DOCSLIB.ORG

U·M·I University Microfilms International a Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
U·M·I University Microfilms International a Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor Testing eigenvalue software. Item Type text; Dissertation-Reproduction (electronic) Authors Henderson, Lehman Edwin, Jr. Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 03/10/2021 22:22:26 Link to Item http://hdl.handle.net/10150/185744 INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each o'riginal is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. U·M·I University Microfilms International A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor. M148106-1346 USA 313/761-4700 800/521-0600 Order Number 9213693 Testing eigenvalue· software Henderson, Lehman Edwin, Jr., Ph.D. The University of Arizona, 1991 Copyright ®1991 by Henderson, Lehman Edwin, Jr. All rights reserved. U·M·I 300 N. Zeeb Rd. Ann Arbor, MI48106 Testing Eigenvalue Software by Lehman Edwin Henderson, Jr. Copyright © 1991, Lehman Edwin Henderson, Jr. A Dissertation Submitted to the Faculty of the Department of Computer Science in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Graduate College of THE UNIVERSITY OF ARIZONA 199 1 2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Final Examination Committee, we certify that we have read the dissertation prepared by Lehman Edwin Henderson, Jr. entitled _________T~E~S~T~I~N~G~EI~G~E~NV~AL~UE~S~O~F~TW~AR~E~ ______________________ and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy ID/31/'11 Date lolso/cr.1 Date Date' I Date Date Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. rd D. Schlichting Date Webb Miller 3 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under the rules of the Library. Brief quotations from this dissertation are allowed without special permission, provided that accurate acknowledgement of source is made. Requests for permis­ sion for extended quotation from, or reproduction of this manuscript in whole or in part may be granted by the copyright holder. I wish to acknowledge the help of my parents, Lehman and Vivian Henderson, for their early and continued support and encouragement of my interest in the sciences. ·J TABLE OF CONTENTS LIST OF FIGURES . .. 10 LIST OF TABLES. .. 11 ABSTRACT ...... .. 13 CHAPTER 1: Introduction 1-1 1.1 What I Have Done 1-1 1.2 What It Is Based Upon . 1.5 1.3 Other Related Work Ii 1.4 My Contribution 19 1.5 Conclusions ... 21 1.6 Organization of This Dissertation 2:3 CHAPTER 2: Definitions and Relevant Theorems. 2.5 2.1 Reduction to Tridiagonal Form :32 2.2 Plane Rotations . :33 6 CHAPTER 3: Testing Methods Reported in the Literature. 3·5 3.1 The Handbook ..... :3i 3.2 EISPACK test matrices. 60 3.3 Other Recent Papers .. 61 CHAPTER 4: The QR Algorithm. 69 4.1 Rotations ... iO 4.2 Why QL Instead of QR? 80 4.3 Generic QL Algorithm, Non-optimized 81 4.4 Choice of Shift ......... 8.5 4.5 Explicitly Shifted QL Algorithm. 8i 4.6 Optimized QL Algorithm. 88 4.7 Convergence Criteria 9.5 4.8 Error of Result ... 96 4.9 Programs That Were Tested 96 CHAPTER 5: Measuring How Well a Program Performs 102 5.1 Nearness: Vector and Matrix Norms. 10:3 5.2 Instability score . 106 5.3 An Estimate of w lOi CHAPTER 6: Random Testing 11-1 6.1 Introduction...... 115 7 11.j 6.2 Statement of the Problem . " 6.3 Experimental Method. 116 6.4 Results......... 118 CHAPTER 7: Hill-Climbing Software 129 7.1 The Compiler ........ 1:31 7.2 Straight-Line Code Generator 138 7.3 Maximizer.......... 1:39 7.4 Operation of the Maximizer 140 CHAPTER 8: Results from Hill Climbing . 142 8.1 Comparing Hill Climbing to Random Testing. 143 8.2 Will a Few Matrices Do? 14.j 8.3 Finding the Instability 148 8.4 Conclusion. 16:3 CHAPTER 9: Conclusions and Recommendations. 16.j 9.1 Conclusions ............. 165 9.2 Recommendations for future research 166 APPENDIX A: Data from Random Testing 169 A.1 IMTQL2. 170 A.2 IMTQLV. 170 8 A.3 OKRATV . ............... 173 AA OKWRAV ................................ Ii.j A.5 PWK ................................... 178 A.6 PWKU. 179 A.7 SKRAT ................................. 179 A.S SKRATN ............ ................... 181 A.9 SKRATW . 182 A.10 TQL1 .................................. 182 A.ll TQL1MB .............. 18.5 A.12 TQLIMA . 0 0 0 • 0 0 • 0 0 0 0 0 0 0 00 ••• 0 0 0 0 • 0 0 0 • 0 0 • 186 A.13 TQLIMB ................................ 187 A.14 TQL1NB ................................. 187 A.15 TQL1 U ................................. 188 A.16 TQL2 .................................. 190 A.17 TQLRAT ................................ 191 APPENDIX B: Data from Hill Climbing .................... 19.j Bol OKRATV 0 • 0 • 0 0 0 0 •••••• 0 • 0 ••• 0 •••• 0 ••• 0 •• 19i B.2 OKWRAV .. 0 ••••••••••••••••••••••••••••• 203 B.3 PWK ................................... 210 B.4 PWKU. 0 • • • • • • • • • • • • • • • • • 217 B.5 SKRAT ........ ............. 22-1 9 B.6 SKRATN ... 2:31 B.7 TQL1 .................................. 238 B.S TQL1MA . 245 B.9 TQL1MB ....................... ........ 2.j1 B.lO TQL1NA ................................ 258 B.11 TQL1NB ................................. 26.5 B.12 TQL1S .................................. 272 B.13 TQL1 U ........................... ...... 279 B.14 TQLRAT ................................ 286 REFERENCES .................................. 294 10 LIST OF FIGURES 2.1 Elimination of Off-Diagonal Element by Rotation . .. :34 4.1 Elements of the Matrix A ..... 84 4.2 Elements of the Matrix A during iteration 84 4.3 Two Rows During Rotation 84 4.4 Two Rows After Rotation . 85 4.5 Inner Loop of EISPACK TQLl Code 89 5.1 Typical relation of w to w 108 5.2 Linear Transform of Outputs With Respect to Inputs 110 5.3 Linear Transform of Outputs \Vith Respect to Rounding Errors 111 5.4 Geometric Interpretation of w . 112 7.1 Block Diagram of the Hill-climbing Software ............. 130 11 LIST OF TABLES 1.1 Results of testing 4 x 4 Linear Equation Solvers 17 1.2 Results of testing 4 x 4 Eigenvalue Solvers ... 20 5.1 Instabilities in Linear Equation Software Found By Hill-Climbing . 109 6.1 Maximum instability score, unmodified software 119 6.2 Comparing the Effects of Re-Orthogonalization 120 6.3 Maximum w Using Different Constants for Machine Precision. 121 6.4 Comparing the Effects of Scoring Precision . : . 123 6.5 Comparing the Effects of Scoring Precision of TQLRAT versus the Size of the Test Matrix . 12-1 6.6 Comparing the Effects of the Magnitude of the Input Data 12.5 6.7 Maximum Instability Score Varying the Size of the Test Matrix 126 6.8 Distribution of Maximum Instability Scores for OKWRAV 127 8.1 Summary of Instabilities Found by Random Testing . 1-1:3 8.2 Instabilities Found By Hill-climbing in Eigenvalue Software . 1-1-1 12 8.3 Using Selected Matrices on Unstable Routines 1-16 8.4 Using Selected Matrices on Stable Routines . 1-18 ABSTRACT This dissertation describes a significant advance in automated testing of eigenvalue software. Several programs are described that assist the re­ searcher in verifying that a new program is stable. Using backwards error techniques popularized by Wilkinson, a maximizer or "hill climber" system­ atically searches for instabilities in the program being tested. This work builds on software first reported by Miller and removes the restriction of not being able to work on iterative methods. Testing eigenvalue solver programs with sets of small random input data can often find instabilities, but the described hill climbing technique is more efficient. Using only ten sets of starting points, the maximizer will often find the instability, if it exists, in only a few tries. 1-1 CHAPTER 1 Introduction "The fundamental principle of science, the definition almost, is this: the sole test of the validity of any idea is experiment." R. P. Feynman 1.1 What I Have Done This dissertation is about testing eigenvalue software. Its high point is the de­ scription of a software package that will systematically search for input data that causes an unstable eigenvalue routine to fail.
