DOCSLIB.ORG

Dense Matrix Computations : Communication Cost and Numerical Stability

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

No d’ordre : No attribué par la bibliothèque : -UNIVERSITÉ PARIS-SUD - Laboratoire de Recherche en Informatique - UMR 427 - LRI THÈSE en vue d’obtenir le grade de Docteur de l’Université Paris-Sud Spécialité : Informatique au titre de l’École Doctorale Informatique Paris-Sud présentée et soutenue publiquement le 11 Février 2013 par Amal KHABOU Dense matrix computations : communication cost and numerical stability. Directeur de thèse : Docteur Laura GRIGORI Après avis de : Professeur Nicholas J. HIGHAM Rapporteur Professeur Yves ROBERT Rapporteur Devant la commission d’examen formée de : Professeur Iain DUFF Examinateur Professeur Yannis MANOUSSAKIS Examinateur Enseignant-chercheur Jean-Louis ROCH Examinateur ii Abstract: This dissertation focuses on a widely used linear algebra kernel to solve linear systems, that is the LU decomposition. Usually, to perform such a computation one uses the Gaussian elimination with partial pivoting (GEPP). The backward stability of GEPP depends on a quantity which is referred to as the growth factor, it is known that in general GEPP leads to modest element growth in practice. However its parallel version does not attain the communication lower bounds. Indeed the panel factorization rep- resents a bottleneck in terms of communication. To overcome this communication bottleneck, Grigori et al [60] have developed a communication avoiding LU factorization (CALU), which is asymptotically optimal in terms of communication cost at the cost of some redundant computation. In theory, the upper bound of the growth factor is larger than that of Gaussian elimination with partial pivoting, however CALU is stable in practice. To improve the upper bound of the growth factor, we study a new pivoting strategy based on strong rank revealing QR factorization. Thus we develop a new block algorithm for the LU factorization. This algorithm has a smaller growth factor upper bound compared to Gaussian elimination with partial pivoting. The strong rank revealing pivoting is then combined with tournament pivoting strategy to produce a communication avoiding LU factorization that is more stable than CALU. For hierarchical systems, multiple levels of parallelism are available. However, none of the previously cited methods fully exploit these hierarchical systems. We propose and study two recursive algorithms based on the communication avoiding LU algorithm, which are more suitable for architectures with multiple levels of parallelism. For an accurate and realistic cost analysis of these hierarchical algo- rithms, we introduce a hierarchical parallel performance model that takes into account processor and network hierarchies. This analysis enables us to accurately predict the performance of the hierarchical LU factorization on an exascale platform. Keywords: LU factorization, Gaussian elimination with partial pivoting, growth factor, strong rank re- vealing QR factorization, minimizing the communication cost, parallel algorithms, hierarchi- cal systems, performance models, pivoting strategies. Résumé : Cette thèse traite d’une routine d’algèbre linéaire largement utilisée pour la résolution des systèmes li- néaires, il s’agit de la factorisation LU. Habituellement, pour calculer une telle décomposition, on utilise l’élimination de Gauss avec pivotage partiel (GEPP). La stabilité numérique de l’élimination de Gauss avec pivotage partiel est caractérisée par un facteur de croissance qui est reste assez petit en pratique. Toutefois, la version parallèle de cet algorithme ne permet pas d’atteindre les bornes inférieures qui ca- ractérisent le coût de communication pour un algorithme donné. En effet, la factorisation d’un bloc de colonnes constitue un goulot d’étranglement en termes de communication. Pour remédier à ce problème, Grigori et al [60] ont développé une factorisation LU qui minimise la communication(CALU) au prix de quelques calculs redondants. En théorie la borne supérieure du facteur de croissance de CALU est plus grande que celle de l’élimination de Gauss avec pivotage partiel, cependant CALU est stable en pratique. Pour améliorer la borne supérieure du facteur de croissance, nous étudions une nouvelle stra- tégie de pivotage utilisant la factorisation QR avec forte révélation de rang. Ainsi nous développons un nouvel algorithme pour la factorisation LU par blocs. La borne supérieure du facteur de croissance de cet algorithme est plus petite que celle de l’élimination de Gauss avec pivotage partiel. Cette stratégie de pivotage est ensuite combinée avec le pivotage basé sur un tournoi pour produire une factorisation LU qui minimise la communication et qui est plus stable que CALU. Pour les systèmes hiérarchiques, plusieurs niveaux de parallélisme sont disponibles. Cependant, aucune