DOCSLIB.ORG

Finding Eigenvalues: Arnoldi Iteration and the QR Algorithm

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Finding Eigenvalues: Arnoldi Iteration and the QR Algorithm Finding Eigenvalues: Arnoldi Iteration and the QR Algorithm Will Wheeler Algorithms Interest Group Jul 13, 2016 Outline Finding the largest eigenvalue I Largest eigenvalue power method I So much work for one eigenvector. What about others? I More eigenvectors orthogonalize Krylov matrix I build orthogonal list Arnoldi iteration Solving the eigenvalue problem I Eigenvalues diagonal of triangular matrix I Triangular matrix QR algorithm I QR algorithm QR decomposition I Better QR decomposition Hessenberg matrix I Hessenberg matrix Arnoldi algorithm Power Method How do we find the largest eigenvalue of an m × m matrix A? I Start with a vector b and make a power sequence: b; Ab; A2b;::: I Higher eigenvalues dominate: n n n n A (v1 + v2) = λ1v1 + λ2v2 ≈ λ1v1 Vector sequence (normalized) converged? I Yes: Eigenvector I No: Iterate some more Krylov Matrix Power method throws away information along the way. I Put the sequence into a matrix K = [b; Ab; A2b;:::; An−1b] (1) = [xn; xn−1; xn−2;:::; x1] (2) I Gram-Schmidt approximates first n eigenvectors v1 = x1 (3) x2 · x1 v2 = x2 − x1 (4) x1 · x1 x3 · x1 x3 · x2 v3 = x3 − x1 − x2 (5) x1 · x1 x2 · x2 ::: (6) I Why not orthogonalize as we build the list? Arnoldi Iteration I Start with a normalized vector q1 xk = Aqk−1 (next power) (7) k−1 X yk = xk − (qj · xk ) qj (Gram-Schmidt) (8) j=1 qk = yk = jyk j (normalize) (9) I Orthonormal vectors fq1;:::; qng span Krylov subspace n−1 (Kn = spanfq1; Aq1;:::; A q1g) I These are not the eigenvectors of A, but make a similarity y (unitary) transformation Hn = QnAQn Arnoldi Iteration I Start with a normalized vector q1 xk = Aqk−1 (next power) (10) k−1 X yk = xk − (qj · xk ) qj (Gram-Schmidt) (11) j=1 qk = yk =jyk j (normalize) (12) I Orthonormal vectors fq1;:::; qng span Krylov subspace n−1 (Kn = spanfq1; Aq1;:::; A q1g) I These are not the eigenvectors of A, but make a similarity y (unitary) transformation Hn = QnAQn Arnoldi Iteration - Construct Hn We can construct the matrix Hn along the way. y I For k ≥ j: hj;k−1 = qj · xk = qj Aqk−1 I For k = j: hj;k−1 = jyk j (equivalent) I For k < j: hj;k−1 = 0 I ) Hn is upper Hessenberg 0 1 h1;1 h1;2 h1;3 h1;4 h1;5 h1;6 Bh2;1 h2;2 h2;3 h2;4 h2;5 h2;6C B C B 0 h3;2 h3;3 h3;4 h3;5 h3;6C Hn = B C (13) B 0 0 h4;3 h4;4 h4;5 h4;6C B C @ 0 0 0 h5;4 h5;5 h5;6A 0 0 0 0 h6;5 h6;6 y I Why is this useful? Hn = QnAQn ! same eigenvalues as A Finding Eigenvalues How do we find eigenvalues of a matrix? I Start with a triangular matrix A I The diagonal elements are the eigenvalues 0 1 a11 a12 a13 a14 a15 a16 B 0 a22 a23 a24 a25 a26C B C B 0 0 a33 a34 a35 a36C A = B C (14) B 0 0 0 a44 a45 a46C B C @ 0 0 0 0 a55 a56A 0 0 0 0 0 a66 What if A isn't triangular? QR algorithm Factorize A = QR I Q is orthonormal (sorry this is a different Q) I R is upper triangular (aka right triangular) Algorithm: 1. Start A1 = A 2. Factorize Ak = Qk Rk 3. Construct Ak+1 = Rk Qk −1 −1 = Qk Qk Rk Qk = Qk Ak Qk similarity transform 4. Ak converges to upper triangular QR Algorithm