Section 4.5 Eigenvalues of Symmetric Tridiagonal Matrices
|
|
- Rolf White
- 6 years ago
- Views:
Transcription
1 Section 4.5 Eigenvalues of Symmetric Tridiagonal Matrices Key Terms Symmetric matrix Tridiagonal matrix Orthogonal matrix QR-factorization Rotation matrices (plane rotations) Eigenvalues
2 We will now complete our study of the matrix eigenvalue problem for symmetric matrices by developing an algorithm to compute all of the eigenvalues of a symmetric tridiagonal matrix. The technique which we will develop in this section is called the QR algorithm. Unlike the technique developed for reduction to tridiagonal form, the QR algorithm is iterative in nature. The sequence of matrices generated by the algorithm converges to a diagonal matrix, so the eigenvalues of the final" matrix in the sequence are just the elements along main diagonal. As we will continue to use similarity transformations, these diagonal elements are also the eigenvalues of the original matrix. But we will not use Householder matrices. Overview of eigenvalue approximation using the QR-method: Let A be an n n symmetric matrix which has been reduced to symmetric tridiagonal form, denoted A H, using Householder matrices. This preserves the eigenvalues and provides a matrix which is closer to diagonal form, hence fewer arithmetic operations will be required at later steps. Next we apply a piece of theory to A H Theorem: Any m n complex matrix A can be factored into a product of an m n matrix Q with orthonormal columns and n n upper triangular matrix R. (This is called a QR-factorization; we discuss this later.) In the case that matrix A is real and square, then the matrix Q will be an orthogonal matrix.
3 We use the QR-factorization to generate a sequence of similarity transformations so that the limit of the sequence will be a diagonal matrix. The eigenvalues of A and A H will be preserved and will appear on the diagonal of the diagonal matrix. The sequence is constructed as follows: Let A 0 = A H ; find the QR-factorization of A 0 ; A 0 = Q 0 R 0. (Matrix Q 0 is orthogonal so its inverse is Q 0T.) Define A 1 = R 0 Q 0 = Q 0T A 0 Q 0 since R 0 = Q 0T A 0. Find the QR-factorization of A 1 ; A 1 = Q 1 R 1. Define A 2 = R 1 Q 1 = Q 1T A 1 Q 1 since R 1 = Q 1T A 1. Thus A 2 = Q 1 T Q 0T A 0 Q 0 Q 1. Apply the steps again to A 2 and successively repeat the process. We call iteration process the BASIC QR-method. It generates a sequence A 0, A 1, A 2,, A k, D, a diagonal matrix. The diagonal entries of A k approximate the eigenvalues of matrix A. This process preserves the tridiagonal form and the numerical values of the first sub diagonal and first super diagonal decrease in value as k gets large. We stop the iterative process when the entries of these off-diagonals are sufficiently small. In this case the sequence appears to be converging to a diagonal matrix.
4 Example: Apply the basicqr.m iteration to the symmetric tridiagonal matrix Observe that the off-diagonal elements are converging toward zero, while the diagonal elements are converging toward the eigenvalues of A, which, to three decimal places, are 5.951, and Furthermore, the eigenvalues appear along the diagonal of A k in decreasing order of magnitude. This example demonstrates the general performance of the QR algorithm and that the offdiagonal entries converge toward zero and the rate of convergence O( λ j /λ j-1 ) where j is the row number and j-1 the column number of an off- diagonal entry. (No proof about the rate.)
5 In the example we used the m-file basicqr.m. >> help basicqr basicqr Apply the basicqr method to matrix A iter times. Use in the form ==> basicqr(a,iter) <== There are choices for various display options. MATLAB has an m-file more robust than basicqr.m. The MATLAB file is qr.m. It has various options which appear in the help file, A simple MATLAB code using qr.m which is like for the basic QR-method is given below. for k=1:10, [Q,R]=qr(A); A=R*Q, pause, end Change the number of iterations as needed. We will not prove the QR theorem. A discussion is given in the text. Orthogonal matrices constructed in the QR-algorithm are often called rotation matrices.
