DOCSLIB.ORG

Computing the Moore-Penrose Inverse for Bidiagonal Matrices

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Computing the Moore-Penrose Inverse for Bidiagonal Matrices УДК 519.65 Yu. Hakopian DOI: https://doi.org/10.18523/2617-70802201911-23 COMPUTING THE MOORE-PENROSE INVERSE FOR BIDIAGONAL MATRICES The Moore-Penrose inverse is the most popular type of matrix generalized inverses which has many applications both in matrix theory and numerical linear algebra. It is well known that the Moore-Penrose inverse can be found via singular value decomposition. In this regard, there is the most effective algo- rithm which consists of two stages. In the first stage, through the use of the Householder reflections, an initial matrix is reduced to the upper bidiagonal form (the Golub-Kahan bidiagonalization algorithm). The second stage is known in scientific literature as the Golub-Reinsch algorithm. This is an itera- tive procedure which with the help of the Givens rotations generates a sequence of bidiagonal matrices converging to a diagonal form. This allows to obtain an iterative approximation to the singular value decomposition of the bidiagonal matrix. The principal intention of the present paper is to develop a method which can be considered as an alternative to the Golub-Reinsch iterative algorithm. Realizing the approach proposed in the study, the following two main results have been achieved. First, we obtain explicit expressions for the entries of the Moore-Penrose inverse of bidigonal matrices. Secondly, based on the closed form formulas, we get a finite recursive numerical algorithm of optimal computational complexity. Thus, we can compute the Moore-Penrose inverse of bidiagonal matrices without using the singular value decomposition. Keywords: Moor-Penrose inverse, bidiagonal matrix, inversion formula, finite recursive algorithm. Introduction The most effective procedure to compute the Moore-Penrose inverse involves two main stages As is known, for a real m × n matrix A the [4]. Moore-Penrose inverse A+ is the unique matrix Stage 1. Matrix reduction to the bidiagonal that satisfies the following four properties [1]: form. At this stage an m × n matrix, where m ≥ n, + + + + AA A = A;A AA = A ; by means of the Householder reflections is trans- formed to upper bidiagonal form (A+A)T = A+A; (AA+)T = AA+ : 2 a a 0 ::: 0 3 If A is a square nonsingular matrix, then A+ = 11 12 6 0 a22 a23 ::: 0 7 = A−1. Thus the Moore-Penrose inverse general- 6 7 6 . .. .. .. 7 izes the ordinary matrix inversion. 6 . 7 6 7 : There is well-known formula for the Moore- 6 0 ::: 0 an−1 n−1 an−1 n 7 6 7 Penrose inverse which is obtained by the singular 6 0 ::: 0 0 ann 7 6 7 value decomposition (abbreviated SVD) of the ma- 4 5 trix (see [1; 4], for instance). 0 The singular value decomposition of an m × n (1:3) matrix A with rank r is its factorization of the form The computational process is known as Golub- Kahan bidiagonalization [2]. Thereby the problem T A = UΛV ; (1:1) is reduces to the Moore-Penrose inversion of the bidiagonal matrix (1.3). where U is an m × m orthogonal matrix, Λ = Stage 2. Golub-Reinsch SVD iterative algo- = diag [σ ; σ ; : : : ; σ ] is an m×n diagonal matrix, 1 2 r rithm. and V is an n×n orthogonal matrix. The diagonal Once the bidiagonalization of the initial ma- entries σ ≥ σ ≥ ::: ≥ σ > 0 of Λ are known as 1 2 r trix has been achieved, the next task is to zero the singular values of