des méthodes précédemment ci- tées n’exploite pleinement ces ressources. Nous proposons et étudions alors deux algorithmes récursifs qui utilisent les mêmes principes que CALU mais qui sont plus appropriés pour des architectures à plu- sieurs niveaux de parallélisme. Pour analyser d’une façon précise et réaliste les coûts de ces algorithmes hiérarchiques, nous introduisons un modèle de performance hiérarchique parallèle qui prend en compte aussi bien l’organisation hiérarchique des processeurs et la hiérarchie réseau. Cette analyse nous permet de prédire d’une manière réaliste la performance de ces factorisations hiérarchiques sur une plate-forme exascale. Mots-clés : Factorisation LU, élimination de Gauss avec pivotage partiel, facteur de croissance, facto- risation QR avec forte révélation de rang, minimisation de la communication, algorithmes parallèles, systèmes hiérarchiques, modèles de performance, stratégies de pivotage. Contents Introduction i 1 High performance numerical linear algebra1 1.1 Introduction.........................................1 1.2 Direct methods.......................................1 1.2.1 The LU factorization................................2 1.2.2 The QR factorization................................4 1.3 Communication "avoiding" algorithms...........................6 1.3.1 Communication...................................7 1.3.2 A simplified performance model.........................7 1.3.3 Theoretical lower bounds..............................8 1.3.4 Motivation case: LU factorization as in ScaLAPACK (pdgetrf)..........9 1.3.5 Communication avoiding LU (CALU)....................... 10 1.3.6 Numerical stability................................. 11 1.4 Parallel pivoting strategies................................. 12 1.5 Principles of parallel dense matrix computations...................... 13 1.5.1 Parallel algorithms................................. 14 1.5.2 Tasks decomposition and scheduling techniques.................. 14 1.6 Parallel performance models................................ 18 1.6.1 PRAM model.................................... 19 1.6.2 Network models.................................. 19 1.6.3 Bridging models.................................. 19 1.6.4 Discussion..................................... 21 2 LU factorization with panel rank revealing pivoting 23 2.1 Introduction......................................... 23 2.2 Matrix algebra........................................ 24 2.2.1 Numerical stability................................. 29 2.2.2 Experimental results................................ 30 2.3 Conclusion......................................... 35 3 Communication avoiding LU factorization with panel rank revealing pivoting 37 3.1 Introduction......................................... 37 3.2 Communication avoiding LU_PRRP............................ 39 3.2.1 Matrix algebra................................... 39 3.2.2 Numerical Stability of CALU_PRRP....................... 43 3.2.3 Experimental results................................ 45 iii iv CONTENTS 3.3 Cost analysis of CALU_PRRP............................... 53 3.4 Less stable factorizations that can also minimize communication............. 56 3.5 Conclusion......................................... 59 4 Avoiding communication through a multilevel LU factorization 61 4.1 Introduction......................................... 61 4.2 LU factorization suitable for hierarchical computer systems................ 63 4.2.1 Multilevel CALU algorithm (2D multilevel CALU)............... 64 4.2.2 Numerical stability of multilevel CALU...................... 66 4.3 Recursive CALU (1D multilevel CALU)......................... 75 4.3.1 Recursive CALU algorithm............................ 75 4.4 Performance model of hierarchical LU factorizations................... 77 4.4.1 Cost analysis of 2-level CALU.......................... 78 4.4.2 Cost analysis of 2-recursive CALU........................ 79 4.5 Multilevel CALU performance............................... 80 4.5.1 Implementation on a cluster of multicore processors............... 80 4.5.2 Performance of 2-level CALU........................... 81 4.6 Conclusion......................................... 84 5 On the performance of multilevel CALU 87 5.1 Introduction......................................... 87 5.2 Toward a realistic Hierarchical Cluster Platform model (HCP).............. 88 5.3 Cost analysis of multilevel Cannon algorithm under the HCP model........... 93 5.4 Cost analysis of multilevel CALU under the HCP model................. 95 5.5 Performance predictions.................................. 102 5.5.1 Modeling an Exascale platform with HCP..................... 104 5.5.2 Performance predictions of multilevel CALU................... 105 5.6 Related work.......................................
Recommended publications
  • Numeric-Symbolic Exact Rational Linear System Solver∗