Example (numpy.linalg.qr) 00:313 0:106 0:8991 A1 = @0:381 0:979 0:375A 0:399 0:488 0:876 0 1:649 0:05 −0:2561 A2 = @ 0:035 0:502 0:534 A −0:014 0:004 0:017 01:653e + 00 2:290e − 02 2:305e − 01 1 A3 = @5:966e − 03 5:063e − 01 −5:342e − 01A 8:325e − 05 −9:429e − 05 9:666e − 03 0 1:653e + 00 1:872e − 02 −2:285e − 011 A4 = @ 1:800e − 03 5:063e − 01 5:349e − 01 A −4:812e − 07 1:803e − 06 9:554e − 03 QR Decomposition - Gram-Schmidt How do we get the matrices Q and R? http://www.seas.ucla.edu/ vandenbe/103/lectures/qr.pdf Recursively solve for the first column of Q and the first row of R: I A = QR r11 R12 I a1 A2 = q1 Q2 0 R22 I a1 A2 = r11q1 q1R12 + Q2R22 I r11 = jja1jj, q1 = a1=r11 | | I Note q1q1 = 1 and q1Q2 = 0 | I R12 = q1A2 I Solve A2 − q1R12 = Q2R22 QR Decomposition of Hessenberg Matrix What if we have a matrix in upper Hessenberg form? 0 1 h1;1 h1;2 h1;3 h1;4 h1;5 h1;6 Bh2;1 h2;2 h2;3 h2;4 h2;5 h2;6C B C B 0 h3;2 h3;3 h3;4 h3;5 h3;6C Hn = B C (15) B 0 0 h4;3 h4;4 h4;5 h4;6C B C @ 0 0 0 h5;4 h5;5 h5;6A 0 0 0 0 h6;5 h6;6 We just need to remove the numbers below the diagonal by combining rows ! Givens rotation. Givens Rotation A Givens rotation acts only on two rows and leaves the others unchanged. 0γ −σ 0 0 01 Bσ γ 0 0 0C B C G1 = B0 0 1 0 0C (16) B C @0 0 0 1 0A 0 0 0 0 1 a r p Want G = ; r = a2 + b2 b 0 p 2 2 I γ = a= a + b p 2 2 I σ = −b= a + b QR Decomposition of Hessenberg Matrix 0 1 he11 he12 he13 he14 he15 B C B 0 he22 he23 he24 he25C B C G2G1Hn = B 00 he33 he34 he35C (17) B C @ 0 0 h43 h44 h45A 0 0 0 h54 h55 | | | So Hn = G1G2 ::: Gn−1R = QR QR Decomposition of Hessenberg Matrix QR algorithm is iterative; is RQ also upper Hessenberg? Yes: Acting on the right, the Givens rotations mix two columns instead of rows, but change the same zeros. 0 1 re11 re12 re13 r14 r15 Br21 r22 r23 r24 r25C Be e e C G2G1Hn = B 0 r32 r33 r34 r35C (18) B e e C @ 0 00 r44 r45A 0 0 0 0 r55 Householder Reflections I Arnoldi finds a few eigenvalues of a large matrix I In practice, QR uses Householder reflections I www.math.usm.edu/lambers/mat610/sum10/lecture9.pdf Reflection across the plane perpendicular to a unit vector v: P = I − 2vv| (19) Px = x − 2vv|x (20) Px − x = −v(2v|x) (21) Want to arrange first column into zeros: find v so that Px = αe1. Householder Reflections Want to arrange first column into zeros: find v so that Px = αe1. jjxjj = α (22) | x − 2vv x = αe1 (23) 1 (x − αe ) = v(v|x) (24) 2 1 So we know I v / x − αe1 I v is a unit vector x − αe v = 1 (25) jjx − αe1jj Lanczos What if we apply the Arnoldi algorithm to a Hermitian matrix? I Start with a normalized vector q1 xk = Aqk−1 (next power) (26) k−1 X yk = xk − (qj · xk ) qj (Gram-Schmidt) (27) j=k−2 qk = yk = jyk j (normalize) (28) y Since Hn = QnAQn, I Hn is also Hermitian I Entries of Hn above the first diagonal are 0 (tridiagonal) I We don't need to calculate them! Lanczos What if we apply the Arnoldi algorithm to a Hermitian matrix? y Since Hn = QnAQn, I Hn is also Hermitian I Entries of Hn above the first diagonal are 0 (tridiagonal) I We don't need to calculate them! 0 ∗ 1 h11 h21 0 0 0 ∗ Bh21 h22 h32 0 0 C B ∗ C Hn = B 0 h32 h33 h43 0 C (29) B ∗ C @ 0 0 h43 h44 h54 A 0 0 0 h54 h55.
