7.4 Matrix Norms and Condition Numbers This Section Considers the Accuracy of Computed Solutions to Linear Systems
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Structured Eigenvalue Condition Numbers and Linearizations for Matrix Polynomials
¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ Eidgen¨ossische Ecole polytechnique f´ed´erale de Zurich ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ Technische Hochschule Politecnico federale di Zurigo ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ Zu¨rich Swiss Federal Institute of Technology Zurich Structured eigenvalue condition numbers and linearizations for matrix polynomials B. Adhikari∗, R. Alam† and D. Kressner Research Report No. 2009-01 January 2009 Seminar fu¨r Angewandte Mathematik Eidgen¨ossische Technische Hochschule CH-8092 Zu¨rich Switzerland ∗Department of Mathematics, Indian Institute of Technology Guwahati, India, E-mail: [email protected] †Department of Mathematics, Indian Institute of Technology Guwahati, India, E-mail: rafi[email protected], rafi[email protected], Fax: +91-361-2690762/2582649. Structured eigenvalue condition numbers and linearizations for matrix polynomials Bibhas Adhikari∗ Rafikul Alam† Daniel Kressner‡. Abstract. This work is concerned with eigenvalue problems for structured matrix polynomials, including complex symmetric, Hermitian, even, odd, palindromic, and anti-palindromic matrix poly- nomials. Most numerical approaches to solving such eigenvalue problems proceed by linearizing the matrix polynomial into a matrix pencil of larger size. Recently, linearizations have been classified for which the pencil reflects the structure of the original polynomial. A question of practical impor- tance is whether this process of linearization increases the sensitivity of the eigenvalue with respect to structured perturbations. For all structures under consideration, we show that this is not the case: there is always a linearization for which the structured condition number of an eigenvalue does not differ significantly. This implies, for example, that a structure-preserving algorithm applied to the linearization fully benefits from a potentially low structured eigenvalue condition number of the original matrix polynomial. Keywords. Eigenvalue problem, matrix polynomial, linearization, structured condition num- ber. -
Condition Number Bounds for Problems with Integer Coefficients*
Condition Number Bounds for Problems with Integer Coefficients* Gregorio Malajovich Departamento de Matem´atica Aplicada, Universidade Federal do Rio de Janeiro. Caixa Postal 68530, CEP 21945-970, Rio de Janeiro, RJ, Brasil. E-mail: [email protected] An explicit a priori bound for the condition number associated to each of the following problems is given: general linear equation solving, least squares, non-symmetric eigenvalue problems, solving univariate polynomials, and solv- ing systems of multivariate polynomials. It is assumed that the input has integer coefficients and is not on the degeneracy locus of the respective prob- lem (i.e., the condition number is finite). Our bounds are stated in terms of the dimension and of the bit-size of the input. In the same setting, bounds are given for the speed of convergence of the following iterative algorithms: QR iteration without shift for the symmetric eigenvalue problem, and Graeffe iteration for univariate polynomials. Key Words: condition number, height, complexity 1. INTRODUCTION In most of the numerical analysis literature, complexity and stability of numerical algorithms are usually estimated in terms of the problem instance dimension and of a \condition number". For instance, the complexity of solving an n n linear system Ax = b is usually estimated in terms of the dimension n×(when the input actually consists of n(n + 1) real numbers) and the condition number κ(A) (see section 2.1 for the definition of the condition number). There is a set of problems instances with κ(A) = , and in most cases it makes no sense to solve these instances. -