Load more

Recommended publications
  • 2 Homework Solutions 18.335 " Fall 2004
    2 Homework Solutions 18.335 - Fall 2004 2.1 Count the number of ‡oating point operations required to compute the QR decomposition of an m-by-n matrix using (a) Householder re‡ectors (b) Givens rotations. 2 (a) See Trefethen p. 74-75. Answer: 2mn2 n3 ‡ops. 3 (b) Following the same procedure as in part (a) we get the same ‘volume’, 1 1 namely mn2 n3: The only di¤erence we have here comes from the 2 6 number of ‡opsrequired for calculating the Givens matrix. This operation requires 6 ‡ops (instead of 4 for the Householder re‡ectors) and hence in total we need 3mn2 n3 ‡ops. 2.2 Trefethen 5.4 Let the SVD of A = UV : Denote with vi the columns of V , ui the columns of U and i the singular values of A: We want to …nd x = (x1; x2) and such that: 0 A x x 1 = 1 A 0 x2 x2 This gives Ax2 = x1 and Ax1 = x2: Multiplying the 1st equation with A 2 and substitution of the 2nd equation gives AAx2 = x2: From this we may conclude that x2 is a left singular vector of A: The same can be done to see that x1 is a right singular vector of A: From this the 2m eigenvectors are found to be: 1 v x = i ; i = 1:::m p2 ui corresponding to the eigenvalues = i. Therefore we get the eigenvalue decomposition: 1 0 A 1 VV 0 1 VV = A 0 p2 U U 0 p2 U U 3 2.3 If A = R + uv, where R is upper triangular matrix and u and v are (column) vectors, describe an algorithm to compute the QR decomposition of A in (n2) time.
    [Show full text]
  • A Fast Algorithm for the Recursive Calculation of Dominant Singular Subspaces N
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 218 (2008) 238–246 www.elsevier.com/locate/cam A fast algorithm for the recursive calculation of dominant singular subspaces N. Mastronardia,1, M. Van Barelb,∗,2, R. Vandebrilb,2 aIstituto per le Applicazioni del Calcolo, CNR, via Amendola122/D, 70126, Bari, Italy bDepartment of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium Received 26 September 2006 Abstract In many engineering applications it is required to compute the dominant subspace of a matrix A of dimension m × n, with m?n. Often the matrix A is produced incrementally, so all the columns are not available simultaneously. This problem arises, e.g., in image processing, where each column of the matrix A represents an image of a given sequence leading to a singular value decomposition-based compression [S. Chandrasekaran, B.S. Manjunath, Y.F. Wang, J. Winkeler, H. Zhang, An eigenspace update algorithm for image analysis, Graphical Models and Image Process. 59 (5) (1997) 321–332]. Furthermore, the so-called proper orthogonal decomposition approximation uses the left dominant subspace of a matrix A where a column consists of a time instance of the solution of an evolution equation, e.g., the flow field from a fluid dynamics simulation. Since these flow fields tend to be very large, only a small number can be stored efficiently during the simulation, and therefore an incremental approach is useful [P. Van Dooren, Gramian based model reduction of large-scale dynamical systems, in: Numerical Analysis 1999, Chapman & Hall, CRC Press, London, Boca Raton, FL, 2000, pp.
    [Show full text]
  • High-Performance Parallel Computations Using Python As High-Level Language
    Aachen Institute for Advanced Study in Computational Engineering Science Preprint: AICES-2010/08-01 23/August/2010 High-Performance Parallel Computations using Python as High-Level Language Stefano Masini and Paolo Bientinesi Financial support from the Deutsche Forschungsgemeinschaft (German Research Association) through grant GSC 111 is gratefully acknowledged. ©Stefano Masini and Paolo Bientinesi 2010. All rights reserved List of AICES technical reports: http://www.aices.rwth-aachen.de/preprints High-Performance Parallel Computations using Python as High-Level Language Stefano Masini⋆ and Paolo Bientinesi⋆⋆ RWTH Aachen, AICES, Aachen, Germany Abstract. High-performance and parallel computations have always rep- resented a challenge in terms of code optimization and memory usage, and have typically been tackled with languages that allow a low-level management of resources, like Fortran, C and C++. Nowadays, most of the implementation effort goes into constructing the bookkeeping logic that binds together functionalities taken from standard libraries. Because of the increasing complexity of this kind of codes, it becomes more and more necessary to keep it well organized through proper software en- gineering practices. Indeed, in the presence of chaotic implementations, reasoning about correctness is difficult, even when limited to specific aspects like concurrency; moreover, due to the lack in flexibility of the code, making substantial changes for experimentation becomes a grand challenge. Since the bookkeeping logic only accounts for a tiny fraction of the total execution time, we believe that for such a task it can be afforded to introduce an overhead due to a high-level language. We consider Python as a preliminary candidate with the intent of improving code readability, flexibility and, in turn, the level of confidence with respect to correctness.
