DOCSLIB.ORG

A Fast Parallel Gauss Jordan Algorithm for Matrix Inversion Using CUDA

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Fast Parallel Gauss Jordan Algorithm for Matrix Inversion Using CUDA Computers and Structures 128 (2013) 31–37 Contents lists available at SciVerse ScienceDirect Computers and Structures journal homepage: www.elsevier.com/locate/compstruc Technical Note A fast parallel Gauss Jordan algorithm for matrix inversion using CUDA ⇑ Girish Sharma a, Abhishek Agarwala b, Baidurya Bhattacharya a, a Department of Civil Engineering, Indian Institute of Technology, Kharagpur 721302, India b Archayne Labs, Gurgaon 122001, India article info abstract Article history: The ability to invert large matrices quickly and accurately determines the effectiveness of a computa- Received 5 April 2012 tional tool. Current literature suggests that time complexity of matrix inversion is 2 or higher. This paper Accepted 24 June 2013 redesigns the Gauss Jordan algorithm for matrix inversion on a CUDA platform to exploit the large scale Available online 13 September 2013 parallelization feature of a massively multithreaded GPU. The algorithm is tested for various types of matrices and the performance metrics are studied and compared with CPU based parallel methods. Keywords: We show that the time complexity of matrix inversion scales as n as long as n2 threads can be supported Graphics processing unit by the GPU. Compute unified development architecture Ó 2013 Elsevier Ltd. All rights reserved. Matrix inversion Gauss Jordan Parallelization 1. Introduction introduced the notion of border rank and obtained w < 2.78. Schönhage [25] generalized this notion in 1981, proving Matrix inversion is an essential step in a wide range of numer- w < 2.548. In the same paper, combining his work with ideas by ical problems – starting with solving linear equations [1–4], struc- Pan [23], he also showed w < 2.522. The following year, Romani tural analyses using finite element method [5–7], 3D rendering [8], [26] found that w < 2.517. The first result to break 2.5 was by digital filtering [9], image filtering [10,11] and image processing Coppersmith and Winograd [16] who obtained w < 2.496. In [12] – and constitutes an indispensable component in almost all 1988, Strassen introduced his laser method [27] which was a novel mathematical/statistical software suites. Some of the common approach for matrix multiplication, and thus decreased the bound available algorithms for computing the inverse of a matrix are to w < 2.479. Two years later, Coppersmith and Winograd [28] Strassen [13], Strassen-Newton [14], Gaussian elimination [15], combined Strassen’s technique with a novel form of analysis based Gauss–Jordan [15], Coppersmith and Winograd [16], LUP Decom- on large sets avoiding arithmetic progressions and obtained the position [17], Cholesky decomposition [18], QR decomposition famous bound of w < 2.376 which has remained unchanged for [19], RRQR factorization [20], Monte Carlo Methods for inverse more than 22 years (a very recent unpublished work [29] claims [21,22], etc. to have brought down the limit to w < 2.3727). While most activity Until the late 1960s matrix inversion was believed to require a focuses on trying to reduce the exponent w, both Coppersmith and cubic number of operations, as the fastest algorithm known was Winograd [28] and Cohn et al. [30] presented conjectures which if Gaussian elimination method, or rather Gauss Jordan [15] method true would imply w = 2, but never less than that. which runs in O(n3) time where n is the size of the matrix. In 1969, Inversion methods for specific types of matrices sometimes Strassen [13] excited the research community by giving the first with no set time complexity, like Monte Carlo Methods [21,22] sub cubic time algorithm for matrix multiplication, running in for inverting Hermitian matrix and positive definite matrix, a fast O(n2.808) time. This also reduced the time complexity, w, of Matrix algorithm [31] for the inversion of general Toeplitz matrices and Inversion using Strassen Multiplication to O(n2.808) time. This dis- various matrix decomposition methods [16–19] also exist. covery triggered a long line of research that gradually reduced Recent developments in parallel architecture and its use in the time complexity w over time. In 1978, Pan [23] presented a computation have brought about the prospect of massively parallel method that proved w < 2.796 and the next year, Bini et al. [24] systems capable of reducing running times of codes below the limit of w = 2. The amount of performance boost of course depends largely upon the scope of parallelization that the algorithm pro- vides. Owing to its unique architecture, the graphics processing unit ⇑ Corresponding author. Tel./fax: +91 3222 283422. (GPU) enables massive parallelization unlike anything possible on a E-mail addresses: [email protected], baidurya.bhattacharya@jhu. edu, [email protected] (B. Bhattacharya). CPU based network, as described later. The Gauss Jordan method is 0045-7949/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruc.2013.06.015 32 G. Sharma et al. / Computers and Structures 128 (2013) 31–37 one of the oldest methods for matrix inversion. It is straightforward Within a GPU, there are a couple of techniques for performing and robust and is particularly suitable for massive parallelization parallel computation. In the early years, General-purpose comput- unlike many of the more advanced methods. Nevertheless, the ing on graphics processing units (GPGPU) [38] was the technique available literature either on the scope of parallelization of the used to harness the computational power of a GPU by program- Gauss Jordan algorithm or its optimization appear rather insuffi- mers, until CUDA (Compute Unified Device Architecture) was made cient. This paper tailors and implements the Gauss–Jordan publically available in 2007 as the platform to replace GPGPU. algorithm for matrix inversion on a CUDA (Compute Unified Device CUDA has several advantages [39] over GPGPU and CPU which Architecture [32]) based GPU platform and studies the performance includes faster memory sharing and read backs, minimal threads metrics of the algorithm. creation overhead, and flexibility in the choice of programming language, and has been adopted for this work. More extensive 2. Parallelization and challenges information about Nvidia and CUDA model is explained in the Programming Guide published by Nvidia [32]. The Strassen approach [14,33] for matrix inversion reduces the problem of inverting an n  n matrix into seven n/2  n/2 multipli- 3. Parallel Gauss–Jordan algorithm cations and inversion of two n/2  n/2 matrices, each of which then can be solved recursively. The inverse is given as: Gauss Jordan method for computation of inverse of a matrix is À1 one of the oldest methods. It is robust, accurate over range of matri- A11 A12 C11 C12 AÀ1 ¼ ¼ ð1Þ ces and does not require multiple checks depending upon the type A21 A22 C21 C22 of matrix involved. Existing work on the parallelization of the Gauss Jordan algorithm (e.g. [33]) has been limited in the scope for The four partitions of C may be computed in the following steps: parallelization mainly due to the hardware limitations of the time. À1 This paper redesigns the Gauss Jordan method so as to make full use 1: P1 ¼ A11 2: P2 ¼ A21  P1 3: P3 ¼ P1  A12 of the massive parallelization feature of a modern GPU. 4: P ¼ A  P 5: P ¼ P À A 6: P ¼ PÀ1 4 21 3 5 4 22 6 5 ð2Þ The standard Gauss Jordan method for computation of inverse 7: C12 ¼ P3  P6 8: C21 ¼ P6  P2 9: C11 ¼ P1 À P3  C21 of a matrix A of size n starts by augmenting the matrix with the 10: C22 ¼P6 identity matrix of size n: For large n, the individual matrices will be large enough to pro- ½C¼½AIj ð3Þ vide scope for parallelization of the matrix multiplication steps (steps 2, 3, 4, 7, 8 and 9) while the inverse calculation steps (steps Then, performing elementary row transformations on matrix C, the 0 1 and 6) can be performed recursively. It is this inherent recursive left half of C is transformed column by column into the unit matrix. nature of the algorithm that reduces the scope of large scale This step is broken down into 2 steps per column of the left half parallelization. matrix, the first of which is to convert element aii into 1 by the An additional issue faced by the Strassen approach is that the transformation: accuracy of the resultant inverse depends highly on the quarter Ri Ri=aii ð4Þ matrix A11. If the quarter matrix chosen in step 1 is singular, then pivoting is required and the matrix with the larger determinant be- If aii is zero, then any non-zero row is added to the ith row before tween A11 and A22 is chosen for step 1. This check is performed in applying Eq. (4). The second step is to reduce all the other elements each level of recursion in steps 1 as well as 6, making this algo- of the jth column to zero by the following transformation of every rithm cumbersome. While certain parallel implementations for row except for the jth one: specific types of matrices already have been studied before Ri Ri À Rj  aij ð5Þ [34,35], developing an algorithm to comply with all types of matri- ces is a difficult task. A much easier and robust parallel implemen- After transforming the first column, the matrix reduces to: 2 3 1 a =a a =a 12 11 13 11 ÁÁÁ ÁÁÁ 1=a11 00ÁÁÁ 0 6 7 6 0 a22 À a21  a12=a11 a23 À a21  a13=a11 ÁÁÁ ÁÁÁ 7 6 Àa21=a11 10ÁÁÁ 0 7 0 6 0 a a a =a a a a =a 7 6 32 À 31  12 11 33 À 31  13 11 ÁÁÁ ÁÁÁ Àa31=a11 01ÁÁÁ 0 7 ½C ¼6 7 ð6Þ 6 .