    Numeric-Symbolic Exact Rational Linear System Solver∗

    Numeric-Symbolic Exact Rational Linear System Solver∗ B. David Saunders, David Harlan Wood, and Bryan S. Youse Dept. of Computer and Information Sciences, University of Delaware Newark, Delaware, USA [email protected], [email protected], [email protected] ABSTRACT 1. INTRODUCTION An iterative refinement approach is taken to rational linear We address the problem of solving Ax = b for a vector x 2 n m×n m system solving. Such methods produce, for each entry of the Q ; given A 2 Q and b 2 Q : We will restrict ourselves solution vector, a rational approximation with denominator to square (m = n), nonsingular matrices with integer entries a power of 2. From this the correct rational entry can be of length d bits or fewer. This is the core problem. Also, in reconstructed. Our iteration is a numeric-symbolic hybrid this paper we are concerned with dense matrices, which is in that it uses an approximate numeric solver at each step to say, matrices that do not have so many zero entries that together with a symbolic (exact arithmetic) residual com- more specialized sparse matrix techniques should be applied. putation and symbolic rational reconstruction. The rational We do anticipate that the refined numeric-symbolic iterative solution may be checked symbolically (exactly). However, approach presented here will also apply effectively to sparse there is some possibility of failure of convergence, usually systems. due to numeric ill-conditioning. Alternatively, the algorithm We present a method which is a refinement of the numeric- may be used to obtain an extended precision floating point symbolic method of Wan[27, 26].
  • A Survey of Direct Methods for Sparse Linear Systems

    A Survey of Direct Methods for Sparse Linear Systems

    A survey of direct methods for sparse linear systems Timothy A. Davis, Sivasankaran Rajamanickam, and Wissam M. Sid-Lakhdar Technical Report, Department of Computer Science and Engineering, Texas A&M Univ, April 2016, http://faculty.cse.tamu.edu/davis/publications.html To appear in Acta Numerica Wilkinson defined a sparse matrix as one with enough zeros that it pays to take advantage of them.1 This informal yet practical definition captures the essence of the goal of direct methods for solving sparse matrix problems. They exploit the sparsity of a matrix to solve problems economically: much faster and using far less memory than if all the entries of a matrix were stored and took part in explicit computations. These methods form the backbone of a wide range of problems in computational science. A glimpse of the breadth of applications relying on sparse solvers can be seen in the origins of matrices in published matrix benchmark collections (Duff and Reid 1979a) (Duff, Grimes and Lewis 1989a) (Davis and Hu 2011). The goal of this survey article is to impart a working knowledge of the underlying theory and practice of sparse direct methods for solving linear systems and least-squares problems, and to provide an overview of the algorithms, data structures, and software available to solve these problems, so that the reader can both understand the methods and know how best to use them. 1 Wilkinson actually defined it in the negation: \The matrix may be sparse, either with the non-zero elements concentrated ... or distributed in a less systematic manner. We shall refer to a matrix as dense if the percentage of zero elements or its distribution is such as to make it uneconomic to take advantage of their presence." (Wilkinson and Reinsch 1971), page 191, emphasis in the original.
  • Arxiv:2006.15226V1 [Math.OC]

    Arxiv:2006.15226V1 [Math.OC]