the matrix A. Having the fac- superdiagonal entries in the matrix (1.3). With torization (1.1), the Moore-Penrose inverse can be this purpose the Golub-Reinsch algorithm is im- written as plemented [3]. The algorithm, with the help of the A+ = V Λ+U T ; (1:2) Givens rotations generates a sequence of bidiago- + −1 −1 −1 where Λ = diag [σ1 ; σ2 ; : : : ; σr ] is n × m di- nal matrices that converge to a diagonal form. As agonal matrix. a result, at a certain step of the iterative process we c Yu. Hakopian, 2019 12 ISSN 2617-7080. Могилянський математичний журнал. 2019. Том 2 get an approximation to the SVD of the bidiagonal Case 1: a11 6= 0; ann 6= 0. matrix (1.3). Having the SVD, the Moore-Penrose Let zero diagonal entries of the matrix A are inverse of the matrix is computed (see [1; 4], for ai1 i1 ; ai2 i2 ; : : : ; aip−1 ip−1 , where 1 < i1 < i2 < instance). < ··· < ip−1 < n and p > 1. We split the matrix The objective of the present work is to de- into blocks drawing dividing lines after the rows velop a method which allows to deduce formulas i1 − 1; i2 − 1; : : : ; ip−1 − 1 and after the columns for the entries of the Moore-Penrose inverse of i1; i2; : : : ; ip−1. As a result, the matrix (2.1) takes upper bidiagonal matrices. The obtained closed a block diagonal form. The first and the last diag- form solution to the Moore-Penrose inversion may onal blocks are rectangular bidiagonal matrices of be considered as an alternative to sufficiently the sizes (i1 −1)×i1 and (n−ip−1 +1)×(n−ip−1), labour-consuming Golub-Reinsch iterative proce- respectively, while the remaining blocks are square dure briefly described in the Stage 2 of this sec- lower bidiagonal matrices. As an illustration, a tion. Moreover, explicit expressions for the entries pattern of the matrix (for n = 10), the partition- of the Moore-Penrose inverse lead to fairly simple ing procedure and resulting block diagonal struc- finite numerical algorithm with optimal volume of ture are shown in Fig. 2.1 (stars represent nonzero computational expenditures. entries). Partition of a bidiagonal matrix into blocks ?? ?? Let us consider a real n × n upper bidiagonal ?? matrix 0 ? 2 3 a11 a12 0 ? 6 a22 a23 0 7 ?? 6 7 ?? A = 6 .. .. 7 : 6 . 7 0 ? 6 7 4 0 an−1 n−1 an−1 n 5 ?? ann ? (2:1) Note that it suffices to consider square upper bidi- ?? agonal matrices since for rectangular upper bidi- ?? agonal matrices the problem can be easily reduced ?? to our case. Indeed, if m > n then according to ? (1.3) we have the block structure ? ?? A ; ?? 0 ? where A is a square upper bidiagonal matrix of the ?? ? form (2.1). It can be seen that in this case + Figure 2.1. Partition of the matrix (case 1). A = A+ 0T : 0 Case 2: a11 = 0; ann 6= 0. We allocate the first column of the matrix We assume that the matrix A is singular, i.e. A, as a separate zero block of the size n × a11a22 : : : ann = 0. Next, we assume that × 1. Next, we partition the remaining subma- trix into diagonal blocks as follows. If there are ai i+1 6= 0; i = 1; 2; : : : ; n − 1: (2:2) other zero diagonal entries of the matrix A, say Otherwise, if some of superdiagonal entries of the ai1 i1 ; ai2 i2 ; : : : ; aip−1 ip−1 , where 1 < i1 < i2 < matrix A are equal to zero, the problem of comput- < ··· < ip−1 < n and p > 1, then the subma- ing the Moore-Penrose inverse is decomposed into trix is subdivided according to the rule drscribed several similar problems for bidiagonal matrices of in the Case 