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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]. -
![Arxiv:2006.15226V1 [Math.OC]](https://docslib.b-cdn.net/cover/7812/arxiv-2006-15226v1-math-oc-1307812.webp)
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. -
Dense Matrix Computations : Communication Cost and Numerical Stability
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. -
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. -
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. -
Matrix Depot Documentation Release 0.1.0
Matrix Depot Documentation Release 0.1.0 Weijian Zhang August 14, 2015 Contents 1 Examples 41 2 User Defined Properties 45 3 More Examples 49 4 Matrix Properties 51 4.1 Main Properties............................................. 51 4.2 Other Properties............................................. 51 5 Interface to Test Matrix Collections 53 5.1 Interface to the UF Sparse Matrix Collection.............................. 53 5.2 Interface to NIST Matrix Market (temporarily suspend)........................ 54 6 MatrixDepot.jl 55 6.1 Install................................................... 55 6.2 Usage................................................... 55 Bibliography 59 i ii Matrix Depot Documentation, Release 0.1.0 • binomial • cauchy • chebspec • chow • circul • clement • dingdong • fiedler • forsythe • frank • grcar • hadamard • hilb • invhilb • invol • kahan • kms • lehmer • lotkin • magic • minij • moler • neumann • oscillate • parter • pascal • pei • poisson • prolate • randcorr • rando • randsvd • rohess • rosser • sampling • toeplitz Contents 1 Matrix Depot Documentation, Release 0.1.0 • tridiag • triw • vand • wathen • wilkinson binomial A binomial matrix that arose from the example in [bmsz01]. The matrix is a multiple of involutory matrix. cauchy The Cauchy matrix is an m-by-n matrix with (i; j) element 1 ; xi − yi 6= 0; xi − yi where xi and yi are elements of vectors x and y. 2 Contents Matrix Depot Documentation, Release 0.1.0 chebspec Chebyshev spectral differentiation matrix. If k = 0,the generated matrix is nilpotent and a vector with all one entries is a null vector. If k = 1, the generated matrix is nonsingular and well-conditioned. Its eigenvalues have negative real parts. Contents 3 Matrix Depot Documentation, Release 0.1.0 chow The Chow matrix is a singular Toeplitz lower Hessenberg matrix. -
Geometrical Inverse Preconditioning for Symmetric Positive Definite
Geometrical inverse preconditioning for symmetric positive definite matrices Jean-Paul Chehab∗ Marcos Raydan † November 18, 2015 Abstract We focus on inverse preconditioners based on minimizing F (X)=1 cos(XA, I), where XA is the preconditioned matrix and A is symmetric and positive− definite. We present and analyze gradient-type methods to minimize F (X) on a suitable com- pact set. For that we use the geometrical properties of the non-polyhedral cone of symmetric and positive definite matrices, and also the special properties of F (X) on the feasible set. Preliminary and encouraging numerical results are also presented in which dense and sparse approximations are included. Key words: Preconditioning, cones of matrices, gradient method, minimal residual method. 1 Introduction Algebraic inverse preconditioning play a key role in a wide variety of applications that involve the solution of large and sparse linear systems of equations; see e.g., [4, 6, 7, 9, 14, 22, 23]. For a given square matrix A, there exist several proposals for constructing sparse inverse approximations which are based on optimization techniques, mainly based on minimizing the Frobenius norm of the residual (I XA) over a set P of matrices with a − certain sparsity pattern; see e.g., [2, 7, 8, 9, 10, 12, 18, 19, 13]. However, we must remark that when A is symmetric and positive definite, minimizing the Frobenius norm of the arXiv:1511.07694v1 [math.NA] 24 Nov 2015 residual in general will produce an inverse preconditioner which is neither symmetric nor positive definite; see, e.g., [2]. There is currently a growing interest, and understanding, in the rich geometrical struc- ture of the non-polyhedral cone of symmetric and positive semidefinite matrices (P SD); see e.g., [1, 5, 12, 19, 16, 17, 24]. -