Load more

Recommended publications
  • A Distributed and Parallel Asynchronous Unite and Conquer Method to Solve Large Scale Non-Hermitian Linear Systems Xinzhe Wu, Serge Petiton
    A Distributed and Parallel Asynchronous Unite and Conquer Method to Solve Large Scale Non-Hermitian Linear Systems Xinzhe Wu, Serge Petiton To cite this version: Xinzhe Wu, Serge Petiton. A Distributed and Parallel Asynchronous Unite and Conquer Method to Solve Large Scale Non-Hermitian Linear Systems. HPC Asia 2018 - International Conference on High Performance Computing in Asia-Pacific Region, Jan 2018, Tokyo, Japan. 10.1145/3149457.3154481. hal-01677110 HAL Id: hal-01677110 https://hal.archives-ouvertes.fr/hal-01677110 Submitted on 8 Jan 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A Distributed and Parallel Asynchronous Unite and Conquer Method to Solve Large Scale Non-Hermitian Linear Systems Xinzhe WU Serge G. Petiton Maison de la Simulation, USR 3441 Maison de la Simulation, USR 3441 CRIStAL, UMR CNRS 9189 CRIStAL, UMR CNRS 9189 Université de Lille 1, Sciences et Technologies Université de Lille 1, Sciences et Technologies F-91191, Gif-sur-Yvette, France F-91191, Gif-sur-Yvette, France [email protected] [email protected] ABSTRACT the iteration number. These methods are already well implemented Parallel Krylov Subspace Methods are commonly used for solv- in parallel to profit from the great number of computation cores ing large-scale sparse linear systems.
    [Show full text]
  • Implicitly Restarted Arnoldi/Lanczos Methods for Large Scale Eigenvalue Calculations
    NASA Contractor Report 198342 /" ICASE Report No. 96-40 J ICA IMPLICITLY RESTARTED ARNOLDI/LANCZOS METHODS FOR LARGE SCALE EIGENVALUE CALCULATIONS Danny C. Sorensen NASA Contract No. NASI-19480 May 1996 Institute for Computer Applications in Science and Engineering NASA Langley Research Center Hampton, VA 23681-0001 Operated by Universities Space Research Association National Aeronautics and Space Administration Langley Research Center Hampton, Virginia 23681-0001 IMPLICITLY RESTARTED ARNOLDI/LANCZOS METHODS FOR LARGE SCALE EIGENVALUE CALCULATIONS Danny C. Sorensen 1 Department of Computational and Applied Mathematics Rice University Houston, TX 77251 sorensen@rice, edu ABSTRACT Eigenvalues and eigenfunctions of linear operators are important to many areas of ap- plied mathematics. The ability to approximate these quantities numerically is becoming increasingly important in a wide variety of applications. This increasing demand has fu- eled interest in the development of new methods and software for the numerical solution of large-scale algebraic eigenvalue problems. In turn, the existence of these new methods and software, along with the dramatically increased computational capabilities now avail- able, has enabled the solution of problems that would not even have been posed five or ten years ago. Until very recently, software for large-scale nonsymmetric problems was virtually non-existent. Fortunately, the situation is improving rapidly. The purpose of this article is to provide an overview of the numerical solution of large- scale algebraic eigenvalue problems. The focus will be on a class of methods called Krylov subspace projection methods. The well-known Lanczos method is the premier member of this class. The Arnoldi method generalizes the Lanczos method to the nonsymmetric case.
    [Show full text]
  • Krylov Subspaces
    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 is the amount of memory needed to store a large matrix and the amount of time needed to read its entries. Methods using Krylov subspaces avoid this difficulty by studying how a matrix acts on vectors, making it unnecessary in many cases to create the matrix itself. More specifically, we can construct a Krylov subspace just by knowing how a linear transformation acts on vectors, and with these subspaces we can closely approximate eigenvalues of the transformation and solutions to associated linear systems. The Arnoldi iteration is an algorithm for finding an orthonormal basis of a Krylov subspace. Its outputs can also be used to approximate the eigenvalues of the original matrix. The Arnoldi Iteration The order-N Krylov subspace of A generated by x is 2 n−1 Kn(A; x) = spanfx;Ax;A x;:::;A xg: If the vectors fx;Ax;A2x;:::;An−1xg are linearly independent, then they form a basis for Kn(A; x). However, this basis is usually far from orthogonal, and hence computations using this basis will likely be ill-conditioned. One way to find an orthonormal basis for Kn(A; x) would be to use the modi- fied Gram-Schmidt algorithm from Lab TODO on the set fx;Ax;A2x;:::;An−1xg. More efficiently, the Arnold iteration integrates the creation of fx;Ax;A2x;:::;An−1xg with the modified Gram-Schmidt algorithm, returning an orthonormal basis for Kn(A; x).