1. As an illustration, see a pattern lower order. of the matrix given in Fig. 2.2 . The last diagonal To compute the Moore-Penrose inverse of the block of the submatrix is rectangular bidiagonal matrix A, we apply a special partition of this ma- matrix of the size (n − ip−1 + 1) × (n − ip−1); the trix into blocks. The partitioning procedure uses remaining diagonal blocks are square lower bidiag- the arrangement of zeros on the main diagonal onal matrices. If there are no other zero diagonal of the matrix. We distinguish the following four entries, except the first one, then the submatrix is cases. not subdivided. Yu. Hakopian. Computing the Moore-Penrose inverse for bidiagonal matrices 13 0 ? ?? ?? ?? ?? ?? ?? ? 0 ? ?? 0 ? ? ?? ? 0 ? ?? ?? ?? ? 0 ? Figure 2.3. Partition of the matrix (case 3). ?? ?? ?? Case 4: a11 = 0; ann = 0. ? The allocation of the first column and the last 0 ? row of the matrix A gives us three zero blocks of ?? the sizes (n − 1) × 1, 1 × (n − 1) and 1 × 1 (see ? Fig. 2.4). Then we partition the remaining subma- ?? trix. If there are other zero diagonal entries of the ? matrix A, say ai1 i1 ; ai2 i2 ; : : : ; aip−1 ip−1 , where 1 < < i1 < i2 < ··· < ip−1 < n and p > 1, then the submatrix is subdivided by the rule drscribed in the Case 1. The diagonal blocks of this subdivi- Figure 2.2. Partition of the matrix (case 2). sion are square lower bidiagonal matrices. If there are no other zero diagonal entries except the first and last, then the submatrix is not subdivided. Case 3: a11 6= 0; ann = 0. First, we allocate the last row of the matrix A, as a separate zero block of the size 1 × n. 0 ? Next, we partition the remaining submatrix into ?? diagonal blocks using the same idea. If there are 0 ? other zero diagonal entries of the matrix A, say ?? ?? ai i ; ai i ; : : : ; ai i , where 1 < i1 < i2 < 1 1 2 2 p−1 p−1 0 ? < ··· < i < n and p > 1, then the submatrix p−1 0 ? is subdivided by the rule drscribed in the Case 1 ?? (see a pattern of the matrix given in Fig. 2.3). The ?? first diagonal block of the submatrix is rectangu- 0 lar bidiagonal matrix of the size (i1 − 1) × i1; the remaining diagonal blocks are square lower bidiag- ? onal matrices.
Load more

Recommended publications
  • 18.06 Linear Algebra, Problem Set 2 Solutions
    18.06 Problem Set 2 Solution Total: 100 points Section 2.5. Problem 24: Use Gauss-Jordan elimination on [U I] to find the upper triangular −1 U : 2 3 2 3 2 3 1 a b 1 0 0 −1 4 5 4 5 4 5 UU = I 0 1 c x1 x2 x3 = 0 1 0 : 0 0 1 0 0 1 −1 Solution (4 points): Row reduce [U I] to get [I U ] as follows (here Ri stands for the ith row): 2 3 2 3 1 a b 1 0 0 (R1 = R1 − aR2) 1 0 b − ac 1 −a 0 4 5 4 5 0 1 c 0 1 0 −! (R2 = R2 − cR2) 0 1 0 0 1 −c 0 0 1 0 0 1 0 0 1 0 0 1 ( ) 2 3 R1 = R1 − (b − ac)R3 1 0 0 1 −a ac − b −! 40 1 0 0 1 −c 5 : 0 0 1 0 0 1 Section 2.5. Problem 40: (Recommended) A is a 4 by 4 matrix with 1's on the diagonal and −1 −a; −b; −c on the diagonal above. Find A for this bidiagonal matrix. −1 Solution (12 points): Row reduce [A I] to get [I A ] as follows (here Ri stands for the ith row): 2 3 1 −a 0 0 1 0 0 0 6 7 60 1 −b 0 0 1 0 07 4 5 0 0 1 −c 0 0 1 0 0 0 0 1 0 0 0 1 2 3 (R1 = R1 + aR2) 1 0 −ab 0 1 a 0 0 6 7 (R2 = R2 + bR2) 60 1 0 −bc 0 1 b 07 −! 4 5 (R3 = R3 + cR4) 0 0 1 0 0 0 1 c 0 0 0 1 0 0 0 1 2 3 (R1 = R1 + abR3) 1 0 0 0 1 a ab abc (R = R + bcR ) 60 1 0 0 0 1 b bc 7 −! 2 2 4 6 7 : 40 0 1 0 0 0 1 c 5 0 0 0 1 0 0 0 1 Alternatively, write A = I − N.