Load more

Recommended publications
  • Realizing Euclidean Distance Matrices by Sphere Intersection
    Realizing Euclidean distance matrices by sphere intersection Jorge Alencara,∗, Carlile Lavorb, Leo Libertic aInstituto Federal de Educa¸c~ao,Ci^enciae Tecnologia do Sul de Minas Gerais, Inconfidentes, MG, Brazil bUniversity of Campinas (IMECC-UNICAMP), 13081-970, Campinas-SP, Brazil cCNRS LIX, Ecole´ Polytechnique, 91128 Palaiseau, France Abstract This paper presents the theoretical properties of an algorithm to find a re- alization of a (full) n × n Euclidean distance matrix in the smallest possible embedding dimension. Our algorithm performs linearly in n, and quadratically in the minimum embedding dimension, which is an improvement w.r.t. other algorithms. Keywords: Distance geometry, sphere intersection, euclidean distance matrix, embedding dimension. 2010 MSC: 00-01, 99-00 1. Introduction Euclidean distance matrices and sphere intersection have a strong mathe- matical importance [1, 2, 3, 4, 5, 6], in addition to many applications, such as navigation problems, molecular and nanostructure conformation, network 5 localization, robotics, as well as other problems of distance geometry [7, 8, 9]. Before defining precisely the problem of interest, we need the formal defini- tion for Euclidean distance matrices. Let D be a n × n symmetric hollow (i.e., with zero diagonal) matrix with non-negative elements. We say that D is a (squared) Euclidean Distance Matrix (EDM) if there are points x1; x2; : : : ; xn 2 RK (for a positive integer K) such that 2 D(i; j) = Dij = kxi − xjk ; i; j 2 f1; : : : ; ng; where k · k denotes the Euclidean norm. The smallest K for which such a set of points exists is called the embedding dimension of D, denoted by dim(D).
    [Show full text]
  • Arxiv:1701.03378V1 [Math.RA] 12 Jan 2017 Setao Apihe Rusadfe Probability” Free and Groups T Lamplighter by on Supported Technology
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by TUGraz OPEN Library Linearizing the Word Problem in (some) Free Fields Konrad Schrempf∗ January 13, 2017 Abstract We describe a solution of the word problem in free fields (coming from non- commutative polynomials over a commutative field) using elementary linear algebra, provided that the elements are given by minimal linear representa- tions. It relies on the normal form of Cohn and Reutenauer and can be used more generally to (positively) test rational identities. Moreover we provide a construction of minimal linear representations for the inverse of non-zero elements. Keywords: word problem, minimal linear representation, linearization, realiza- tion, admissible linear system, rational series AMS Classification: 16K40, 16S10, 03B25, 15A22 Introduction arXiv:1701.03378v1 [math.RA] 12 Jan 2017 Free (skew) fields arise as universal objects when it comes to embed the ring of non-commutative polynomials, that is, polynomials in (a finite number of) non- commuting variables, into a skew field [Coh85, Chapter 7]. The notion of “free fields” goes back to Amitsur [Ami66]. A brief introduction can be found in [Coh03, Section 9.3], for details we refer to [Coh95, Section 6.4]. In the present paper we restrict the setting to commutative ground fields, as a special case. See also [Rob84]. In [CR94], Cohn and Reutenauer introduced a normal form for elements in free fields in order to extend results from the theory of formal languages. In particular they characterize minimality of linear representations in terms of linear independence of ∗Contact: [email protected], Department of Discrete Mathematics (Noncommutative Structures), Graz University of Technology.