6 Notes: The eig command in MATLAB uses a more robust implementation than that given in the basic QR-(eigen)method discussed here. This command is quite dependable. If a matrix has eigenvalues very close in magnitude, then the convergence of the eigen method based on QR-factorizations can quite slow. If a matrix is defective or nearly defective the convergence of QR-(eigen)method can be slow. The basic QR-(eigen)method can fail. When asked to compute QR-factorizations involving symmetric matrices it is recommended that the matrix first be transformed to symmetric tridiagonal form. QR-factorizations are often implemented using another technique called (Givens) plane rotations. If A is not symmetric how can you approximate its eigenvalues?
7 Eigenvalues of non-symmetric matrices As opposed to the symmetric problem, the eigenvalues a of non-symmetric matrix do not form an orthogonal set of vectors. Moreover, eigenvectors may not form a linearindependent vector system (this is possible, although not necessarily, in case of multiple eigenvalues - a subspace with size less than k can correspond to the eigenvalue of multiplicity k). The second distinction from the symmetric problem is that a non-symmetric matrix could not be easily reduced to a tridiagonal matrix or a matrix in other compact form - matrix asymmetry causes the fact that after zeroing all the elements below the first subdiagonal (using an orthogonal transformation) all the elements in the upper triangle are not zeroed, so we get a matrix in upper Hessenberg form. This slows an algorithm down, because it is necessary to update all the upper triangles of the matrix after each iteration of the QR algorithm. At last, the third distinction is that the eigenvalues of a non-symmetric matrix could be complex (as are their corresponding eigenvectors).
8 Algorithm description Solving a non-symmetric problem of finding eigenvalues is performed in some steps. In the first step, the matrix is reduced to upper Hessenberg form by using an orthogonal transformation. In the second step, which takes the most amount of time, the matrix is reduced to upper Schur form by using an orthogonal transformation. If we only have to find the eigenvalues, this step is the last because the matrix eigenvalues are located in the diagonal blocks of a quasi-triangular matrix from the canonical Schur form. If we have to find the eigenvectors as well, it is necessary to perform a backward substitution with Schur vectors and quasi-triangular vectors (in fact - solving a system of linear equations; the process of backward substitution itself takes a small amount of time, but the necessity to save all the transformations makes the algorithm twice as slow). The Schur decomposition, which takes the most amount of time, is performed by using the QR algorithm with multiple shifts. The algorithm is taken from the LAPACK library. This algorithm is a block-matrix analog of the ordinary QR-algorithm with double shift. As all other block-matrix algorithms, this algorithm requires adjustment to achieve optimal performance. (Thankfully all this is incorporated in MATLAB s eig command.) More details at
632 CHAP. 11 EIGENVALUES AND EIGENVECTORS. QR Method
632 CHAP 11 EIGENVALUES AND EIGENVECTORS QR Method Suppose that A is a real symmetric matrix In the preceding section we saw how Householder s method is used to construct a similar tridiagonal matrix The
More informationOrthogonal iteration to QR
Notes for 2016-03-09 Orthogonal iteration to QR The QR iteration is the workhorse for solving the nonsymmetric eigenvalue problem. Unfortunately, while the iteration itself is simple to write, the derivation
More informationLinear algebra & Numerical Analysis
Linear algebra & Numerical Analysis Eigenvalues and Eigenvectors Marta Jarošová http://homel.vsb.cz/~dom033/ Outline Methods computing all eigenvalues Characteristic polynomial Jacobi method for symmetric
More informationECS130 Scientific Computing Handout E February 13, 2017
ECS130 Scientific Computing Handout E February 13, 2017 1. The Power Method (a) Pseudocode: Power Iteration Given an initial vector u 0, t i+1 = Au i u i+1 = t i+1 / t i+1 2 (approximate eigenvector) θ
More informationLecture # 11 The Power Method for Eigenvalues Part II. The power method find the largest (in magnitude) eigenvalue of. A R n n.