    RIEMANNIAN OPTIMIZATION ON THE SYMPLECTIC STIEFEL MANIFOLD∗ BIN GAO†, NGUYEN THANH SON‡, P.-A. ABSIL†, AND TATJANA STYKEL§ Abstract. The symplectic Stiefel manifold, denoted by Sp(2p, 2n), is the set of linear symplectic maps between the standard symplectic spaces R2p and R2n. When p = n, it reduces to the well-known set of 2n × 2n symplectic matrices. Optimization problems on Sp(2p, 2n) find applications in various areas, such as optics, quantum physics, numerical linear algebra and model order reduction of dynamical systems. The purpose of this paper is to propose and analyze gradient-descent methods on Sp(2p, 2n), where the notion of gradient stems from a Riemannian metric. We consider a novel Riemannian metric on Sp(2p, 2n) akin to the canonical metric of the (standard) Stiefel manifold. In order to perform a feasible step along the antigradient, we develop two types of search strategies: one is based on quasi-geodesic curves, and the other one on the symplectic Cayley transform. The resulting optimization algorithms are proved to converge globally to critical points of the objective function. Numerical experiments illustrate the efficiency of the proposed methods. Key words. Riemannian optimization, symplectic Stiefel manifold, quasi-geodesic, Cayley transform AMS subject classifications. 65K05, 70G45, 90C48 1. Introduction. We consider the following optimization problem with symplectic con- straints: min f(X) (1.1) X∈R2n×2p ⊤ s. t. X J2nX = J2p, 0 Im where p n, J2m = , and Im is the m m identity matrix for any positive ≤ −Im 0 × integer m. When there is no confusion, we omit the subscript of J2m and Im for simplicity.
  • Parallel Codes for Computing the Numerical Rank '

    Parallel Codes for Computing the Numerical Rank '

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector LINEAR ALGEBRA AND ITS APPLICATIONS ELSEIVIER Linear Algebra and its Applications 275~ 276 (1998) 451 470 Parallel codes for computing the numerical rank ’ Gregorio Quintana-Orti ***, Enrique S. Quintana-Orti ’ Abstract In this paper we present an experimental comparison of several numerical tools for computing the numerical rank of dense matrices. The study includes the well-known SVD. the URV decomposition, and several rank-revealing QR factorizations: the QR factorization with column pivoting, and two QR factorizations with restricted column pivoting. Two different parallel programming methodologies are analyzed in our paper. First, we present block-partitioned algorithms for the URV decomposition and rank-re- vealing QR factorizations which provide efficient implementations on shared memory environments. Furthermore, we also present parallel distributed algorithms, based on the message-passing paradigm, for computing rank-revealing QR factorizations on mul- ticomputers. 0 1998 Elsevier Science Inc. All rights reserved. 1. Introduction Consider the matrix A E KY’“” (w.1.o.g. m 3 n), and define its rank as 1’ := rank(il) = dim(range(A)). The rank of a matrix is related to its singular values a(A) = {crt, 02,. , o,,}, p = min(m, H), since *Corresponding author. E-mail: [email protected]. ’ All authors were partially supported by the Spanish CICYT project TIC 96-1062~CO3-03. ’ This work was partially carried out while the author was visiting the Mathematics and Computer Science Division at Argonne National Laboratory. ’ E-mail: [email protected].
  • Efficient Scalable Algorithms for Hierarchically Semiseparable Matrices

    Efficient Scalable Algorithms for Hierarchically Semiseparable Matrices

    EFFICIENT SCALABLE ALGORITHMS FOR HIERARCHICALLY SEMISEPARABLE MATRICES SHEN WANG¤, XIAOYE S. LIy , JIANLIN XIAz , YINGCHONG SITUx , AND MAARTEN V. DE HOOP{ Abstract. Hierarchically semiseparable (HSS) matrix structures are very useful in the develop- ment of fast direct solvers for both dense and sparse linear systems. For example, they can built into parallel sparse solvers for Helmholtz equations arising from frequency domain wave equation mod- eling prevailing in the oil and gas industry. Here, we study HSS techniques in large-scale scienti¯c computing by considering the scalability of HSS structures and by developing massively parallel HSS algorithms. These algorithms include parallel rank-revealing QR factorizations, HSS constructions with hierarchical compression, ULV HSS factorizations, and HSS solutions. BLACS and ScaLAPACK are used as our portable libraries. We construct contexts (sub-communicators) on each node of the HSS tree and exploit the governing 2D-block-cyclic data distribution scheme widely used in ScaLAPACK. Some tree techniques for symmetric positive de¯nite HSS matrices are generalized to nonsymmetric problems in parallel. These parallel HSS methods are very useful for sparse solutions when they are used as kernels for intermediate dense matrices. Numerical examples (including 3D ones) con¯rm the performance of our implementations, and demonstrate the potential of structured methods in parallel computing, especially parallel direction solutions of large linear systems. Related techniques are also useful for the parallelization of other rank structured methods. Key words. HSS matrix, parallel HSS algorithm, low-rank property, compression, HSS con- struction, direct solver AMS subject classi¯cations. 15A23, 65F05, 65F30, 65F50 1. Introduction. In recent years, rank structured matrices have attracted much attention and have been widely used in the fast solutions of various partial di®erential equations, integral equations, and eigenvalue problems.
  • A Parallel Butterfly Algorithm