    [Show full text]
  • [Math.RA] 19 Jun 2003 Two Linear Transformations Each Tridiagonal with Respect to an Eigenbasis of the Ot
    Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array∗ Paul Terwilliger Abstract Let K denote a field. Let d denote a nonnegative integer and consider a sequence ∗ K p = (θi,θi , i = 0...d; ϕj , φj, j = 1...d) consisting of scalars taken from . We call p ∗ ∗ a parameter array whenever: (PA1) θi 6= θj, θi 6= θj if i 6= j, (0 ≤ i, j ≤ d); (PA2) i−1 θh−θd−h ∗ ∗ ϕi 6= 0, φi 6= 0 (1 ≤ i ≤ d); (PA3) ϕi = φ1 + (θ − θ )(θi−1 − θ ) h=0 θ0−θd i 0 d i−1 θh−θd−h ∗ ∗ (1 ≤ i ≤ d); (PA4) φi = ϕ1 + (Pθ − θ )(θd−i+1 − θ0) (1 ≤ i ≤ d); h=0 θ0−θd i 0 −1 ∗ ∗ ∗ ∗ −1 (PA5) (θi−2 − θi+1)(θi−1 − θi) P, (θi−2 − θi+1)(θi−1 − θi ) are equal and independent of i for 2 ≤ i ≤ d − 1. In [13] we showed the parameter arrays are in bijection with the isomorphism classes of Leonard systems. Using this bijection we obtain the following two characterizations of parameter arrays. Assume p satisfies PA1, PA2. Let ∗ ∗ A, B, A ,B denote the matrices in Matd+1(K) which have entries Aii = θi, Bii = θd−i, ∗ ∗ ∗ ∗ ∗ ∗ Aii = θi , Bii = θi (0 ≤ i ≤ d), Ai,i−1 = 1, Bi,i−1 = 1, Ai−1,i = ϕi, Bi−1,i = φi (1 ≤ i ≤ d), and all other entries 0. We show the following are equivalent: (i) p satisfies −1 PA3–PA5; (ii) there exists an invertible G ∈ Matd+1(K) such that G AG = B and G−1A∗G = B∗; (iii) for 0 ≤ i ≤ d the polynomial i ∗ ∗ ∗ ∗ ∗ ∗ (λ − θ0)(λ − θ1) · · · (λ − θn−1)(θi − θ0)(θi − θ1) · · · (θi − θn−1) ϕ1ϕ2 · · · ϕ nX=0 n is a scalar multiple of the polynomial i ∗ ∗ ∗ ∗ ∗ ∗ (λ − θd)(λ − θd−1) · · · (λ − θd−n+1)(θ − θ )(θ − θ ) · · · (θ − θ ) i 0 i 1 i n−1 .
    [Show full text]
  • Self-Interlacing Polynomials Ii: Matrices with Self-Interlacing Spectrum
    SELF-INTERLACING POLYNOMIALS II: MATRICES WITH SELF-INTERLACING SPECTRUM MIKHAIL TYAGLOV Abstract. An n × n matrix is said to have a self-interlacing spectrum if its eigenvalues λk, k = 1; : : : ; n, are distributed as follows n−1 λ1 > −λ2 > λ3 > ··· > (−1) λn > 0: A method for constructing sign definite matrices with self-interlacing spectra from totally nonnegative ones is presented. We apply this method to bidiagonal and tridiagonal matrices. In particular, we generalize a result by O. Holtz on the spectrum of real symmetric anti-bidiagonal matrices with positive nonzero entries. 1. Introduction In [5] there were introduced the so-called self-interlacing polynomials. A polynomial p(z) is called self- interlacing if all its roots are real, semple and interlacing the roots of the polynomial p(−z). It is easy to see that if λk, k = 1; : : : ; n, are the roots of a self-interlacing polynomial, then the are distributed as follows n−1 (1.1) λ1 > −λ2 > λ3 > ··· > (−1) λn > 0; or n (1.2) − λ1 > λ2 > −λ3 > ··· > (−1) λn > 0: The polynomials whose roots are distributed as in (1.1) (resp. in (1.2)) are called self-interlacing of kind I (resp. of kind II). It is clear that a polynomial p(z) is self-interlacing of kind I if, and only if, the polynomial p(−z) is self-interlacing of kind II. Thus, it is enough to study self-interlacing of kind I, since all the results for self-interlacing of kind II will be obtained automatically. Definition 1.1. An n × n matrix is said to possess a self-interlacing spectrum if its eigenvalues λk, k = 1; : : : ; n, are real, simple, are distributed as in (1.1).