Lecture # 11 The Power Method for Eigenvalues Part II The power method find the largest (in magnitude) eigenvalue of It makes two assumptions. 1. A is diagonalizable. That is, A R n n. A = XΛX 1 for some
More informationTHE QR METHOD A = Q 1 R 1
THE QR METHOD Given a square matrix A, form its QR factorization, as Then define A = Q 1 R 1 A 2 = R 1 Q 1 Continue this process: for k 1(withA 1 = A), A k = Q k R k A k+1 = R k Q k Then the sequence {A
More informationEigenvalue and Eigenvector Problems
Eigenvalue and Eigenvector Problems An attempt to introduce eigenproblems Radu Trîmbiţaş Babeş-Bolyai University April 8, 2009 Radu Trîmbiţaş ( Babeş-Bolyai University) Eigenvalue and Eigenvector Problems
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J Olver 8 Numerical Computation of Eigenvalues In this part, we discuss some practical methods for computing eigenvalues and eigenvectors of matrices Needless to
More informationMath 405: Numerical Methods for Differential Equations 2016 W1 Topics 10: Matrix Eigenvalues and the Symmetric QR Algorithm
Math 405: Numerical Methods for Differential Equations 2016 W1 Topics 10: Matrix Eigenvalues and the Symmetric QR Algorithm References: Trefethen & Bau textbook Eigenvalue problem: given a matrix A, find
More informationMatrices, Moments and Quadrature, cont d
Jim Lambers CME 335 Spring Quarter 2010-11 Lecture 4 Notes Matrices, Moments and Quadrature, cont d Estimation of the Regularization Parameter Consider the least squares problem of finding x such that
More informationOrthogonal iteration revisited
Week 10 11: Friday, Oct 30 and Monday, Nov 2 Orthogonal iteration revisited Last time, we described a generalization of the power methods to compute invariant subspaces. That is, starting from some initial
More information13-2 Text: 28-30; AB: 1.3.3, 3.2.3, 3.4.2, 3.5, 3.6.2; GvL Eigen2
The QR algorithm The most common method for solving small (dense) eigenvalue problems. The basic algorithm: QR without shifts 1. Until Convergence Do: 2. Compute the QR factorization A = QR 3. Set A :=
More informationEIGENVALUE PROBLEMS. EIGENVALUE PROBLEMS p. 1/4
EIGENVALUE PROBLEMS EIGENVALUE PROBLEMS p. 1/4 EIGENVALUE PROBLEMS p. 2/4 Eigenvalues and eigenvectors Let A C n n. Suppose Ax = λx, x 0, then x is a (right) eigenvector of A, corresponding to the eigenvalue
More informationLast Time. Social Network Graphs Betweenness. Graph Laplacian. Girvan-Newman Algorithm. Spectral Bisection
Eigenvalue Problems Last Time Social Network Graphs Betweenness Girvan-Newman Algorithm Graph Laplacian Spectral Bisection λ 2, w 2 Today Small deviation into eigenvalue problems Formulation Standard eigenvalue
More informationOrthogonal iteration to QR
Week 10: Wednesday and Friday, Oct 24 and 26 Orthogonal iteration to QR On Monday, we went through a somewhat roundabout algbraic path from orthogonal subspace iteration to the QR iteration. Let me start
More informationBindel, Spring 2016 Numerical Analysis (CS 4220) Notes for
Notes for 2016-03-11 Woah, We re Halfway There Last time, we showed that the QR iteration maps upper Hessenberg matrices to upper Hessenberg matrices, and this fact allows us to do one QR sweep in O(n
More information4.8 Arnoldi Iteration, Krylov Subspaces and GMRES
48 Arnoldi Iteration, Krylov Subspaces and GMRES We start with the problem of using a similarity transformation to convert an n n matrix A to upper Hessenberg form H, ie, A = QHQ, (30) with an appropriate
More informationEXAM. Exam 1. Math 5316, Fall December 2, 2012
EXAM Exam Math 536, Fall 22 December 2, 22 Write all of your answers on separate sheets of paper. You can keep the exam questions. This is a takehome exam, to be worked individually. You can use your notes.