    A Parallel Butterfly Algorithm

    A Parallel Butterfly Algorithm The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Poulson, Jack, Laurent Demanet, Nicholas Maxwell, and Lexing Ying. “A Parallel Butterfly Algorithm.” SIAM Journal on Scientific Computing 36, no. 1 (February 4, 2014): C49–C65.© 2014, Society for Industrial and Applied Mathematics. As Published http://dx.doi.org/10.1137/130921544 Publisher Society for Industrial and Applied Mathematics Version Final published version Citable link http://hdl.handle.net/1721.1/88176 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. SIAM J. SCI. COMPUT. c 2014 Society for Industrial and Applied Mathematics Vol. 36, No. 1, pp. C49–C65 ∗ A PARALLEL BUTTERFLY ALGORITHM JACK POULSON†, LAURENT DEMANET‡, NICHOLAS MAXWELL§, AND ¶ LEXING YING Abstract. The butterfly algorithm is a fast algorithm which approximately evaluates a dis- crete analogue of the integral transform Rd K(x, y)g(y)dy at large numbers of target points when the kernel, K(x, y), is approximately low-rank when restricted to subdomains satisfying a certain d simple geometric condition. In d dimensions with O(N ) quasi-uniformly distributed source and target points, when each appropriate submatrix of K is approximately rank-r, the running time 2 d of the algorithm is at most O(r N log N). A parallelization of the butterfly algorithm is intro- duced which, assuming a message latency of α and per-process inverse bandwidth of β, executes 2 Nd Nd in at most O(r p log N +(βr p + α)logp)timeusingp processes.
  • Solving Linear Systems Arising from Reservoirs Modeling Hussam Al Daas

    Solving Linear Systems Arising from Reservoirs Modeling Hussam Al Daas

    Solving linear systems arising from reservoirs modeling Hussam Al Daas To cite this version: Hussam Al Daas. Solving linear systems arising from reservoirs modeling. Mathematics [math]. Inria Paris; Sorbonne Université, UPMC University of Paris 6, Laboratoire Jacques-Louis Lions, 2018. English. tel-01984047v1 HAL Id: tel-01984047 https://hal.inria.fr/tel-01984047v1 Submitted on 16 Jan 2019 (v1), last revised 11 Jun 2020 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Sorbonne Université Inria Doctoral School École Doctorale Sciences Mathématiques de Paris Centre University Department Inria Thesis defended by Hussam Al Daas Defended on 13th December, 2018 In order to become Doctor from Sorbonne Université Academic Field Applied Mathematics Speciality Numerical Linear Algebra Solving linear systems arising from reservoirs modelling Thesis supervised by Laura Grigori Supervisor Pascal Hénon Co-Supervisor Committee members Referees Serge Gratton Professor at INP ENSEEIHT Julien Langou Professor at UC Denver Examiners Frédéric Nataf Senior Researcher
  • Rank Revealing Factorizations and Low Rank Approximations