Matrix Depot Documentation Release 0.1.0
Matrix Depot Documentation Release 0.1.0 Weijian Zhang Nov 30, 2018 Contents 1 Installation 3 1.1 Usage...................................................3 1.2 Matrices.................................................5 1.3 Random Graphs............................................. 46 1.4 Test Problems for Regularization Methods............................... 47 1.5 Groups.................................................. 51 1.6 Interface to Test Collections....................................... 52 1.7 Adding New Matrix Generators..................................... 55 1.8 Examples................................................. 58 Bibliography 65 i ii Matrix Depot Documentation, Release 0.1.0 Matrix Depot is an extensible test matrix collection for Julia. It provides a diverse collection of test matrices, including parametrized matrices and real-life matrices. • Source at Github • Release Notes Contents 1 Matrix Depot Documentation, Release 0.1.0 2 Contents CHAPTER 1 Installation To install the release version, type: julia>] pkg> add MatrixDepot 1.1 Usage Every matrix in the collection is represented by a string "matrix_name", for example, the Cauchy matrix is repre- sented by "cauchy" and the Hilbert matrix is represented by "hilb". The matrix groups are noted as symbols. For example, the class of the symmetric matrices is symbolized by :symmetric. mdinfo() Return a list of all the matrices in the collection: julia> matrixdepot() Matrices: 1) baart2) binomial3) blur4) cauchy 5) chebspec6) chow7) circul8) clement 9) companion 10) deriv2 -
A COMPARISON of METHODS for COMPUTING the EIGENVALUES and EIGENVECTORS of a MATRIX by I
.»«Vt. ·.} ·. ISHI Joint Nuclear Research Center Ispra Establishment - Italy Scientific Data Processing Center - CETIS ocument was prepared under the sponsorship of the Commission e European Communities. ««lllllÄill accuracy, completeness, or usefulness of the information contained in this document, or that the use of any information, apparatus, method, or process disclosed in this rtnrumpn. mnv nn. τητΓτησ.» τγτ.νη.<*ϊν ruam«-*, rinhfc· nr Assume any Iiahility with respect to the use of. or for damages resulting f tl Γ .Γ .. .TT TTT siliäville h»**tf "it»';·. This document was reproduced on the basis of the best available copy. iffiieaiai^ mmm Srffia !f ΙΡΰi lift iïlfiu Κ .ή! (í.i¡:.s¡i."-tíufcí. 3^_8 UUi3' Bij "9' iiiflVf Piíirllv n»Sft_HFUÎS ϋ*ΚΪ3ίΓ'*{ι tipi Irfï.'i*:' Mi EUR 4055 e A COMPARISON OF METHODS FOR COMPUTING THE EIGENVALUES AND EIGENVECTORS OF A MATRIX by I. GALL1GANI European Atomic Energy Community - EURATOM Joint Nuclear Center - lspra Establishment (Italy) Scientific Data Processing Center - CET1S Brussels, September 1968 - 102 Pages - 2 Figures - FB 150 In this report a comparison of some methods for solving the eigen- problem of a matrix is given. An attempt has been made to establish the «efficiency», on the basis of computing time and accuracy, of each method by carrying out «experimental, calculations on «representative> problems for which exact results are known. EUR 4055 e A COMPARISON OF METHODS FOR COMPUTING THE EIGENVALUES AND EIGENVECTORS OF A MATRIX by 1. GALL1GAN1 European Atomic Energy Community - EURATOM Joint Nuclear Center - lspra Establishment (Italy) Scientific Data Processing Center - CET1S Brussels, September 1968 - 102 Pages - 2 Figures - FB 150 In this report a comparison of some methods for solving the eigen- prohlem of a matrix is given. -
Calu: a Communication Optimal Lu Factorization Algorithm∗