    [Show full text]
  • Iterative Methods for Linear System
    Iterative methods for Linear System JASS 2009 Student: Rishi Patil Advisor: Prof. Thomas Huckle Outline ●Basics: Matrices and their properties Eigenvalues, Condition Number ●Iterative Methods Direct and Indirect Methods ●Krylov Subspace Methods Ritz Galerkin: CG Minimum Residual Approach : GMRES/MINRES Petrov-Gaelerkin Method: BiCG, QMR, CGS 1st April JASS 2009 2 Basics ●Linear system of equations Ax = b ●A Hermitian matrix (or self-adjoint matrix ) is a square matrix with complex entries which is equal to its own conjugate transpose, 3 2 + i that is, the element in the ith row and jth column is equal to the A = complex conjugate of the element in the jth row and ith column, for 2 − i 1 all indices i and j • Symmetric if a ij = a ji • Positive definite if, for every nonzero vector x XTAx > 0 • Quadratic form: 1 f( x) === xTAx −−− bTx +++ c 2 ∂ f( x) • Gradient of Quadratic form: ∂x 1 1 1 f' (x) = ⋮ = ATx + Ax − b ∂ 2 2 f( x) ∂xn 1st April JASS 2009 3 Various quadratic forms Positive-definite Negative-definite matrix matrix Singular Indefinite positive-indefinite matrix matrix 1st April JASS 2009 4 Various quadratic forms 1st April JASS 2009 5 Eigenvalues and Eigenvectors For any n×n matrix A, a scalar λ and a nonzero vector v that satisfy the equation Av =λv are said to be the eigenvalue and the eigenvector of A. ●If the matrix is symmetric , then the following properties hold: (a) the eigenvalues of A are real (b) eigenvectors associated with distinct eigenvalues are orthogonal ●The matrix A is positive definite (or positive semidefinite) if and only if all eigenvalues of A are positive (or nonnegative).
    [Show full text]
  • Implicitly Restarted Arnoldi/Lanczos Methods for Large Scale Eigenvalue Calculations
    https://ntrs.nasa.gov/search.jsp?R=19960048075 2020-06-16T03:31:45+00:00Z NASA Contractor Report 198342 /" ICASE Report No. 96-40 J ICA IMPLICITLY RESTARTED ARNOLDI/LANCZOS METHODS FOR LARGE SCALE EIGENVALUE CALCULATIONS Danny C. Sorensen NASA Contract No. NASI-19480 May 1996 Institute for Computer Applications in Science and Engineering NASA Langley Research Center Hampton, VA 23681-0001 Operated by Universities Space Research Association National Aeronautics and Space Administration Langley Research Center Hampton, Virginia 23681-0001 IMPLICITLY RESTARTED ARNOLDI/LANCZOS METHODS FOR LARGE SCALE EIGENVALUE CALCULATIONS Danny C. Sorensen 1 Department of Computational and Applied Mathematics Rice University Houston, TX 77251 sorensen@rice, edu ABSTRACT Eigenvalues and eigenfunctions of linear operators are important to many areas of ap- plied mathematics. The ability to approximate these quantities numerically is becoming increasingly important in a wide variety of applications. This increasing demand has fu- eled interest in the development of new methods and software for the numerical solution of large-scale algebraic eigenvalue problems. In turn, the existence of these new methods and software, along with the dramatically increased computational capabilities now avail- able, has enabled the solution of problems that would not even have been posed five or ten years ago. Until very recently, software for large-scale nonsymmetric problems was virtually non-existent. Fortunately, the situation is improving rapidly. The purpose of this article is to provide an overview of the numerical solution of large- scale algebraic eigenvalue problems. The focus will be on a class of methods called Krylov subspace projection methods. The well-known Lanczos method is the premier member of this class.