    [Show full text]
  • Accurate Singular Values of Bidiagonal Matrices
    d d Accurate Singular Values of Bidiagonal Matrices (Appeared in the SIAM J. Sci. Stat. Comput., v. 11, n. 5, pp. 873-912, 1990) James Demmel W. Kahan Courant Institute Computer Science Division 251 Mercer Str. University of California New York, NY 10012 Berkeley, CA 94720 Abstract Computing the singular values of a bidiagonal matrix is the ®nal phase of the standard algo- rithm for the singular value decomposition of a general matrix. We present a new algorithm which computes all the singular values of a bidiagonal matrix to high relative accuracy indepen- dent of their magnitudes. In contrast, the standard algorithm for bidiagonal matrices may com- pute sm all singular values with no relative accuracy at all. Numerical experiments show that the new algorithm is comparable in speed to the standard algorithm, and frequently faster. Keywords: singular value decomposition, bidiagonal matrix, QR iteration AMS(MOS) subject classi®cations: 65F20, 65G05, 65F35 1. Introduction The standard algorithm for computing the singular value decomposition (SVD ) of a gen- eral real matrix A has two phases [7]: = T 1) Compute orthogonal matrices P 11and Q such that B PAQ 11is in bidiagonal form, i.e. has nonzero entries only on its diagonal and ®rst superdiagonal. Σ= T 2) Compute orthogonal matrices P 22and Q such that PBQ 22is diagonal and nonnega- σ Σ tive. The diagonal entries i of are the singular values of A. We will take them to be σ≥σ = sorted in decreasing order: ii+112. The columns of Q QQ are the right singular vec- = tors,andthecolumnsofP PP12are the left singular vectors.
    [Show full text]
  • Design and Evaluation of Tridiagonal Solvers for Vector and Parallel Computers Universitat Politècnica De Catalunya
    Design and Evaluation of Tridiagonal Solvers for Vector and Parallel Computers Author: Josep Lluis Larriba Pey Advisor: Juan José Navarro Guerrero Barcelona, January 1995 UPC Universitat Politècnica de Catalunya Departament d'Arquitectura de Computadors UNIVERSITAT POLITÈCNICA DE CATALU NYA B L I O T E C A X - L I B R I S Tesi doctoral presentada per Josep Lluis Larriba Pey per tal d'aconseguir el grau de Doctor en Informàtica per la Universitat Politècnica de Catalunya UNIVERSITAT POLITÈCNICA DE CATAl U>'YA ADA'UNÍIEÏRACÍÓ ;;//..;,3UMPf';S AO\n¿.-v i S -i i::-.« c:¿fCM or,re: iïhcc'a a la pàgina .rf.S# a;; 3 b el r;ú¡; Barcelona, 6 N L'ENCARREGAT DEL REGISTRE. Barcelona, de de 1995 To my wife Marta To my parents Elvira and José Luis "A journey of a thousand miles must begin with a single step" Lao-Tse Acknowledgements I would like to thank my parents, José Luis and Elvira, for their unconditional help and love to me. I will always be indebted to you. I also want to thank my wife, Marta, for being the sparkle in my life, for her support and for understanding my (good) moods. I thank Juanjo, my advisor, for his friendship, patience and good advice. Also, I want to thank Àngel Jorba for his unconditional collaboration, work and support in some of the contributions of this work. I thank the members of "Comissió de Doctorat del DAG" for their comments and specially Miguel Valero for his suggestions on the topics of chapter 4. I thank Mateo Valero for his friendship and always good advice.