    [Show full text]
  • Left Eigenvector of a Stochastic Matrix
    Advances in Pure Mathematics, 2011, 1, 105-117 doi:10.4236/apm.2011.14023 Published Online July 2011 (http://www.SciRP.org/journal/apm) Left Eigenvector of a Stochastic Matrix Sylvain Lavalle´e Departement de mathematiques, Universite du Quebec a Montreal, Montreal, Canada E-mail: [email protected] Received January 7, 2011; revised June 7, 2011; accepted June 15, 2011 Abstract We determine the left eigenvector of a stochastic matrix M associated to the eigenvalue 1 in the commu- tative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenvalue 0 is (,,NN1 n ), where Ni is the ith principal minor of NMI= n , where In is the 11 identity matrix of dimension n . In the noncommutative case, this eigenvector is (,P1 ,Pn ), where Pi is the sum in aij of the corresponding labels of nonempty paths starting from i and not passing through i in the complete directed graph associated to M . Keywords: Generic Stochastic Noncommutative Matrix, Commutative Matrix, Left Eigenvector Associated To The Eigenvalue 1, Skew Field, Automata 1. Introduction stochastic free field and that the vector 11 (,,PP1 n ) is fixed by our matrix; moreover, the sum 1 It is well known that 1 is one of the eigenvalue of a of the Pi is equal to 1, hence they form a kind of stochastic matrix (i.e. the sum of the elements of each noncommutative limiting probability. row is equal to 1) and its associated right eigenvector is These results have been proved in [1] but the proof the vector (1,1, ,1)T .
    [Show full text]
  • ON EUCLIDEAN DISTANCE MATRICES of GRAPHS∗ 1. Introduction. a Matrix D ∈ R N×N Is a Euclidean Distance Matrix (EDM), If Ther
    Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 26, pp. 574-589, August 2013 ELA ON EUCLIDEAN DISTANCE MATRICES OF GRAPHS∗ GASPERˇ JAKLICˇ † AND JOLANDA MODIC‡ Abstract. In this paper, a relation between graph distance matrices and Euclidean distance matrices (EDM) is considered. It is proven that distance matrices of paths and cycles are EDMs. The proofs are constructive and the generating points of studied EDMs are given in a closed form. A generalization to weighted graphs (networks) is tackled. Key words. Graph, Euclidean distance matrix, Distance, Eigenvalue. AMS subject classifications. 15A18, 05C50, 05C12. n n 1. Introduction. A matrix D R × is a Euclidean distance matrix (EDM), ∈ if there exist points x Rr, i =1, 2,...,n, such that d = x x 2. The minimal i ∈ ij k i − j k possible r is called an embedding dimension (see, e.g., [3]). Euclidean distance matri- ces were introduced by Menger in 1928, later they were studied by Schoenberg [13], and other authors. They have many interesting properties, and are used in various applications in linear algebra, graph theory, and bioinformatics. A natural problem is to study configurations of points xi, where only distances between them are known. n n Definition 1.1. A matrix D = (d ) R × is hollow, if d = 0 for all ij ∈ ii i =1, 2,...,n. There are various characterizations of EDMs. n n Lemma 1.2. [8, Lemma 5.3] Let a symmetric hollow matrix D R × have only ∈ one positive eigenvalue λ and the corresponding eigenvector e := [1, 1,..., 1]T Rn.