More information(Linear equations) Applied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB (Linear equations) Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots
More informationMath 411 Preliminaries
Math 411 Preliminaries Provide a list of preliminary vocabulary and concepts Preliminary Basic Netwon s method, Taylor series expansion (for single and multiple variables), Eigenvalue, Eigenvector, Vector
More informationNumerical Methods I Eigenvalue Problems
Numerical Methods I Eigenvalue Problems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fall 2014 October 2nd, 2014 A. Donev (Courant Institute) Lecture
More informationEigenvalues and eigenvectors
Chapter 6 Eigenvalues and eigenvectors An eigenvalue of a square matrix represents the linear operator as a scaling of the associated eigenvector, and the action of certain matrices on general vectors
More informationQR-decomposition. The QR-decomposition of an n k matrix A, k n, is an n n unitary matrix Q and an n k upper triangular matrix R for which A = QR
QR-decomposition The QR-decomposition of an n k matrix A, k n, is an n n unitary matrix Q and an n k upper triangular matrix R for which In Matlab A = QR [Q,R]=qr(A); Note. The QR-decomposition is unique
More informationNotes on Eigenvalues, Singular Values and QR
Notes on Eigenvalues, Singular Values and QR Michael Overton, Numerical Computing, Spring 2017 March 30, 2017 1 Eigenvalues Everyone who has studied linear algebra knows the definition: given a square
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 16: Reduction to Hessenberg and Tridiagonal Forms; Rayleigh Quotient Iteration Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical
More informationSolving linear equations with Gaussian Elimination (I)
Term Projects Solving linear equations with Gaussian Elimination The QR Algorithm for Symmetric Eigenvalue Problem The QR Algorithm for The SVD Quasi-Newton Methods Solving linear equations with Gaussian
More information1 Number Systems and Errors 1
Contents 1 Number Systems and Errors 1 1.1 Introduction................................ 1 1.2 Number Representation and Base of Numbers............. 1 1.2.1 Normalized Floating-point Representation...........
More informationEECS 275 Matrix Computation
EECS 275 Matrix Computation Ming-Hsuan Yang Electrical Engineering and Computer Science University of California at Merced Merced, CA 95344 http://faculty.ucmerced.edu/mhyang Lecture 12 1 / 18 Overview
More information6.4 Krylov Subspaces and Conjugate Gradients
6.4 Krylov Subspaces and Conjugate Gradients Our original equation is Ax = b. The preconditioned equation is P Ax = P b. When we write P, we never intend that an inverse will be explicitly computed. P
More informationAPPLIED NUMERICAL LINEAR ALGEBRA
APPLIED NUMERICAL LINEAR ALGEBRA James W. Demmel University of California Berkeley, California Society for Industrial and Applied Mathematics Philadelphia Contents Preface 1 Introduction 1 1.1 Basic Notation
More informationEigenvalue problems. Eigenvalue problems
Determination of eigenvalues and eigenvectors Ax x, where A is an N N matrix, eigenvector x 0, and eigenvalues are in general complex numbers In physics: - Energy eigenvalues in a quantum mechanical system
More informationExercise Set 7.2. Skills
Orthogonally diagonalizable matrix Spectral decomposition (or eigenvalue decomposition) Schur decomposition Subdiagonal Upper Hessenburg form Upper Hessenburg decomposition Skills Be able to recognize
More informationLecture 3: QR-Factorization
Lecture 3: QR-Factorization This lecture introduces the Gram Schmidt orthonormalization process and the associated QR-factorization of matrices It also outlines some applications of this factorization
More informationMatrix Algorithms. Volume II: Eigensystems. G. W. Stewart H1HJ1L. University of Maryland College Park, Maryland
Matrix Algorithms Volume II: Eigensystems G. W. Stewart University of Maryland College Park, Maryland H1HJ1L Society for Industrial and Applied Mathematics Philadelphia CONTENTS Algorithms Preface xv xvii
More informationFast Hessenberg QR Iteration for Companion Matrices
Fast Hessenberg QR Iteration for Companion Matrices David Bindel Ming Gu David Garmire James Demmel Shivkumar Chandrasekaran Fast Hessenberg QR Iteration for Companion Matrices p.1/20 Motivation: Polynomial
More information22.4. Numerical Determination of Eigenvalues and Eigenvectors. Introduction. Prerequisites. Learning Outcomes
Numerical Determination of Eigenvalues and Eigenvectors 22.4 Introduction In Section 22. it was shown how to obtain eigenvalues and eigenvectors for low order matrices, 2 2 and. This involved firstly solving
More informationComputing Eigenvalues and/or Eigenvectors;Part 2, The Power method and QR-algorithm
Computing Eigenvalues and/or Eigenvectors;Part 2, The Power method and QR-algorithm Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo November 13, 2009 Today
More informationIndex. for generalized eigenvalue problem, butterfly form, 211
Index ad hoc shifts, 165 aggressive early deflation, 205 207 algebraic multiplicity, 35 algebraic Riccati equation, 100 Arnoldi process, 372 block, 418 Hamiltonian skew symmetric, 420 implicitly restarted,
More informationComputation of eigenvalues and singular values Recall that your solutions to these questions will not be collected or evaluated.