    [Show full text]
  • The Influence of Orthogonality on the Arnoldi Method
    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 309 (2000) 307±323 www.elsevier.com/locate/laa The in¯uence of orthogonality on the Arnoldi method T. Braconnier *, P. Langlois, J.C. Rioual IREMIA, Universite de la Reunion, DeÂpartement de Mathematiques et Informatique, BP 7151, 15 Avenue Rene Cassin, 97715 Saint-Denis Messag Cedex 9, La Reunion, France Received 1 February 1999; accepted 22 May 1999 Submitted by J.L. Barlow Abstract Many algorithms for solving eigenproblems need to compute an orthonormal basis. The computation is commonly performed using a QR factorization computed using the classical or the modi®ed Gram±Schmidt algorithm, the Householder algorithm, the Givens algorithm or the Gram±Schmidt algorithm with iterative reorthogonalization. For the eigenproblem, although textbooks warn users about the possible instability of eigensolvers due to loss of orthonormality, few theoretical results exist. In this paper we prove that the loss of orthonormality of the computed basis can aect the reliability of the computed eigenpair when we use the Arnoldi method. We also show that the stopping criterion based on the backward error and the value computed using the Arnoldi method can dier because of the loss of orthonormality of the computed basis of the Krylov subspace. We also give a bound which quanti®es this dierence in terms of the loss of orthonormality. Ó 2000 Elsevier Science Inc. All rights reserved. Keywords: QR factorization; Backward error; Eigenvalues; Arnoldi method 1. Introduction The aim of the most commonly used eigensolvers is to approximate a basis of an invariant subspace associated with some eigenvalues.
    [Show full text]
  • Power Method and Krylov Subspaces
    Power method and Krylov subspaces Introduction When calculating an eigenvalue of a matrix A using the power method, one starts with an initial random vector b and then computes iteratively the sequence Ab, A2b,...,An−1b normalising and storing the result in b on each iteration. The sequence converges to the eigenvector of the largest eigenvalue of A. The set of vectors 2 n−1 Kn = b, Ab, A b,...,A b , (1) where n < rank(A), is called the order-n Krylov matrix, and the subspace spanned by these vectors is called the order-n Krylov subspace. The vectors are not orthogonal but can be made so e.g. by Gram-Schmidt orthogonalisation. For the same reason that An−1b approximates the dominant eigenvector one can expect that the other orthogonalised vectors approximate the eigenvectors of the n largest eigenvalues. Krylov subspaces are the basis of several successful iterative methods in numerical linear algebra, in particular: Arnoldi and Lanczos methods for finding one (or a few) eigenvalues of a matrix; and GMRES (Generalised Minimum RESidual) method for solving systems of linear equations. These methods are particularly suitable for large sparse matrices as they avoid matrix-matrix opera- tions but rather multiply vectors by matrices and work with the resulting vectors and matrices in Krylov subspaces of modest sizes. Arnoldi iteration Arnoldi iteration is an algorithm where the order-n orthogonalised Krylov matrix Qn of a matrix A is built using stabilised Gram-Schmidt process: • start with a set Q = {q1} of one random normalised vector q1 • repeat for k =2 to n : – make a new vector qk = Aqk−1 † – orthogonalise qk to all vectors qi ∈ Q storing qi qk → hi,k−1 – normalise qk storing qk → hk,k−1 – add qk to the set Q By construction the matrix Hn made of the elements hjk is an upper Hessenberg matrix, h1,1 h1,2 h1,3 ··· h1,n h2,1 h2,2 h2,3 ··· h2,n 0 h3,2 h3,3 ··· h3,n Hn = , (2) .
    [Show full text]
  • LARGE-SCALE COMPUTATION of PSEUDOSPECTRA USING ARPACK and EIGS∗ 1. Introduction. the Matrices in Many Eigenvalue Problems
    SIAM J. SCI. COMPUT. c 2001 Society for Industrial and Applied Mathematics Vol. 23, No. 2, pp. 591–605 LARGE-SCALE COMPUTATION OF PSEUDOSPECTRA USING ARPACK AND EIGS∗ THOMAS G. WRIGHT† AND LLOYD N. TREFETHEN† Abstract. ARPACK and its Matlab counterpart, eigs, are software packages that calculate some eigenvalues of a large nonsymmetric matrix by Arnoldi iteration with implicit restarts. We show that at a small additional cost, which diminishes relatively as the matrix dimension increases, good estimates of pseudospectra in addition to eigenvalues can be obtained as a by-product. Thus in large- scale eigenvalue calculations it is feasible to obtain routinely not just eigenvalue approximations, but also information as to whether or not the eigenvalues are likely to be physically significant. Examples are presented for matrices with dimension up to 200,000. Key words. Arnoldi, ARPACK, eigenvalues, implicit restarting, pseudospectra AMS subject classifications. 65F15, 65F30, 65F50 PII. S106482750037322X 1. Introduction. The matrices in many eigenvalue problems are too large to allow direct computation of their full spectra, and two of the iterative tools available for computing a part of the spectrum are ARPACK [10, 11]and its Matlab counter- part, eigs.1 For nonsymmetric matrices, the mathematical basis of these packages is the Arnoldi iteration with implicit restarting [11, 23], which works by compressing the matrix to an “interesting” Hessenberg matrix, one which contains information about the eigenvalues and eigenvectors of interest. For general information on large-scale nonsymmetric matrix eigenvalue iterations, see [2, 21, 29, 31]. For some matrices, nonnormality (nonorthogonality of the eigenvectors) may be physically important [30].