    [Show full text]
  • Basic Matrices
    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 373 (2003) 143–151 www.elsevier.com/locate/laa Basic matricesୋ Miroslav Fiedler Institute of Computer Science, Czech Academy of Sciences, 182 07 Prague 8, Czech Republic Received 30 April 2002; accepted 13 November 2002 Submitted by B. Shader Abstract We define a basic matrix as a square matrix which has both subdiagonal and superdiagonal rank at most one. We investigate, partly using additional restrictions, the relationship of basic matrices to factorizations. Special basic matrices are also mentioned. © 2003 Published by Elsevier Inc. AMS classification: 15A23; 15A33 Keywords: Structure rank; Tridiagonal matrix; Oscillatory matrix; Factorization; Orthogonal matrix 1. Introduction and preliminaries For m × n matrices, structure (cf. [2]) will mean any nonvoid subset of M × N, M ={1,...,m}, N ={1,...,n}. Given a matrix A (over a field, or a ring) and a structure, the structure rank of A is the maximum rank of any submatrix of A all entries of which are contained in the structure. The most important examples of structure ranks for square n × n matrices are the subdiagonal rank (S ={(i, k); n i>k 1}), the superdiagonal rank (S = {(i, k); 1 i<k n}), the off-diagonal rank (S ={(i, k); i/= k,1 i, k n}), the block-subdiagonal rank, etc. All these ranks enjoy the following property: Theorem 1.1 [2, Theorems 2 and 3]. If a nonsingular matrix has subdiagonal (su- perdiagonal, off-diagonal, block-subdiagonal, etc.) rank k, then the inverse also has subdiagonal (...)rank k.
    [Show full text]
  • Moore–Penrose Inverse of Bidiagonal Matrices. I
    PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical and Mathematical Sciences 2015, № 2, p. 11–20 Mathematics MOORE–PENROSE INVERSE OF BIDIAGONAL MATRICES.I 1 ∗ 2∗∗ Yu. R. HAKOPIAN , S. S. ALEKSANYAN ; 1Chair of Numerical Analysis and Mathematical Modeling YSU, Armenia 2Chair of Mathematics and Mathematical Modeling RAU, Armenia In the present paper we deduce closed form expressions for the entries of the Moore–Penrose inverse of a special type upper bidiagonal matrices. On the base of the formulae obtained, a finite algorithm with optimal order of computational complexity is constructed. MSC2010: 15A09; 65F20. Keywords: generalized inverse, Moore–Penrose inverse, bidiagonal matrix. Introduction. For a real m × n matrix A, the Moore–Penrose inverse A+ is the unique n × m matrix that satisfies the following four properties: AA+A = A; A+AA+ = A+ ; (A+A)T = A+A; (AA+)T = AA+ (see [1], for example). If A is a square nonsingular matrix, then A+ = A−1. Thus, the Moore–Penrose inversion generalizes ordinary matrix inversion. The idea of matrix generalized inverse was first introduced in 1920 by E. Moore [2], and was later redis- covered by R. Penrose [3]. The Moore–Penrose inverses have numerous applications in least-squares problems, regressive analysis, linear programming and etc. For more information on the generalized inverses see [1] and its extensive bibliography. In this work we consider the problem of the Moore–Penrose inversion of square upper bidiagonal matrices 2 3 d1 b1 6 d2 b2 0 7 6 7 6 .. .. 7 A = 6 . 7 : (1) 6 7 4 0 dn−1 bn−1 5 dn A natural question arises: what caused an interest in pseudoinversion of such matrices ? The reason is as follows.