Math 504, Homework 5 Computation of eigenvalues and singular values Recall that your solutions to these questions will not be collected or evaluated 1 Find the eigenvalues and the associated eigenspaces
More informationNumerical Methods I: Eigenvalues and eigenvectors
1/25 Numerical Methods I: Eigenvalues and eigenvectors Georg Stadler Courant Institute, NYU stadler@cims.nyu.edu November 2, 2017 Overview 2/25 Conditioning Eigenvalues and eigenvectors How hard are they
More informationWe will discuss matrix diagonalization algorithms in Numerical Recipes in the context of the eigenvalue problem in quantum mechanics, m A n = λ m
Eigensystems We will discuss matrix diagonalization algorithms in umerical Recipes in the context of the eigenvalue problem in quantum mechanics, A n = λ n n, (1) where A is a real, symmetric Hamiltonian
More informationIntroduction to Applied Linear Algebra with MATLAB
Sigam Series in Applied Mathematics Volume 7 Rizwan Butt Introduction to Applied Linear Algebra with MATLAB Heldermann Verlag Contents Number Systems and Errors 1 1.1 Introduction 1 1.2 Number Representation
More informationApplied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in
More informationMAA507, Power method, QR-method and sparse matrix representation.
,, and representation. February 11, 2014 Lecture 7: Overview, Today we will look at:.. If time: A look at representation and fill in. Why do we need numerical s? I think everyone have seen how time consuming
More informationEigenvalues and Eigenvectors
Chapter 1 Eigenvalues and Eigenvectors Among problems in numerical linear algebra, the determination of the eigenvalues and eigenvectors of matrices is second in importance only to the solution of linear
More informationSection 6.4. The Gram Schmidt Process
Section 6.4 The Gram Schmidt Process Motivation The procedures in 6 start with an orthogonal basis {u, u,..., u m}. Find the B-coordinates of a vector x using dot products: x = m i= x u i u i u i u i Find
More informationSolving large scale eigenvalue problems
arge scale eigenvalue problems, Lecture 4, March 14, 2018 1/41 Lecture 4, March 14, 2018: The QR algorithm http://people.inf.ethz.ch/arbenz/ewp/ Peter Arbenz Computer Science Department, ETH Zürich E-mail:
More informationIntroduction to Mathematical Programming
Introduction to Mathematical Programming Ming Zhong Lecture 6 September 12, 2018 Ming Zhong (JHU) AMS Fall 2018 1 / 20 Table of Contents 1 Ming Zhong (JHU) AMS Fall 2018 2 / 20 Solving Linear Systems A
More informationScientific Computing with Case Studies SIAM Press, Lecture Notes for Unit VII Sparse Matrix
Scientific Computing with Case Studies SIAM Press, 2009 http://www.cs.umd.edu/users/oleary/sccswebpage Lecture Notes for Unit VII Sparse Matrix Computations Part 1: Direct Methods Dianne P. O Leary c 2008
More informationNumerical Methods - Numerical Linear Algebra
Numerical Methods - Numerical Linear Algebra Y. K. Goh Universiti Tunku Abdul Rahman 2013 Y. K. Goh (UTAR) Numerical Methods - Numerical Linear Algebra I 2013 1 / 62 Outline 1 Motivation 2 Solving Linear
More informationCOMPUTER SCIENCE 515 Numerical Linear Algebra SPRING 2006 ASSIGNMENT # 4 (39 points) February 27
Due Friday, March 1 in class COMPUTER SCIENCE 1 Numerical Linear Algebra SPRING 26 ASSIGNMENT # 4 (9 points) February 27 1. (22 points) The goal is to compare the effectiveness of five different techniques
More informationNUMERICAL COMPUTATION IN SCIENCE AND ENGINEERING
NUMERICAL COMPUTATION IN SCIENCE AND ENGINEERING C. Pozrikidis University of California, San Diego New York Oxford OXFORD UNIVERSITY PRESS 1998 CONTENTS Preface ix Pseudocode Language Commands xi 1 Numerical
More informationApplied Linear Algebra
Applied Linear Algebra Peter J. Olver School of Mathematics University of Minnesota Minneapolis, MN 55455 olver@math.umn.edu http://www.math.umn.edu/ olver Chehrzad Shakiban Department of Mathematics University