C H a P T E R § Methods Related to the Norm Al Equations
¡ ¢ £ ¤ ¥ ¦ § ¨ © © © © ¨ ©! "# $% &('*),+-)(./+-)0.21434576/),+98,:%;<)>=,'41/? @%3*)BAC:D8E+%=B891GF*),+H;I?H1KJ2.21K8L1GMNAPO45Q5R)S;S+T? =RU ?H1*)>./+EAVOKAPM ;<),5 ?H1G;P8W.XAVO4575R)S;S+Y? =Z8L1*)/[%\Z1*)]AB3*=,'Q;<)B=,'41/? @%3*)RAP8LU F*)BA(;S'*)Z)>@%3/? F*.4U ),1G;^U ?H1*)>./+ APOKAP;_),5a`Rbc`^dfeg`Rbih,jk=>.4U U )Blm;B'*)n1K84+o5R.EUp)B@%3*.K;q? 8L1KA>[r\0:D;_),1Ejp;B'/? A2.4s4s/+t8/.,=,' b ? A7.KFK84? l4)>lu?H1vs,+D.*=S;q? =>)m6/)>=>.43KA<)W;B'*)w=B8E)IxW=K? ),1G;W5R.K;S+Y? yn` `z? AW5Q3*=,'|{}84+-A_) =B8L1*l9? ;I? 891*)>lX;B'*.41Q`7[r~k8K{i)IF*),+NjL;B'*)71K8E+o5R.4U4)>@%3*.G;I? 891KA0.4s4s/+t8.*=,'w5R.BOw6/)Z.,l4)IM @%3*.K;_)7?H17A_8L5R)QAI? ;S3*.K;q? 8L1KA>[p1*l4)>)>lj9;B'*),+-)Q./+-)Q)SF*),1.Es4s4U ? =>.K;q? 8L1KA(?H1Q{R'/? =,'W? ;R? A s/+-)S:Y),+o+D)Blm;_8|;B'*)W3KAB3*.4Urc+HO4U 8,F|AS346KABs*.,=B)Q;_)>=,'41/? @%3*)BAB[&('/? A]=*'*.EsG;_),+}=B8*F*),+tA7? ;PM ),+-.K;q? F*)75R)S;S'K8ElAi{^'/? =,'Z./+-)R)K? ;B'*),+pl9?+D)B=S;BU OR84+k?H5Qs4U ? =K? ;BU O2+-),U .G;<)Bl7;_82;S'*)Q1K84+o5R.EU )>@%3*.G;I? 891KA>[ mWX(fQ - uK In order to solve the linear system 7 when is nonsymmetric, we can solve the equivalent system b b [¥K¦ ¡¢ £N¤ which is Symmetric Positive Definite. This system is known as the system of the normal equations associated with the least-squares problem, [ ¦ £N¤ minimize §* R¨©7ª§*«9¬ Note that (8.1) is typically used to solve the least-squares problem (8.2) for over- ®°¯± ±²³® determined systems, i.e., when is a rectangular matrix of size , . -
The Column and Row Hilbert Operator Spaces
The Column and Row Hilbert Operator Spaces Roy M. Araiza Department of Mathematics Purdue University Abstract Given a Hilbert space H we present the construction and some properties of the column and row Hilbert operator spaces Hc and Hr, respectively. The main proof of the survey is showing that CB(Hc; Kc) is completely isometric to B(H ; K ); in other words, the completely bounded maps be- tween the corresponding column Hilbert operator spaces is completely isometric to the bounded operators between the Hilbert spaces. A similar theorem for the row Hilbert operator spaces follows via duality of the operator spaces. In particular there exists a natural duality between the column and row Hilbert operator spaces which we also show. The presentation and proofs are due to Effros and Ruan [1]. 1 The Column Hilbert Operator Space Given a Hilbert space H , we define the column isometry C : H −! B(C; H ) given by ξ 7! C(ξ)α := αξ; α 2 C: Then given any n 2 N; we define the nth-matrix norm on Mn(H ); by (n) kξkc;n = C (ξ) ; ξ 2 Mn(H ): These are indeed operator space matrix norms by Ruan's Representation theorem and thus we define the column Hilbert operator space as n o Hc = H ; k·kc;n : n2N It follows that given ξ 2 Mn(H ) we have that the adjoint operator of C(ξ) is given by ∗ C(ξ) : H −! C; ζ 7! (ζj ξ) : Suppose C(ξ)∗(ζ) = d; c 2 C: Then we see that cd = (cj C(ξ)∗(ζ)) = (cξj ζ) = c(ζj ξ) = (cj (ζj ξ)) : We are also able to compute the matrix norms on rectangular matrices. -
Matrices and Linear Algebra
Chapter 26 Matrices and linear algebra ap,mat Contents 26.1 Matrix algebra........................................... 26.2 26.1.1 Determinant (s,mat,det)................................... 26.2 26.1.2 Eigenvalues and eigenvectors (s,mat,eig).......................... 26.2 26.1.3 Hermitian and symmetric matrices............................. 26.3 26.1.4 Matrix trace (s,mat,trace).................................. 26.3 26.1.5 Inversion formulas (s,mat,mil)............................... 26.3 26.1.6 Kronecker products (s,mat,kron).............................. 26.4 26.2 Positive-definite matrices (s,mat,pd) ................................ 26.4 26.2.1 Properties of positive-definite matrices........................... 26.5 26.2.2 Partial order......................................... 