SIAM J. MATRIX ANAL. APPL. c 2011 Society for Industrial and Applied Mathematics Vol. 32, No. 4, pp. 1317–1350 CALU: A COMMUNICATION OPTIMAL LU FACTORIZATION ALGORITHM∗ † ‡ § LAURA GRIGORI , JAMES W. DEMMEL , AND HUA XIANG Abstract. Since the cost of communication (moving data) greatly exceeds the cost of doing arithmetic on current and future computing platforms, we are motivated to devise algorithms that communicate as little as possible, even if they do slightly more arithmetic, and as long as they still get the right answer. This paper is about getting the right answer for such an algorithm. It discusses CALU, a communication avoiding LU factorization algorithm based on a new pivoting strategy, that we refer to as tournament pivoting. The reason to consider CALU is that it does an optimal amount of communication, and asymptotically less than Gaussian elimination with partial pivoting (GEPP), and so will be much faster on platforms where communication is expensive, as shown in previous work. We show that the Schur complement obtained after each step of performing CALU on a matrix A is the same as the Schur complement obtained after performing GEPP on a larger matrix whose entries are the same as the entries of A (sometimes slightly perturbed) and zeros. More generally, the entire CALU process is equivalent to GEPP on a large, but very sparse matrix, formed by entries of A and zeros. Hence we expect that CALU will behave as GEPP and it will also be very stable in practice. In addition, extensive experiments on random matrices and a set of special matrices show that CALU is stable in practice. -
CALU: a Communication Optimal LU Factorization Algorithm
CALU: A Communication Optimal LU Factorization Algorithm James Demmel Laura Grigori Hua Xiang Electrical Engineering and Computer Sciences University of California at Berkeley Technical Report No. UCB/EECS-2010-29 http://www.eecs.berkeley.edu/Pubs/TechRpts/2010/EECS-2010-29.html March 15, 2010 Copyright © 2010, by the author(s). All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission. Acknowledgement This work has been supported in part by the French National Research Agency (ANR) through the COSINUS program (project PETAL no. ANR- 08-COSI-009), by Microsoft (Award #024263), by Intel (Award #024894) and U.C. Discovery (Award #DIG07-10227). CALU: A COMMUNICATION OPTIMAL LU FACTORIZATION ALGORITHM LAURA GRIGORI∗, JAMES W. DEMMEL†, AND HUA XIANG ‡ Abstract. Since the cost of communication (moving data) greatly exceeds the cost of doing arithmetic on current and future computing platforms, we are motivated to devise algorithms that communicate as little as possible, even if they do slightly more arithmetic, and as long as they still get the right answer. This paper is about getting the right answer for such an algorithm. It discusses CALU, a communication avoiding LU factorization algorithm based on a new pivoting strategy, that we refer to as ca-pivoting. -
Infinitely Divisible Matrices
isid/ms/2005/04 April 11, 2005 http://www.isid.ac.in/ statmath/eprints e Infinitely Divisible Matrices Rajendra Bhatia Indian Statistical Institute, Delhi Centre 7, SJSS Marg, New Delhi–110 016, India Infinitely Divisible Matrices Rajendra Bhatia Indian Statistical Institute, New Delhi 110 016, India [email protected] 1 Introduction 1.1 Positive semidefinite matrices Let A be an n × n matrix with complex entries aij. We write this as A = [aij] . A is Hermitian ∗ A = A if aij = aji, a condition that is readily verified by inspection. A Hermitian matrix is positive semidefinite (psd for short) if it satisfies any of the following equivalent conditions (i) For every vector x ∈ Cn, the inner product hx,Axi≥ 0. (ii) All principal minors of A are nonnegative. (iii) All eigenvalues of A are nonnegative. (iv) A = BB∗ for some matrix B. (v) A = LL∗ for some lower triangular matrix L. (vi) There exist vectors u1,...,un in some inner product space such that aij = hui, uji. The matrix A is then said to be the Gram matrix associated with the vectors {u1,...,un}. It is not easy to verify any of the conditions (i)–(vi) and a little ingenuity is often needed in proving that a certain matrix is psd. 1.2 The Hadamard product Let A = [aij] and B = [bij] be two n × n matrices. The Hadamard product (or the entrywise product) of A and B is the matrix A ◦ B = [aijbij]. The most interesting theorem about Hadamard products was proved by I. Schur. This says that if A and B are psd, then so is A ◦ B.