More informationEIGENVALUE PROBLEMS (EVP)
EIGENVALUE PROBLEMS (EVP) (Golub & Van Loan: Chaps 7-8; Watkins: Chaps 5-7) X.-W Chang and C. C. Paige PART I. EVP THEORY EIGENVALUES AND EIGENVECTORS Let A C n n. Suppose Ax = λx with x 0, then x is a
More informationRoundoff Error. Monday, August 29, 11
Roundoff Error A round-off error (rounding error), is the difference between the calculated approximation of a number and its exact mathematical value. Numerical analysis specifically tries to estimate
More informationLINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS
LINEAR ALGEBRA, -I PARTIAL EXAM SOLUTIONS TO PRACTICE PROBLEMS Problem (a) For each of the two matrices below, (i) determine whether it is diagonalizable, (ii) determine whether it is orthogonally diagonalizable,
More informationPresentation of XLIFE++
Presentation of XLIFE++ Eigenvalues Solver & OpenMP Manh-Ha NGUYEN Unité de Mathématiques Appliquées, ENSTA - Paristech 25 Juin 2014 Ha. NGUYEN Presentation of XLIFE++ 25 Juin 2014 1/19 EigenSolver 1 EigenSolver
More informationApplied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in
More informationSTAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 9
STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 9 1. qr and complete orthogonal factorization poor man s svd can solve many problems on the svd list using either of these factorizations but they
More informationAssignment #9: Orthogonal Projections, Gram-Schmidt, and Least Squares. Name:
Assignment 9: Orthogonal Projections, Gram-Schmidt, and Least Squares Due date: Friday, April 0, 08 (:pm) Name: Section Number Assignment 9: Orthogonal Projections, Gram-Schmidt, and Least Squares Due
More informationLeast squares and Eigenvalues
Lab 1 Least squares and Eigenvalues Lab Objective: Use least squares to fit curves to data and use QR decomposition to find eigenvalues. Least Squares A linear system Ax = b is overdetermined if it has
More informationLinear Algebra in Numerical Methods. Lecture on linear algebra MATLAB/Octave works well with linear algebra
Linear Algebra in Numerical Methods Lecture on linear algebra MATLAB/Octave works well with linear algebra Linear Algebra A pseudo-algebra that deals with a system of equations and the transformations
More informationQR Decomposition. When solving an overdetermined system by projection (or a least squares solution) often the following method is used:
(In practice not Gram-Schmidt, but another process Householder Transformations are used.) QR Decomposition When solving an overdetermined system by projection (or a least squares solution) often the following
More informationANONSINGULAR tridiagonal linear system of the form
Generalized Diagonal Pivoting Methods for Tridiagonal Systems without Interchanges Jennifer B. Erway, Roummel F. Marcia, and Joseph A. Tyson Abstract It has been shown that a nonsingular symmetric tridiagonal
More informationEigenvalue Problems. Eigenvalue problems occur in many areas of science and engineering, such as structural analysis
Eigenvalue Problems Eigenvalue problems occur in many areas of science and engineering, such as structural analysis Eigenvalues also important in analyzing numerical methods Theory and algorithms apply
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 4 Eigenvalue Problems Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction
More informationA Cholesky LR algorithm for the positive definite symmetric diagonal-plus-semiseparable eigenproblem
A Cholesky LR algorithm for the positive definite symmetric diagonal-plus-semiseparable eigenproblem Bor Plestenjak Department of Mathematics University of Ljubljana Slovenia Ellen Van Camp and Marc Van
More informationIn this section again we shall assume that the matrix A is m m, real and symmetric.