26.5 26.2.3 Diagonal dominance.................................... 26.5 26.2.4 Diagonal majorizers..................................... 26.5 26.2.5 Simultaneous diagonalization................................ 26.6 26.3 Vector norms (s,mat,vnorm) .................................... 26.6 26.3.1 Examples of vector norms.................................. 26.6 26.3.2 Inequalities......................................... 26.7 26.3.3 Properties.......................................... 26.7 26.4 Inner products (s,mat,inprod) ................................... 26.7 26.4.1 Examples.......................................... 26.7 26.4.2 Properties.......................................... 26.8 26.5 Matrix norms (s,mat,mnorm) .................................... 26.8 26.5.1 -
The Eigenvalue Problem: Perturbation Theory
Jim Lambers MAT 610 Summer Session 2009-10 Lecture 13 Notes These notes correspond to Sections 7.2 and 8.1 in the text. The Eigenvalue Problem: Perturbation Theory The Unsymmetric Eigenvalue Problem Just as the problem of solving a system of linear equations Ax = b can be sensitive to pertur- bations in the data, the problem of computing the eigenvalues of a matrix can also be sensitive to perturbations in the matrix. We will now obtain some results concerning the extent of this sensitivity. Suppose that A is obtained by perturbing a diagonal matrix D by a matrix F whose diagonal entries are zero; that is, A = D + F . If is an eigenvalue of A with corresponding eigenvector x, then we have (D − I)x + F x = 0: If is not equal to any of the diagonal entries of A, then D − I is nonsingular and we have x = −(D − I)−1F x: Taking 1-norms of both sides, we obtain −1 −1 kxk1 = k(D − I) F xk1 ≤ k(D − I) F k1kxk1; which yields n −1 X jfijj k(D − I) F k1 = max ≥ 1: 1≤i≤n jdii − j j=1;j6=i It follows that for some i, 1 ≤ i ≤ n, satisfies n X jdii − j ≤ jfijj: j=1;j6=i That is, lies within one of the Gerschgorin circles in the complex plane, that has center aii and radius n X ri = jaijj: j=1;j6=i 1 This is result is known as the Gerschgorin Circle Theorem. Example The eigenvalues of the matrix 2 −5 −1 1 3 A = 4 −2 2 −1 5 1 −3 7 are (A) = f6:4971; 2:7930; −5:2902g: The Gerschgorin disks are D1 = fz 2 C j jz − 7j ≤ 4g;D2 = fz 2 C j jz − 2j ≤ 3g;D3 = fz 2 C j jz + 5j ≤ 2g: We see that each disk contains one eigenvalue. -
Lecture 5 Ch. 5, Norms for Vectors and Matrices
KTH ROYAL INSTITUTE OF TECHNOLOGY Norms for vectors and matrices — Why? Lecture 5 Problem: Measure size of vector or matrix. What is “small” and what is “large”? Ch. 5, Norms for vectors and matrices Problem: Measure distance between vectors or matrices. When are they “close together” or “far apart”? Emil Björnson/Magnus Jansson/Mats Bengtsson April 27, 2016 Answers are given by norms. Also: Tool to analyze convergence and stability of algorithms. 2 / 21 Vector norm — axiomatic definition Inner product — axiomatic definition Definition: LetV be a vector space over afieldF(R orC). Definition: LetV be a vector space over afieldF(R orC). A function :V R is a vector norm if for all A function , :V V F is an inner product if for all || · || → �· ·� × → x,y V x,y,z V, ∈ ∈ (1) x 0 nonnegative (1) x,x 0 nonnegative || ||≥ � �≥ (1a) x =0iffx= 0 positive (1a) x,x =0iffx=0 positive || || � � (2) cx = c x for allc F homogeneous (2) x+y,z = x,z + y,z additive || || | | || || ∈ � � � � � � (3) x+y x + y triangle inequality (3) cx,y =c x,y for allc F homogeneous || ||≤ || || || || � � � � ∈ (4) x,y = y,x Hermitian property A function not satisfying (1a) is called a vector � � � � seminorm. Interpretation: “Angle” (distance) between vectors. Interpretation: Size/length of vector. 3 / 21 4 / 21 Connections between norm and inner products Examples / Corollary: If , is an inner product, then x =( x,x ) 1 2 � The Euclidean norm (l ) onC n: �· ·� || || � � 2 is a vector norm. 2 2 2 1/2 x 2 = ( x1 + x 2 + + x n ) . Called: Vector norm derived from an inner product. -