    [Show full text]
  • Iterative Refinement for Symmetric Eigenvalue Decomposition II
    Japan Journal of Industrial and Applied Mathematics (2019) 36:435–459 https://doi.org/10.1007/s13160-019-00348-4 ORIGINAL PAPER Iterative refinement for symmetric eigenvalue decomposition II: clustered eigenvalues Takeshi Ogita1 · Kensuke Aishima2 Received: 22 October 2018 / Revised: 5 February 2019 / Published online: 22 February 2019 © The Author(s) 2019 Abstract We are concerned with accurate eigenvalue decomposition of a real symmetric matrix A. In the previous paper (Ogita and Aishima in Jpn J Ind Appl Math 35(3): 1007–1035, 2018), we proposed an efficient refinement algorithm for improving the accuracy of all eigenvectors, which converges quadratically if a sufficiently accurate initial guess is given. However, since the accuracy of eigenvectors depends on the eigenvalue gap, it is difficult to provide such an initial guess to the algorithm in the case where A has clustered eigenvalues. To overcome this problem, we propose a novel algorithm that can refine approximate eigenvectors corresponding to clustered eigenvalues on the basis of the algorithm proposed in the previous paper. Numerical results are presented showing excellent performance of the proposed algorithm in terms of convergence rate and overall computational cost and illustrating an application to a quantum materials simulation. Keywords Accurate numerical algorithm · Iterative refinement · Symmetric eigenvalue decomposition · Clustered eigenvalues Mathematics Subject Classification 65F15 · 15A18 · 15A23 This study was partially supported by CREST, JST and JSPS KAKENHI Grant numbers 16H03917, 25790096. B Takeshi Ogita [email protected] Kensuke Aishima [email protected] 1 Division of Mathematical Sciences, School of Arts and Sciences, Tokyo Woman’s Christian University, 2-6-1 Zempukuji, Suginami-ku, Tokyo 167-8585, Japan 2 Faculty of Computer and Information Sciences, Hosei University, 3-7-2 Kajino-cho, Koganei-shi, Tokyo 184-8584, Japan 123 436 T.
    [Show full text]
  • Numerical Linear Algebra Revised February 15, 2010 4.1 the LU
    Numerical Linear Algebra Revised February 15, 2010 4.1 The LU Decomposition The Elementary Matrices and badgauss In the previous chapters we talked a bit about the solving systems of the form Lx = b and Ux = b where L is lower triangular and U is upper triangular. In the exercises for Chapter 2 you were asked to write a program x=lusolver(L,U,b) which solves LUx = b using forsub and backsub. We now address the problem of representing a matrix A as a product of a lower triangular L and an upper triangular U: Recall our old friend badgauss. function B=badgauss(A) m=size(A,1); B=A; for i=1:m-1 for j=i+1:m a=-B(j,i)/B(i,i); B(j,:)=a*B(i,:)+B(j,:); end end The heart of badgauss is the elementary row operation of type 3: B(j,:)=a*B(i,:)+B(j,:); where a=-B(j,i)/B(i,i); Note also that the index j is greater than i since the loop is for j=i+1:m As we know from linear algebra, an elementary opreration of type 3 can be viewed as matrix multipli- cation EA where E is an elementary matrix of type 3. E looks just like the identity matrix, except that E(j; i) = a where j; i and a are as in the MATLAB code above. In particular, E is a lower triangular matrix, moreover the entries on the diagonal are 1's. We call such a matrix unit lower triangular.