    [Show full text]
  • Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods1
    Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods1 Richard Barrett2, Michael Berry3, Tony F. Chan4, James Demmel5, June M. Donato6, Jack Dongarra3,2, Victor Eijkhout7, Roldan Pozo8, Charles Romine9, and Henk Van der Vorst10 This document is the electronic version of the 2nd edition of the Templates book, which is available for purchase from the Society for Industrial and Applied Mathematics (http://www.siam.org/books). 1This work was supported in part by DARPA and ARO under contract number DAAL03-91-C-0047, the National Science Foundation Science and Technology Center Cooperative Agreement No. CCR-8809615, the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract DE-AC05-84OR21400, and the Stichting Nationale Computer Faciliteit (NCF) by Grant CRG 92.03. 2Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37830- 6173. 3Department of Computer Science, University of Tennessee, Knoxville, TN 37996. 4Applied Mathematics Department, University of California, Los Angeles, CA 90024-1555. 5Computer Science Division and Mathematics Department, University of California, Berkeley, CA 94720. 6Science Applications International Corporation, Oak Ridge, TN 37831 7Texas Advanced Computing Center, The University of Texas at Austin, Austin, TX 78758 8National Institute of Standards and Technology, Gaithersburg, MD 9Office of Science and Technology Policy, Executive Office of the President 10Department of Mathematics, Utrecht University, Utrecht, the Netherlands. ii How to Use This Book We have divided this book into five main chapters. Chapter 1 gives the motivation for this book and the use of templates. Chapter 2 describes stationary and nonstationary iterative methods.
    [Show full text]
  • Block Bidiagonal Decomposition and Least Squares Problems
    Block Bidiagonal Decomposition and Least Squares Problems Ake˚ Bjorck¨ Department of Mathematics Linkoping¨ University Perspectives in Numerical Analysis, Helsinki, May 27–29, 2008 Outline Bidiagonal Decomposition and LSQR Block Bidiagonalization Methods Multiple Right-Hand Sides in LS and TLS Rank Deficient Blocks and Deflation Summary The Bidiagonal Decomposition In 1965 Golub and Kahan gave two algorithms for computing a bidiagonal decomposition (BD) L A = U V T ∈ Rm×n, 0 where B is lower bidiagonal, α1 β α 2 2 .. (n+1)×n B = β3 . ∈ R , .. . αn βn+1 and U and V are square orthogonal matrices. Given u1 = Q1e1, the decomposition is uniquely determined if αk βk+1 6= 0, k = 1 : n, The Bidiagonal Decomposition The Householder Algorithm For given Householder matrix Q1, set A0 = Q1A, and apply Householder transformations alternately from right and left: Ak = Qk+1(Ak−1Pk ), k = 1, 2,.... Pk zeros n − k elements in kth row; Qk+1 zeros m − k − 1 elements in the kth column. If Q1 is chosen so that Q1b = β1e1, then β e B UT b AV = 1 1 , 0 0 where B is lower bidiagonal and U = Q1Q2 ··· Qn+1, V = P1P2 ··· Pn−1. The Bidiagonal Decomposition Lanczos algorithm In the Lanczos process, successive columns of U = (u1, u2,..., un+1), V = (v1, v2,..., vn), with β1u1 = b, β1 = kbk2, are generated recursively. Equating columns in AT U = VBT and AV = UB, gives the coupled two-term recurrence relations (v0 ≡ 0): T αk vk = A uk − βk vk−1, βk+1uk+1 = Avk − αk uk , k = 1 : n.
    [Show full text]
  • Computation-And Space-Efficient Implementation Of
    Computation- and Space-Efficient Implementation of SSA Anton Korobeynikov Department of Statistical Modelling, Saint Petersburg State University 198504, Universitetskiy pr. 28, Saint Petersburg, Russia [email protected] April 20, 2010 Abstract The computational complexity of different steps of the basic SSA is discussed. It is shown that the use of the general-purpose “blackbox” routines (e.g. found in packages like LAPACK) leads to huge waste of time resources since the special Hankel structure of the trajectory matrix is not taken into account. We outline several state-of-the-art algorithms (for example, Lanczos- based truncated SVD) which can be modified to exploit the structure of the trajectory matrix. The key components here are Hankel matrix-vector multiplication and the Hankelization operator. We show that both can be computed efficiently by the means of Fast Fourier Transform. The use of these methods yields the reduction of the worst-case compu- tational complexity from O(N 3) to O(kN log N), where N is series length and k is the number of eigentriples desired. 1 Introduction Despite the recent growth of SSA-related techniques it seems that little attention arXiv:0911.4498v2 [cs.NA] 13 Apr 2010 was paid to their efficient implementations, e.g. the singular value decomposi- tion of Hankel trajectory matrix, which is a dominating time-consuming step of the basic SSA. In almost all SSA-related applications only a few leading eigentriples are desired, thus the use of general-purpose “blackbox” SVD implementations causes a huge waste of time and memory resources. This almost prevents the use of the optimum window sizes even on moderate length (say, few thousand) time series.