84 3. The QR algorithm without shifts See Chapter 28 of the textbook In this section again we shall assume that the matrix A is m m, real and symmetric. 3.1. Simultaneous Iterations algorithm Suppose we
More informationLinear Least Squares Problems
Linear Least Squares Problems Introduction We have N data points (x 1,y 1 ),...(x N,y N ). We assume that the data values are given by y j = g(x j ) + e j, j = 1,...,N where g(x) = c 1 g 1 (x) + + c n
More informationSUMMARY OF MATH 1600
SUMMARY OF MATH 1600 Note: The following list is intended as a study guide for the final exam. It is a continuation of the study guide for the midterm. It does not claim to be a comprehensive list. You
More informationDirect methods for symmetric eigenvalue problems
Direct methods for symmetric eigenvalue problems, PhD McMaster University School of Computational Engineering and Science February 4, 2008 1 Theoretical background Posing the question Perturbation theory
More informationLinear Algebra, part 3 QR and SVD
Linear Algebra, part 3 QR and SVD Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2012 Going back to least squares (Section 1.4 from Strang, now also see section 5.2). We
More informationEECS 275 Matrix Computation
EECS 275 Matrix Computation Ming-Hsuan Yang Electrical Engineering and Computer Science University of California at Merced Merced, CA 95344 http://faculty.ucmerced.edu/mhyang Lecture 17 1 / 26 Overview
More informationMatrix decompositions
Matrix decompositions Zdeněk Dvořák May 19, 2015 Lemma 1 (Schur decomposition). If A is a symmetric real matrix, then there exists an orthogonal matrix Q and a diagonal matrix D such that A = QDQ T. The
More informationMatrix Algebra: Summary
May, 27 Appendix E Matrix Algebra: Summary ontents E. Vectors and Matrtices.......................... 2 E.. Notation.................................. 2 E..2 Special Types of Vectors.........................
More information11.3 Eigenvalues and Eigenvectors of a Tridiagonal Matrix
11.3 Eigenvalues and Eigenvectors of a ridiagonal Matrix Evaluation of the Characteristic Polynomial Once our original, real, symmetric matrix has been reduced to tridiagonal form, one possible way to
More informationThe Singular Value Decomposition
The Singular Value Decomposition Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) SVD Fall 2015 1 / 13 Review of Key Concepts We review some key definitions and results about matrices that will
More informationEigenvalue Problems and Singular Value Decomposition
Eigenvalue Problems and Singular Value Decomposition Sanzheng Qiao Department of Computing and Software McMaster University August, 2012 Outline 1 Eigenvalue Problems 2 Singular Value Decomposition 3 Software
More informationComputing Eigenvalues and/or Eigenvectors;Part 2, The Power method and QR-algorithm
Computing Eigenvalues and/or Eigenvectors;Part 2, The Power method and QR-algorithm Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo November 19, 2010 Today
More informationMath 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination
Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column
More informationModule 6.6: nag nsym gen eig Nonsymmetric Generalized Eigenvalue Problems. Contents
Eigenvalue and Least-squares Problems Module Contents Module 6.6: nag nsym gen eig Nonsymmetric Generalized Eigenvalue Problems nag nsym gen eig provides procedures for solving nonsymmetric generalized
More informationMath 504 (Fall 2011) 1. (*) Consider the matrices
Math 504 (Fall 2011) Instructor: Emre Mengi Study Guide for Weeks 11-14 This homework concerns the following topics. Basic definitions and facts about eigenvalues and eigenvectors (Trefethen&Bau, Lecture
More informationI. Multiple Choice Questions (Answer any eight)
Name of the student : Roll No : CS65: Linear Algebra and Random Processes Exam - Course Instructor : Prashanth L.A. Date : Sep-24, 27 Duration : 5 minutes INSTRUCTIONS: The test will be evaluated ONLY
More informationLinear least squares problem: Example
Linear least squares problem: Example We want to determine n unknown parameters c,,c n using m measurements where m n Here always denotes the -norm v = (v + + v n) / Example problem Fit the experimental
More informationTEACHING NUMERICAL LINEAR ALGEBRA AT THE UNDERGRADUATE LEVEL by Biswa Nath Datta Department of Mathematical Sciences Northern Illinois University