    [Show full text]
  • Massively Parallel Poisson and QR Factorization Solvers
    Computers Math. Applic. Vol. 31, No. 4/5, pp. 19-26, 1996 Pergamon Copyright~)1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0898-1221/96 $15.00 + 0.00 0898-122 ! (95)00212-X Massively Parallel Poisson and QR Factorization Solvers M. LUCK£ Institute for Control Theory and Robotics, Slovak Academy of Sciences DdbravskA cesta 9, 842 37 Bratislava, Slovak Republik utrrluck@savba, sk M. VAJTERSIC Institute of Informatics, Slovak Academy of Sciences DdbravskA cesta 9, 840 00 Bratislava, P.O. Box 56, Slovak Republic kaifmava©savba, sk E. VIKTORINOVA Institute for Control Theoryand Robotics, Slovak Academy of Sciences DdbravskA cesta 9, 842 37 Bratislava, Slovak Republik utrrevka@savba, sk Abstract--The paper brings a massively parallel Poisson solver for rectangle domain and parallel algorithms for computation of QR factorization of a dense matrix A by means of Householder re- flections and Givens rotations. The computer model under consideration is a SIMD mesh-connected toroidal n x n processor array. The Dirichlet problem is replaced by its finite-difference analog on an M x N (M + 1, N are powers of two) grid. The algorithm is composed of parallel fast sine transform and cyclic odd-even reduction blocks and runs in a fully parallel fashion. Its computational complexity is O(MN log L/n2), where L = max(M + 1, N). A parallel proposal of QI~ factorization by the Householder method zeros all subdiagonal elements in each column and updates all elements of the given submatrix in parallel. For the second method with Givens rotations, the parallel scheme of the Sameh and Kuck was chosen where the disjoint rotations can be computed simultaneously.
    [Show full text]
  • LAPACK WORKING NOTE 195: SCALAPACK's MRRR ALGORITHM 1. Introduction. Since 2005, the National Science Foundation Has Been Fund
    LAPACK WORKING NOTE 195: SCALAPACK’S MRRR ALGORITHM CHRISTOF VOMEL¨ ∗ Abstract. The sequential algorithm of Multiple Relatively Robust Representations, MRRR, can compute numerically orthogonal eigenvectors of an unreduced symmetric tridiagonal matrix T ∈ Rn×n with O(n2) cost. This paper describes the design of ScaLAPACK’s parallel MRRR algorithm. One emphasis is on the critical role of the representation tree in achieving both numerical accuracy and parallel scalability. A second point concerns the favorable properties of this code: subset computation, the use of static memory, and scalability. Unlike ScaLAPACK’s Divide & Conquer and QR, MRRR can compute subsets of eigenpairs at reduced cost. And in contrast to inverse iteration which can fail, it is guaranteed to produce a numerically satisfactory answer while maintaining memory scalability. ParEig, the parallel MRRR algorithm for PLAPACK, uses dynamic memory allocation. This is avoided by our code at marginal additional cost. We also use a different representation tree criterion that allows for more accurate computation of the eigenvectors but can make parallelization more difficult. AMS subject classifications. 65F15, 65Y15. Key words. Symmetric eigenproblem, multiple relatively robust representations, ScaLAPACK, numerical software. 1. Introduction. Since 2005, the National Science Foundation has been funding an initiative [19] to improve the LAPACK [3] and ScaLAPACK [14] libraries for Numerical Linear Algebra computations. One of the ambitious goals of this initiative is to put more of LAPACK into ScaLAPACK, recognizing the gap between available sequential and parallel software support. In particular, parallelization of the MRRR algorithm [24, 50, 51, 26, 27, 28, 29], the topic of this paper, is identified as one key improvement to ScaLAPACK.