    [Show full text]
  • A Submodular Approach
    SPARSEST NETWORK SUPPORT ESTIMATION: A SUBMODULAR APPROACH Mario Coutino, Sundeep Prabhakar Chepuri, Geert Leus Delft University of Technology ABSTRACT recovery of the adjacency and normalized Laplacian matrix, in the In this work, we address the problem of identifying the underlying general case, due to the convex formulation of the recovery problem, network structure of data. Different from other approaches, which solutions that only retrieve the support approximately are obtained, are mainly based on convex relaxations of an integer problem, here i.e., thresholding operations are required. we take a distinct route relying on algebraic properties of a matrix Taking this issue into account, and following arguments closely representation of the network. By describing what we call possible related to the ones used for network topology estimation through ambiguities on the network topology, we proceed to employ sub- spectral templates [10], the main aim of this work is twofold: (i) to modular analysis techniques for retrieving the network support, i.e., provide a proper characterization of a particular set of matrices that network edges. To achieve this we only make use of the network are of high importance within graph signal processing: fixed-support modes derived from the data. Numerical examples showcase the ef- matrices with the same eigenbasis, and (ii) to provide a pure com- fectiveness of the proposed algorithm in recovering the support of binatorial algorithm, based on the submodular machinery that has sparse networks. been proven useful in subset selection problems within signal pro- cessing [18–22], as an alternative to traditional convex relaxations. Index Terms— Graph signal processing, graph learning, sparse graphs, network deconvolution, network topology inference.
    [Show full text]
  • 69 Adjacency Matrix
    Index A-optimality, 439 angle between matrices, 168 absolute error, 484, 494, 526 angle between vectors, 37, 229, 357 ACM Transactions on Mathematical ANSI (standards), 475, 562, 563 Software, 552, 617 Applied Statistics algorithms, 617 adj(·), 69 approximation and estimation, 431 adjacency matrix, 331–334, 390 approximation of a matrix, 174, 257, Exercise 8.20:, 396 339, 437, 533 augmented, 390 approximation of a vector, 41–43 adjoint (see also conjugate transpose), arithmetic mean, 35, 37 59 Arnoldi method, 318 adjoint, classical (see also adjugate), 69 artificial ill-conditioning, 269 adjugate, 69, 598 ASCII code, 460 adjugate and inverse, 118 association matrix, 328–338, 357, 365, affine group, 115, 179 366, 369–371 affine space, 43 adjacency matrix, 331–332, 390 affine transformation, 228 connectivity matrix, 331–332, 390 Aitken’s integral, 217 dissimilarity matrix, 369 AL(·) (affine group), 115 distance matrix, 369 algebraic multiplicity, 144 incidence matrix, 331–332, 390 similarity matrix, 369 algorithm, 264, 499–513 ATLAS (Automatically Tuned Linear batch, 512 Algebra Software), 555 definition, 509 augmented adjacency matrix, 390 direct, 264 augmented connectivity matrix, 390 divide and conquer, 506 Automatically Tuned Linear Algebra greedy, 506 Software (ATLAS), 555 iterative, 264, 508–511, 564 autoregressive process, 447–450 reverse communication, 564 axpy, 12, 50, 83, 554, 555 online, 512 axpy elementary operator matrix, 83 out-of-core, 512 real-time, 512 back substitution, 274, 406 algorithmically singular, 121 backward error analysis,
    [Show full text]
  • The Paving Property for Uniformly Bounded Matrices
    THE PAVING PROPERTY FOR UNIFORMLY BOUNDED MATRICES JOEL A. TROPP Abstract. This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison–Singer problem. The result shows that every unit- norm matrix whose entries are relatively small in comparison with its dimension can be paved by a partition of constant size. That is, the coordinates can be partitioned into a constant number of blocks so that the restriction of the matrix to each block of coordinates has norm less than one half. The original proof of Bourgain and Tzafriri involves a long, delicate calculation. The new proof relies on the systematic use of symmetrization and Khintchine inequalities to estimate the norm of some random matrices. 1. Introduction This note presents a new proof of a result about the paving problem for matrices. Suppose that A is an n × n matrix. We say that A has an (m, ε)-paving if there exists a partition of the set {1, 2, . , n} into m blocks {σ1, σ2, . , σm} so that Xm Pσ APσ ≤ ε kAk j=1 j j where Pσj denotes the diagonal projector onto the coordinates listed in σj. Since every projector in this note is diagonal, we will omit the qualification from here onward. As usual, k·k is the norm n on linear operators mapping `2 to itself. The fundamental question concerns the paving of matrices with a zero diagonal (i.e., hollow matrices). Problem 1 (Paving Problem). Fix ε ∈ (0, 1). Is there a constant m = m(ε) so that, for sufficiently large n, every hollow n × n matrix has an (m, ε)-paving? Anderson [And79] has shown that the Paving Problem is equivalent with the Kadison–Singer problem, which is a major open question in operator theory.