TEACHING NUMERICAL LINEAR ALGEBRA AT THE UNDERGRADUATE LEVEL by Biswa Nath Datta Department of Mathematical Sciences Northern Illinois University DeKalb, IL 60115 E-mail: dattab@math.niu.edu What is Numerical
More informationSolving large scale eigenvalue problems
arge scale eigenvalue problems, Lecture 5, March 23, 2016 1/30 Lecture 5, March 23, 2016: The QR algorithm II http://people.inf.ethz.ch/arbenz/ewp/ Peter Arbenz Computer Science Department, ETH Zürich
More information5.3 The Power Method Approximation of the Eigenvalue of Largest Module
192 5 Approximation of Eigenvalues and Eigenvectors 5.3 The Power Method The power method is very good at approximating the extremal eigenvalues of the matrix, that is, the eigenvalues having largest and
More informationBindel, Fall 2016 Matrix Computations (CS 6210) Notes for
1 Logistics Notes for 2016-08-26 1. Our enrollment is at 50, and there are still a few students who want to get in. We only have 50 seats in the room, and I cannot increase the cap further. So if you are
More informationIndex. Copyright (c)2007 The Society for Industrial and Applied Mathematics From: Matrix Methods in Data Mining and Pattern Recgonition By: Lars Elden
Index 1-norm, 15 matrix, 17 vector, 15 2-norm, 15, 59 matrix, 17 vector, 15 3-mode array, 91 absolute error, 15 adjacency matrix, 158 Aitken extrapolation, 157 algebra, multi-linear, 91 all-orthogonality,
More informationNotes for CS542G (Iterative Solvers for Linear Systems)
Notes for CS542G (Iterative Solvers for Linear Systems) Robert Bridson November 20, 2007 1 The Basics We re now looking at efficient ways to solve the linear system of equations Ax = b where in this course,
More informationApplied Mathematics 205. Unit V: Eigenvalue Problems. Lecturer: Dr. David Knezevic
Applied Mathematics 205 Unit V: Eigenvalue Problems Lecturer: Dr. David Knezevic Unit V: Eigenvalue Problems Chapter V.4: Krylov Subspace Methods 2 / 51 Krylov Subspace Methods In this chapter we give
More informationComputational Methods. Least Squares Approximation/Optimization
Computational Methods Least Squares Approximation/Optimization Manfred Huber 2011 1 Least Squares Least squares methods are aimed at finding approximate solutions when no precise solution exists Find the
More information11.0 Introduction. An N N matrix A is said to have an eigenvector x and corresponding eigenvalue λ if. A x = λx (11.0.1)
Chapter 11. 11.0 Introduction Eigensystems An N N matrix A is said to have an eigenvector x and corresponding eigenvalue λ if A x = λx (11.0.1) Obviously any multiple of an eigenvector x will also be an
More informationMATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.
MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. Adjoint operator and adjoint matrix Given a linear operator L on an inner product space V, the adjoint of L is a transformation
More informationClass notes: Approximation
Class notes: Approximation Introduction Vector spaces, linear independence, subspace The goal of Numerical Analysis is to compute approximations We want to approximate eg numbers in R or C vectors in R
More informationLARGE SPARSE EIGENVALUE PROBLEMS. General Tools for Solving Large Eigen-Problems
LARGE SPARSE EIGENVALUE PROBLEMS Projection methods The subspace iteration Krylov subspace methods: Arnoldi and Lanczos Golub-Kahan-Lanczos bidiagonalization General Tools for Solving Large Eigen-Problems
More informationPreliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012
Instructions Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012 The exam consists of four problems, each having multiple parts. You should attempt to solve all four problems. 1.
More informationKrylov Subspaces. Lab 1. The Arnoldi Iteration
Lab 1 Krylov Subspaces Lab Objective: Discuss simple Krylov Subspace Methods for finding eigenvalues and show some interesting applications. One of the biggest difficulties in computational linear algebra
More informationNumerical Mathematics
Alfio Quarteroni Riccardo Sacco Fausto Saleri Numerical Mathematics Second Edition With 135 Figures and 45 Tables 421 Springer Contents Part I Getting Started 1 Foundations of Matrix Analysis 3 1.1 Vector
More informationLinear Algebra. Carleton DeTar February 27, 2017
Linear Algebra Carleton DeTar detar@physics.utah.edu February 27, 2017 This document provides some background for various course topics in linear algebra: solving linear systems, determinants, and finding
More information