    [Show full text]
  • On Factorizations of Totally Positive Matrices
    On Factorizations of Totally Positive Matrices M. Gasca and J. M. Pe˜na Abstract. Different approaches to the decomposition of a nonsingular totally positive matrix as a product of bidiagonal matrices are studied. Special attention is paid to the interpretation of the factorization in terms of the Neville elimination process of the matrix and in terms of corner cutting algorithms of Computer Aided Geometric Design. Conditions of uniqueness for the decomposition are also given. §1. Introduction Totally positive matrices (TP matrices in the sequel) are real, nonnegative matrices whose all minors are nonnegative. They have a long history and many applications (see the paper by Allan Pinkus in this volume for the early history and motivations) and have been studied mainly by researchers of those applications. In spite of their interesting algebraic properties, they have not yet received much attention from linear algebrists, including those working specifically on nonnegative matrices. One of the aims of the masterful survey [1] by T. Ando, which presents a very complete list of results on TP matrices until 1986, was to attract this attention. In some papers by M. Gasca and G. M¨uhlbach ([13] for example) on the connection between interpolation formulas and elimination techniques it became clear that what they called Neville elimination had special interest for TP matrices. It is a procedure to make zeros in a column of a matrix by adding to each row an appropriate multiple of the precedent one and had been already used in some of the first papers on TP matrices [33]. However, in the last years, we have developed a better knowledge of the properties of Neville elimination which has allowed us to improve many previous results on those matrices [14–22].
    [Show full text]
  • EXPLICIT BOUNDS for the PSEUDOSPECTRA of VARIOUS CLASSES of MATRICES and OPERATORS Contents 1. Introduction 2 2. Pseudospectra 2
    EXPLICIT BOUNDS FOR THE PSEUDOSPECTRA OF VARIOUS CLASSES OF MATRICES AND OPERATORS FEIXUE GONG1, OLIVIA MEYERSON2, JEREMY MEZA3, ABIGAIL WARD4 MIHAI STOICIU5 (ADVISOR) SMALL 2014 - MATHEMATICAL PHYSICS GROUP N×N Abstract. We study the "-pseudospectra σ"(A) of square matrices A ∈ C . We give a complete characterization of the "-pseudospectrum of any 2×2 matrix and describe the asymptotic behavior (as " → 0) of σ"(A) for any square matrix A. We also present explicit upper and lower bounds for the "-pseudospectra of bidiagonal and tridiagonal matrices, as well as for finite rank operators. Contents 1. Introduction 2 2. Pseudospectra 2 2.1. Motivation and Definitions 2 2.2. Normal Matrices 6 2.3. Non-normal Diagonalizable Matrices 7 2.4. Non-diagonalizable Matrices 8 3. Pseudospectra of 2 2 Matrices 10 3.1. Non-diagonalizable× 2 2 Matrices 10 3.2. Diagonalizable 2 2 Matrices× 12 4. Asymptotic Union of Disks× Theorem 15 5. Pseudospectra of N N Jordan Block 20 6. Pseudospectra of bidiagonal× matrices 22 6.1. Asymptotic Bounds for Periodic Bidiagonal Matrices 22 6.2. Examples of Bidiagonal Matrices 29 7. Tridiagonal Matrices 32 Date: October 30, 2014. 1;2;5Williams College 3Carnegie Mellon University 4University of Chicago. 1 2 GONG, MEYERSON, MEZA, WARD, STOICIU 8. Finite Rank operators 33 References 35 1. Introduction The pseudospectra of matrices and operators is an important mathematical object that has found applications in various areas of mathematics: linear algebra, func- tional analysis, numerical analysis, and differential equations. An overview of the main results on pseudospectra can be found in [14].
    [Show full text]