, the product is taken component wise; for instance if X = (x 1, x2, x 3, …, x n) and Y = (y 1, y 2, y 3, … , y n) then X × Y = (x 1y1, x 2y2, x 3y3, …, x nyn) which is also the natural product of X with Y. But we cannot find the product of a n × 1 column matrix with another n × 1 column matrix, infact the product is not defined. Thus if x1 y1 x y X = 2 and Y = 2 xn yn under natural product x1 y 1 x y X × Y = 2 2 . n xn y n Thus by introducing natural product we can find the product of column matrices and product of two rectangular matrices of same order. Further 5 this product is more natural which is just identical with addition replaced by multiplication on these matrices. Another fact about natural product is this enables the product of any two super matrices of same order and with same type of partition.