Matrix Norms 30.4
Matrix Norms 30.4 Introduction A matrix norm is a number defined in terms of the entries of the matrix. The norm is a useful quantity which can give important information about a matrix. ' • be familiar with matrices and their use in $ writing systems of equations • revise material on matrix inverses, be able to find the inverse of a 2 × 2 matrix, and know Prerequisites when no inverse exists Before starting this Section you should ... • revise Gaussian elimination and partial pivoting • be aware of the discussion of ill-conditioned and well-conditioned problems earlier in Section 30.1 #& • calculate norms and condition numbers of % Learning Outcomes small matrices • adjust certain systems of equations with a On completion you should be able to ... view to better conditioning " ! 34 HELM (2008): Workbook 30: Introduction to Numerical Methods ® 1. Matrix norms The norm of a square matrix A is a non-negative real number denoted kAk. There are several different ways of defining a matrix norm, but they all share the following properties: 1. kAk ≥ 0 for any square matrix A. 2. kAk = 0 if and only if the matrix A = 0. 3. kkAk = |k| kAk, for any scalar k. 4. kA + Bk ≤ kAk + kBk. 5. kABk ≤ kAk kBk. The norm of a matrix is a measure of how large its elements are. It is a way of determining the “size” of a matrix that is not necessarily related to how many rows or columns the matrix has. Key Point 6 Matrix Norm The norm of a matrix is a real number which is a measure of the magnitude of the matrix. -
5 Norms on Linear Spaces
5 Norms on Linear Spaces 5.1 Introduction In this chapter we study norms on real or complex linear linear spaces. Definition 5.1. Let F = (F, {0, 1, +, −, ·}) be the real or the complex field. A semi-norm on an F -linear space V is a mapping ν : V −→ R that satisfies the following conditions: (i) ν(x + y) ≤ ν(x)+ ν(y) (the subadditivity property of semi-norms), and (ii) ν(ax)= |a|ν(x), for x, y ∈ V and a ∈ F . By taking a = 0 in the second condition of the definition we have ν(0)=0 for every semi-norm on a real or complex space. A semi-norm can be defined on every linear space. Indeed, if B is a basis of V , B = {vi | i ∈ I}, J is a finite subset of I, and x = i∈I xivi, define νJ (x) as P 0 if x = 0, νJ (x)= ( j∈J |aj | for x ∈ V . We leave to the readerP the verification of the fact that νJ is indeed a seminorm. Theorem 5.2. If V is a real or complex linear space and ν : V −→ R is a semi-norm on V , then ν(x − y) ≥ |ν(x) − ν(y)|, for x, y ∈ V . Proof. We have ν(x) ≤ ν(x − y)+ ν(y), so ν(x) − ν(y) ≤ ν(x − y). (5.1) 160 5 Norms on Linear Spaces Since ν(x − y)= |− 1|ν(y − x) ≥ ν(y) − ν(x) we have −(ν(x) − ν(y)) ≤ ν(x) − ν(y). (5.2) The Inequalities 5.1 and 5.2 give the desired inequality. -
Inverse Littlewood-Offord Theorems and the Condition Number Of
Annals of Mathematics, 169 (2009), 595{632 Inverse Littlewood-Offord theorems and the condition number of random discrete matrices By Terence Tao and Van H. Vu* Abstract Consider a random sum η1v1 + ··· + ηnvn, where η1; : : : ; ηn are indepen- dently and identically distributed (i.i.d.) random signs and v1; : : : ; vn are inte- gers. The Littlewood-Offord problem asks to maximize concentration probabil- ities such as P(η1v1+···+ηnvn = 0) subject to various hypotheses on v1; : : : ; vn. In this paper we develop an inverse Littlewood-Offord theory (somewhat in the spirit of Freiman's inverse theory in additive combinatorics), which starts with the hypothesis that a concentration probability is large, and concludes that almost all of the v1; : : : ; vn are efficiently contained in a generalized arithmetic progression. As an application we give a new bound on the magnitude of the least singular value of a random Bernoulli matrix, which in turn provides upper tail estimates on the condition number. 1. Introduction Let v be a multiset (allowing repetitions) of n integers v1; : : : ; vn. Consider a class of discrete random walks Yµ,v on the integers Z, which start at the origin and consist of n steps, where at the ith step one moves backwards or forwards with magnitude vi and probability µ/2, and stays at rest with probability 1−µ. More precisely: Definition 1.1 (Random walks). For any 0 ≤ µ ≤ 1, let ηµ 2 {−1; 0; 1g denote a random variable which equals 0 with probability 1 − µ and ±1 with probability µ/2 each. In particular, η1 is a random sign ±1, while η0 is iden- tically zero. -