    [Show full text]
  • The Sparse Matrix Transform for Covariance Estimation and Analysis of High Dimensional Signals
    The Sparse Matrix Transform for Covariance Estimation and Analysis of High Dimensional Signals Guangzhi Cao*, Member, IEEE, Leonardo R. Bachega, and Charles A. Bouman, Fellow, IEEE Abstract Covariance estimation for high dimensional signals is a classically difficult problem in statistical signal analysis and machine learning. In this paper, we propose a maximum likelihood (ML) approach to covariance estimation, which employs a novel non-linear sparsity constraint. More specifically, the covariance is constrained to have an eigen decomposition which can be represented as a sparse matrix transform (SMT). The SMT is formed by a product of pairwise coordinate rotations known as Givens rotations. Using this framework, the covariance can be efficiently estimated using greedy optimization of the log-likelihood function, and the number of Givens rotations can be efficiently computed using a cross- validation procedure. The resulting estimator is generally positive definite and well-conditioned, even when the sample size is limited. Experiments on a combination of simulated data, standard hyperspectral data, and face image sets show that the SMT-based covariance estimates are consistently more accurate than both traditional shrinkage estimates and recently proposed graphical lasso estimates for a variety of different classes and sample sizes. An important property of the new covariance estimate is that it naturally yields a fast implementation of the estimated eigen-transformation using the SMT representation. In fact, the SMT can be viewed as a generalization of the classical fast Fourier transform (FFT) in that it uses “butterflies” to represent an orthonormal transform. However, unlike the FFT, the SMT can be used for fast eigen-signal analysis of general non-stationary signals.
    [Show full text]
  • Chapter 3 LEAST SQUARES PROBLEMS
    Chapter 3 LEAST SQUARES PROBLEMS E B D C A the sea F One application is geodesy & surveying. Let z = elevation, and suppose we have the mea- surements: zA ≈ 1., zB ≈ 2., zC ≈ 3., zB − zA ≈ 1., zC − zB ≈ 2., zC − zA ≈ 1. 71 This is overdetermined and inconsistent: 0 1 0 1 100 1. B C 0 1 B . C B 010C z B 2 C B C A B . C B 001C @ z A ≈ B 3 C . B − C B B . C B 110C z B 1 C @ 0 −11A C @ 2. A −101 1. Another application is fitting a curve to given data: y y=ax+b x 2 3 2 3 x1 1 y1 6 7 6 7 6 x2 1 7 a 6 y2 7 6 . 7 ≈ 6 . 7 . 4 . 5 b 4 . 5 xn 1 yn More generally Am×nxn ≈ bm where A is known exactly, m ≥ n,andb is subject to inde- pendent random errors of equal variance (which can be achieved by scaling the equations). Gauss (1821) proved that the “best” solution x minimizes kb − Axk2, i.e., the sum of the squares of the residual components. Hence, we might write Ax ≈2 b. Even if the 2-norm is inappropriate, it is the easiest to work with. 3.1 The Normal Equations 3.2 QR Factorization 3.3 Householder Reflections 3.4 Givens Rotations 3.5 Gram-Schmidt Orthogonalization 3.6 Singular Value Decomposition 72 3.1 The Normal Equations Recall the inner product xTy for x, y ∈ Rm. (What is the geometric interpretation of the 3 m T inner product in R ?) A sequence of vectors x1,x2,...,xn in R is orthogonal if xi xj =0 T if and only if i =6 j and orthonormal if xi xj = δij.Twosubspaces S, T are orthogonal if T T x ∈ S, y ∈ T ⇒ x y =0.IfX =[x1,x2,...,xn], then orthonormal means X X = I.
    [Show full text]
  • SVD TR1690.Pdf
    Computer Sciences Department Computing the Singular Value Decomposition of 3 x 3 matrices with minimal branching and elementary floating point operations Aleka McAdams Andrew Selle Rasmus Tamstorf Joseph Teran Eftychios Sifakis Technical Report #1690 May 2011 Computing the Singular Value Decomposition of 3 × 3 matrices with minimal branching and elementary floating point operations Aleka McAdams1,2 Andrew Selle1 Rasmus Tamstorf1 Joseph Teran2,1 Eftychios Sifakis3,1 1 Walt Disney Animation Studios 2 University of California, Los Angeles 3 University of Wisconsin, Madison Abstract The algorithm first determines the factor V by computing the eige- nanalysis of the matrix AT A = VΣ2VT which is symmetric A numerical method for the computation of the Singular Value De- and positive semi-definite. This is accomplished via a modified composition of 3 × 3 matrices is presented. The proposed method- Jacobi iteration where the Jacobi factors are approximated using ology robustly handles rank-deficient matrices and guarantees or- inexpensive, elementary arithmetic operations as described in sec- thonormality of the computed rotational factors. The algorithm is tion 2. Since the symmetric eigenanalysis also produces Σ2, and tailored to the characteristics of SIMD or vector processors. In par- consequently Σ itself, the remaining factor U can theoretically be ticular, it does not require any explicit branching beyond simple obtained as U = AVΣ−1; however this process is not applica- conditional assignments (as in the C++ ternary operator ?:, or the ble when A (and as a result, Σ) is singular, and can also lead to SSE4.1 instruction VBLENDPS), enabling trivial data-level paral- substantial loss of orthogonality in U for an ill-conditioned, yet lelism for any number of operations.