    [Show full text]
  • On Orthogonal Matrices with Zero Diagonal
    On orthogonal matrices with zero diagonal R. F. Bailey∗ R. Craigeny July 23, 2019 Abstract This paper considers real orthogonal n × n matrices whose diagonal entries are zero and off-diagonal entries nonzero, which are referred to as OMZD(n). It is shown that there exists an OMZD(n) if and only if n 6= 1; 3, and that a symmetric OMZD(n) exists if and only if n is even and n 6= 4. Also, a construction of OMZD(n) obtained from doubly regular tournaments is given. Finally, the results are applied to determine the minimum number of distinct eigenvalues of matrices associated with some families of graphs, and the related notion of orthogonal matrices with partially-zero diagonal is considered. Keywords: Orthogonal matrix; orthogonal pattern; zero diagonal; distinct eigenvalues. MSC2010: 15B10 (primary), 05B20, 05C50, 15A18 (secondary) 1 Introduction T + An n×n real matrix A is orthogonal if and only if AA = cI for some positive constant c 2 R . (We do not insist that c = 1, i.e. that AAT = I.) A number of papers, such as [3, 5, 8, 19], have studied the pattern (or the support) of a matrix, i.e. the arrangement of its zero and nonzero entries, and which patterns admit matrices with given algebraic properties (for instance, orthogonal, positive definite, etc.). In this vein, a pattern is called an orthogonal pattern if it admits an orthogonal matrix. A seemingly natural pattern to consider is where the zero entries are precisely those on the main diagonal; orthogonal matrices with this pattern are the subject of this paper.
    [Show full text]
  • Sign Patterns of Orthogonal Matrices and the Strong Inner Product Property
    Sign Patterns of Orthogonal Matrices and the Strong Inner Product Property Bryan A. Curtisa, Bryan L. Shadera aDepartment of Mathematics, University of Wyoming, Laramie, WY 82071, USA Abstract A new condition, the strong inner product property, is introduced and used to construct sign patterns of row orthogonal matrices. Using this property, infinite families of sign patterns allowing row orthogonality are found. These provide insight into the underlying combinatorial structure of row orthogonal matrices. Algorithmic techniques for verifying that a matrix has the strong inner product property are also presented. These techniques lead to a generalization of the strong inner product property and can be easily implemented using various software. Keywords: Strong inner product property, sign pattern, orthogonality, orthogonal matrix, row orthogonal matrix 2010 MSC: 15B10, 2010 MSC: 15B35 1. Introduction Characterizing the sign patterns of orthogonal matrices has been of interest since the early 1960’s. This endeavor was first proposed by M. Fiedler, in June 1963, at the Symposium on the Theory of Graphs and Its Applications [1]. More recently there has been renewed interest in sign patterns of orthogonal matrices [2], [3], and in related qualitative and combinatorial problems [4], [5], [6]. There has been some success in characterizing the sign patterns of orthogonal matrices for small orders or with additional combinatorial constraints [7], [8], [9]. As of arXiv:1907.09982v1 [math.CO] 23 Jul 2019 the publication of this paper there is no characterization for orders n 6. For many years the only known necessary condition for a sign patte≥ rn to allow orthogonality was potential orthogonality; that is the rows (respectively columns) are nonzero and the sign patterns of each pair of rows (respectively columns) have a realization that are orthogonal.