    Rank Revealing Factorizations and Low Rank Approximations

    Rank revealing factorizations, and low rank approximations L. Grigori Inria Paris, UPMC January 2018 Plan Low rank matrix approximation Rank revealing QR factorization LU CRTP: Truncated LU factorization with column and row tournament pivoting Experimental results, LU CRTP Randomized algorithms for low rank approximation 2 of 70 Plan Low rank matrix approximation Rank revealing QR factorization LU CRTP: Truncated LU factorization with column and row tournament pivoting Experimental results, LU CRTP Randomized algorithms for low rank approximation 3 of 70 Low rank matrix approximation T Problem: given m × n matrix A, compute rank-k approximation ZW , where Z is m × k and W T is k × n. Problem with diverse applications from scientific computing: fast solvers for integral equations, H-matrices to data analytics: principal component analysis, image processing, ... Ax ! ZW T x Flops 2mn ! 2(m + n)k 4 of 70 Singular value decomposition Given A 2 Rm×n, m ≥ n its singular value decomposition is 0 1 Σ1 0 T T A = UΣV = U1 U2 U3 · @ 0 Σ2A · V1 V2 0 0 where U is m × m orthogonal matrix, the left singular vectors of A , U1 is m × k, U2 is m × n − k, U3 is m × m − n Σ is m × n, its diagonal is formed by σ1(A) ≥ ::: ≥ σn(A) ≥ 0 Σ1 is k × k,Σ2 is n − k × n − k V is n × n orthogonal matrix, the right singular vectors of A, V1 is n × k, V2 is n × n − k 5 of 70 Norms v u m n q uX X 2 2 2 jjAjjF = t jaij j = σ1(A) + : : : σn(A) i=1 j=1 jjAjj2 = σmax (A) = σ1(A) Some properties: p jjAjj2 ≤ jjAjjF ≤ min(m; n)jjAjj2 Orthogonal Invariance: If Q 2 Rm×m
  • Some Large-Scale Matrix Computation Problems

    Some Large-Scale Matrix Computation Problems

    JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS ELSEVIER Journal of Computational and Applied Mathematics 74 (1996) 71-89 Some large-scale matrix computation problems Zhaojun Bai a' 1, Mark Fahey a' 1, Gene Golub b'*' 2 aDepartment of Mathematics, University of Kentucky, Lexington, KY 40506, USA bScientific Computing and Computational Mathematics Program, Computer Science Department, Stanford University, Stanford, CA 94305, USA Abstract There are numerous applications in physics, statistics and electrical circuit simulation where it is required to bound entries and the trace of the inverse and the determinant of a large sparse matrix. All these computational tasks are related to the central mathematical problem studied in this paper, namely, bounding the bilinear form uXf(A)v for a given matrix A and vectors u and v, wherefis a given smooth function and is defined on the spectrum of A. We will study a practical numerical algorithm for bounding the bilinear form, where the matrix A is only referenced through matrix-vector multiplications. A Monte Carlo method is also presented to efficiently estimate the trace of the inverse and the determinant of a large sparse matrix. Keywords." Bilinear form; Gaussian quadrature; Trace; Determinant; Matrix inverse; Monte Carlo simulation; Probabil- istic bound AMS classification: 65F 10 1. Introduction The central problem studied in this paper is to estimate a lower bound L and/or an upper bound U, such that L <<. uTf(A) v <~ U, (1) where A is a n × n given real matrix, u and v are given n-vectors, and f is a given smooth function and is defined on the spectrum of the matrix A.
  • MATLAB Language Reference Manual

    MATLAB Language Reference Manual

    MATLAB The Language of Technical Computing Computation Visualization Programming Language Reference Manual Version 5 How to Contact The MathWorks: ☎ 508-647-7000 Phone 508-647-7001 Fax The MathWorks, Inc. Mail ✉ 24 Prime Park Way Natick, MA 01760-1500 http://www.mathworks.com Web ftp.mathworks.com Anonymous FTP server comp.soft-sys.matlab Newsgroup @ [email protected] Technical support [email protected] Product enhancement suggestions [email protected] Bug reports [email protected] Documentation error reports [email protected] Subscribing user registration [email protected] Order status, license renewals, passcodes [email protected] Sales, pricing, and general information MATLAB Language Reference COPYRIGHT 1984 - 1997 by The MathWorks, Inc. All Rights Reserved. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or repro- duced in any form without prior written consent from The MathWorks, Inc. U.S. GOVERNMENT: If Licensee is acquiring the software on behalf of any unit or agency of the U. S. Government, the following shall apply: (a) for units of the Department of Defense: RESTRICTED RIGHTS LEGEND: Use, duplication, or disclosure by the Government is subject to restric- tions as set forth in subparagraph (c)(1)(ii) of the Rights in Technical Data and Computer Software Clause at DFARS 252.227-7013. (b) for any other unit or agency: NOTICE - Notwithstanding any other lease or license agreement that may pertain to, or accompany the delivery of, the computer software and accompanying documentation, the rights of the Government regarding its use, reproduction and disclosure are as set forth in Clause 52.227-19(c)(2) of the FAR.
  • 50 LA15 Abstracts