    [Show full text]
  • Discontinuous Plane Rotations and the Symmetric Eigenvalue Problem
    December 4, 2000 Discontinuous Plane Rotations and the Symmetric Eigenvalue Problem Edward Anderson Lockheed Martin Services Inc. [email protected] Elementary plane rotations are one of the building blocks of numerical linear algebra and are employed in reducing matrices to condensed form for eigenvalue computations and during the QR algorithm. Unfortunately, their implementation in standard packages such as EISPACK, the BLAS and LAPACK lack the continuity of their mathematical formulation, which makes results from software that use them sensitive to perturbations. Test cases illustrating this problem will be presented, and reparations to the standard software proposed. 1.0 Introduction Unitary transformations are frequently used in numerical linear algebra software to reduce dense matrices to bidiagonal, tridiagonal, or Hessenberg form as a preliminary step towards finding the eigenvalues or singular values. Elementary plane rotation matrices (also called Givens rotations) are used to selectively reduce values to zero, such as when reducing a band matrix to tridiagonal form, or during the QR algorithm when attempting to deflate a tridiagonal matrix. The mathematical properties of rota- tions were studied by Wilkinson [9] and are generally not questioned. However, Wilkinson based his analysis on an equation for computing a rotation matrix that is different from those in common use in the BLAS [7] and LAPACK [2]. While the classical formula is continuous in the angle of rotation, the BLAS and LAPACK algorithms have lines of discontinuity that can lead to abrupt changes in the direction of computed eigenvectors. This makes comparing results difficult and proving backward stability impossible. It is easy to restore continuity in the generation of Givens rotations even while scaling to avoid overflow and underflow, and we show how in this report.
    [Show full text]
  • CORDIC-Based LMMSE Equalizer for Software Defined Radio
    CORDIC-Based LMMSE Equalizer for Software Defined Radio Murugappan SENTHILVELAN † §, Javier HORMIGO ‡, Joon Hwa CHUN €, Mihai SIMA *, Daniel IANCU †¶, Michael SCHULTE §, John GLOSSNER ¥ †Optimum Semiconductor Technologies, Inc., Tarrytown, NY, U.S.A. ‡ University of Malaga, Department of Computer Architecture, Malaga, Spain. € Intelligent Automation Inc., Rockville, MD, U.S.A. * University of Victoria, Department of Electrical and Computer Engineering, Victoria, BC, Canada. § University of Wisconsin-Madison, Department of Electrical and Computer Engineering, Madison, WI, U.S.A. ¶ Tampere University of Technology, Korkeakoulunkatu 1, FIN-33720 Tampere, Finland. ¥ Sandbridge Technologies, Inc., Tarrytown, NY, U.S.A. e-mail: [email protected], [email protected], [email protected] [email protected], [email protected], [email protected], [email protected] Abstract — In Code Division Multiple Access (CDMA) interference, and with synchronized transmission of blocks systems, the orthogonality of the spreading codes used to over the channel, multiple users can be multiplexed over the achieve multiple access over a channel is severely degraded same channel by using a different spreading code for each due to multi-path interference. Expensive equalization user. Even though signals from multiple users interfere with techniques are needed to recover the transmitted signal. each other in the channel, due to the orthogonal spreading The Linear Minimum Mean Square Error (LMMSE) codes, each user can extract the transmission intended for equalizer is a sub-optimal equalizer that is a good them from the received signal by despreading it using their compromise between computational complexity and spreading code. communication system performance. It uses While the spreading codes are perfectly orthogonal to computationally-intensive matrix inversion operations to each other, they are not orthogonal to time-shifted versions of perform equalization.
    [Show full text]