Computing the Jordan Structure of an Eigenvalue∗
SIAM J. MATRIX ANAL.APPL. c 2017 Society for Industrial and Applied Mathematics Vol. 38, No. 3, pp. 949{966 COMPUTING THE JORDAN STRUCTURE OF AN EIGENVALUE∗ NICOLA MASTRONARDIy AND PAUL VAN DOORENz Abstract. In this paper we revisit the problem of finding an orthogonal similarity transformation that puts an n × n matrix A in a block upper-triangular form that reveals its Jordan structure at a particular eigenvalue λ0. The obtained form in fact reveals the dimensions of the null spaces of i (A − λ0I) at that eigenvalue via the sizes of the leading diagonal blocks, and from this the Jordan structure at λ0 is then easily recovered. The method starts from a Hessenberg form that already reveals several properties of the Jordan structure of A. It then updates the Hessenberg form in an efficient way to transform it to a block-triangular form in O(mn2) floating point operations, where m is the total multiplicity of the eigenvalue. The method only uses orthogonal transformations and is backward stable. We illustrate the method with a number of numerical examples. Key words. Jordan structure, staircase form, Hessenberg form AMS subject classifications. 65F15, 65F25 DOI. 10.1137/16M1083098 1. Introduction. Finding the eigenvalues and their corresponding Jordan struc- ture of a matrix A is one of the most studied problems in numerical linear algebra. This structure plays an important role in the solution of explicit differential equations, which can be modeled as n×n (1.1) λx(t) = Ax(t); x(0) = x0;A 2 R ; where λ stands for the differential operator. -
A Degeneracy Framework for Scalable Graph Autoencoders
Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence (IJCAI-19) A Degeneracy Framework for Scalable Graph Autoencoders Guillaume Salha1;2 , Romain Hennequin1 , Viet Anh Tran1 and Michalis Vazirgiannis2 1Deezer Research & Development, Paris, France 2Ecole´ Polytechnique, Palaiseau, France [email protected] Abstract In this paper, we focus on the graph extensions of autoen- coders and variational autoencoders. Introduced in the 1980’s In this paper, we present a general framework to [Rumelhart et al., 1986], autoencoders (AE) regained a sig- scale graph autoencoders (AE) and graph varia- nificant popularity in the last decade through neural network tional autoencoders (VAE). This framework lever- frameworks [Baldi, 2012] as efficient tools to learn reduced ages graph degeneracy concepts to train models encoding representations of input data in an unsupervised only from a dense subset of nodes instead of us- way. Furthermore, variational autoencoders (VAE) [Kingma ing the entire graph. Together with a simple yet ef- and Welling, 2013], described as extensions of AE but actu- fective propagation mechanism, our approach sig- ally based on quite different mathematical foundations, also nificantly improves scalability and training speed recently emerged as a successful approach for unsupervised while preserving performance. We evaluate and learning from complex distributions, assuming the input data discuss our method on several variants of existing is the observed part of a larger joint model involving low- graph AE and VAE, providing the first application dimensional latent variables, optimized via variational infer- of these models to large graphs with up to millions ence approximations. [Tschannen et al., 2018] review the of nodes and edges.