    50 LA15 Abstracts

    50 LA15 Abstracts IP1 least-squares regression and principal component analy- Fast Approximation of the Stability Radius and the sis) are widely and routinely used. Devising scalable al- H∞ Norm for Large-Scale Linear Dynamical Sys- gorithms that enable large scale computation of the afore- tems mentioned primitives is crucial for meeting the challenges of Big Data applications. Sketching, which reduces dimen- The stability radius and the H∞ norm are well-known sionality through randomization, has recently emerged as quantities in the robust analysis of linear dynamical sys- a powerful technique for scaling-up these primitives in the tems with output feedback. These two quantities, which presence of massive data, for an extensive class of applica- are reciprocals of each other in the simplest interesting tions. In this talk, we outline how sketching can be used to case, respectively measure how much system uncertainty accelerate these core computations, and elaborate on the can be tolerated without losing stability, and how much tradeoffs involved in their use. We will also demonstrate an input disturbance may be magnified in the output. the utility of the presented algorithms for data analysis and The standard method for computing them, the Boyd- machine learning applications Balakrishnan-Bruinsma-Steinbuch algorithm from 1990, is globally and quadratically convergent, but its cubic cost Haim Avron per iteration makes it inapplicable to large-scale dynami- Business Analytics & Mathematical Sciences cal systems. We present a new class of efficient methods IBM T.J. Watson Research Center for approximating the stability radius and the H∞ norm, [email protected] based on iterative methods to find rightmost points of spec- tral value sets, which are generalizations of pseudospectra IP4 for modeling the linear fractional matrix transformations that arise naturally in analyzing output feedback.
  • An Order Reduction Method for Computing the Finite Eigenvalues of Regular Matrix Pencils Morad Ahmadnasab CERFACS Technical Repo

    An Order Reduction Method for Computing the Finite Eigenvalues of Regular Matrix Pencils Morad Ahmadnasab CERFACS Technical Repo

    An order reduction method for computing the finite eigenvalues of regular matrix pencils Morad Ahmadnasab CERFACS Technical Report TR/PA/08/23 Department of Mathematics, University of Kurdistan, Pasdaran boulevard, Sanandaj, Iran, Postal Code 66177{15175, P. O. Box 416 and CERFACS, 42 Avenue G. Coriolis, 31057 Toulouse Cedex, France E-Mail: [email protected] 1 AN ORDER REDUCTION METHOD FOR COMPUTING THE FINITE EIGENVALUES OF REGULAR MATRIX PENCILS MORAD AHMADNASAB∗ Abstract. Let A; B 2 Cn×n be two given matrices. Under the assumption that at least one of the matrices A or B is nonsingular, we introduce an order reduction method for computing the generalized eigenvalues of the regular matrix pencil A − zB. This method is based on the SVD of the matrix B (resp. A) if the matrix A (resp. B) is nonsingular. When both A and B are nonsingular, the method is based on the SVD of A or B exclusively. The performance of this algorithm is studied and the accuracy of the computed eigenvalues is demonstrated by comparing them with those computed by the QZ method in Matlab. Key words. Homotopic deviation, regular matrix pencils, generalized eigenvalue problem, singular value decomposition AMS subject classifications. 15A18, 15A22, 65F15 1. Introduction. The standard matrix eigenvalue problem for the matrix A 2 Cn×n has the form Ax = λx, where λ is an eigenvalue of A and x 2 Cn, x 6= 0 is an associated eigenvector. Let A and B be two complex n × n matrices. The set of all matrices of the form A − zB with z 2 C is said to be a pencil.