    [Show full text]
  • An Algorithm for Realizing Euclidean Distance Matrices
    An algorithm for realizing Euclidean distance matrices Jorge Alencar 1 Instituto Federal de Educa¸c~ao,Ci^enciae Tecnologia do Sul de Minas Gerais, Inconfidentes, MG, Brazil Tib´eriusBonates 2 Depto de Estat´ıstica e Matem´atica Aplicada, Universidade Federal do Cear´a, Fortaleza, CE, Brazil Carlile Lavor 3 Instituto de Matem´atica, Estat´ıstica e Computa¸c~aoCient´ıfica, Universidade Estadual de Campinas, Campinas, SP, Brazil Leo Liberti 4 CNRS LIX, Ecole´ Polytechnique, 91128 Palaiseau, France Abstract We present an efficient algorithm to find a realization of a (full) n × n squared Euclidean distance matrix in the smallest possible dimension. Most existing algo- rithms work in a given dimension: most of these can be transformed to an algorithm to find the minimum dimension, but gain a logarithmic factor of n in their worst- case running time. Our algorithm performs cubically in n (and linearly when the dimension is fixed, which happens in most applications). Keywords: Distance geometry, sphere interesection, EDM, embedding dimension. 1 Introduction The problem of adjustment of distances among points has been studied since the first decades of the 20th century [2]. It can be formally defined as follows: Let D be a n×n symmetric hollow (i.e., with zero diagonal) matrix with non- negative elements. We say that D is a squared Euclidean Distance Matrix K (EDM) if there are x1; x2; : : : ; xn 2 R , for a positive integer K, such that 2 D(i; j) = Dij = kxi − xjk ; i; j 2 f1; : : : ; ng; where k · k denotes the Euclidean norm. The smallest K for which such a set of points exists is called the embedding dimension of D, denoted by dim(D).
    [Show full text]
  • Appendices a Notation and Definitions
    Appendices A Notation and Definitions All notation used in this work is “standard”, and I have endeavored to use notation consistently. I have opted for simple notation, which, of course, results in a one-to-many map of notation to object classes. Within a given context, however, the overloaded notation is generally unambiguous. This appendix is not intended to be a comprehensive listing of definitions. The Index is a more reliable set of pointers to definitions, except for symbols that are not words. A.1 General Notation Uppercase italic Latin and Greek letters, such as A, B, E, Λ, etc., are generally used to represent either matrices or random variables. Random variables are usually denoted by letters nearer the end of the Latin alphabet, such X, Y ,and Z, and by the Greek letter E. Parameters in models (that is, unobservables in the models), whether or not they are considered to be random variables, are generally represented by lowercase Greek letters. Uppercase Latin and Greek letters are also used to represent cumulative distribution functions. Also, uppercase Latin letters are used to denote sets. Lowercase Latin and Greek letters are used to represent ordinary scalar or vector variables and functions. No distinction in the notation is made between scalars and vectors;thus,β may represent a vector and βi may represent the ith element of the vector β. In another context, however, β may represent a scalar. All vectors are considered to be column vectors, although we may write a vector as x =(x1,x2,...,xn). Transposition of a vector or a matrix is denoted by the superscript “T”.
    [Show full text]
  • Localization: on Division Rings and Tilting Modules
    Localization: On Division Rings and Tilting Modules Javier S´anchez Serd`a Mem`oriapresentada per a aspirar al grau de doctor en Matem`atiques Certifico que la present Mem`oria ha estat realitzada per en Javier S´anchez Serd`a, sota la direcci´ode la Dra. Dolors Herbera Espinal. Bellaterra, Juny 2008 Firmat: Dra. Dolors Herbera Espinal “And here we are again the door is closed behind us and the long road lies ahead where do we go from here?” Fates Warning, Leave the past behind Introduction Let R be a ring. Suppose that R embeds in a division ring E (i.e. E is a not necessarily commutative field). By the division ring of fractions of R inside E we mean the embedding R,→ E(R) where E(R) denotes the intersection of all subdivision rings of E that contain R. When R is a commutative ring, the answer to whether R is embeddable in a division ring is well known: (a) Existence: R has a division ring of fractions Q(R) if and only if R is a domain, i.e. R is a nonzero ring such that xy = 0 implies that x = 0 or y = 0. (b) Uniqueness: If λ: R,→ Q(R) is a division ring of fractions of R, given any embedding ψ : R,→ E in a division ring E, there exists a morphism of rings ψ¯: Q(R) → E such that ψλ¯ = ψ. In particular E(R) =∼ Q(R). (c) Form of the elements: The division ring of fractions Q(R) is constructed in an analogous way as the rationals from the integers.
    [Show full text]