    [Show full text]
  • Comparison of Numerical Methods and Open-Source Libraries for Eigenvalue Analysis of Large-Scale Power Systems
    applied sciences Article Comparison of Numerical Methods and Open-Source Libraries for Eigenvalue Analysis of Large-Scale Power Systems Georgios Tzounas , Ioannis Dassios * , Muyang Liu and Federico Milano School of Electrical and Electronic Engineering, University College Dublin, Belfield, Dublin 4, Ireland; [email protected] (G.T.); [email protected] (M.L.); [email protected] (F.M.) * Correspondence: [email protected] Received: 30 September 2020; Accepted: 24 October 2020; Published: 28 October 2020 Abstract: This paper discusses the numerical solution of the generalized non-Hermitian eigenvalue problem. It provides a comprehensive comparison of existing algorithms, as well as of available free and open-source software tools, which are suitable for the solution of the eigenvalue problems that arise in the stability analysis of electric power systems. The paper focuses, in particular, on methods and software libraries that are able to handle the large-scale, non-symmetric matrices that arise in power system eigenvalue problems. These kinds of eigenvalue problems are particularly difficult for most numerical methods to handle. Thus, a review and fair comparison of existing algorithms and software tools is a valuable contribution for researchers and practitioners that are interested in power system dynamic analysis. The scalability and performance of the algorithms and libraries are duly discussed through case studies based on real-world electrical power networks. These are a model of the All-Island Irish Transmission System with 8640 variables; and, a model of the European Network of Transmission System Operators for Electricity, with 146,164 variables. Keywords: eigenvalue analysis; large non-Hermitian matrices; numerical methods; open-source libraries 1.
    [Show full text]
  • Eigenvalue Problems Last Time …
    Eigenvalue Problems Last Time … Social Network Graphs Betweenness Girvan-Newman Algorithm Graph Laplacian Spectral Bisection 휆2, 푤2 Today … Small deviation into eigenvalue problems … Formulation Standard eigenvalue problem: Given a 푛 × 푛 matrix 퐴, find scalar 휆 and a nonzero vector 푥 such that 퐴푥 = 휆푥 휆 is a eigenvalue, and 푥 is the corresponding eigenvector Spectrum = 휆(퐴) = set of eigenvalues of 퐴 Spectral radius = 퐴 = max 휆 ∶ 휆 ∈ 휆 퐴 Characteristic Polynomial Equation 퐴푥 = 휆푥 is equivalent to 퐴 − 휆퐼 푥 = 0 Eigenvalues of 퐴 are roots of the characteristic polynomial det 퐴 − 휆퐼 = 0 The characteristic polynomial is a powerful theoretical tool but usually not useful computationally Considerations Properties of eigenvalue problem affecting choice of algorithm Are all eigenvalues needed, or only a few? Are only eigenvalues needed, or are corresponding eigenvectors also needed? Is matrix real or complex? Is matrix relatively small and dense, or large and sparse? Does matrix have any special properties, such as symmetry? Problem Transformations Shift: If 퐴푥 = 휆푥 and is any scalar, then 퐴 − 퐼 푥 = 휆 − 푥 1 Inversion: If 퐴 is nonsingular and 퐴푥 = 휆푥, then 휆 ≠ 0 and 퐴−1푥 = 푥 휆 Powers: If 퐴푥 = 휆푥, then 퐴푘푥 = 휆푘푥 Polynomial: If 퐴푥 = 휆푥 and 푝(푡) is a polynomial, then 푝 퐴 푥 = 푝 휆 푥 Similarity Transforms 퐵 is similar to 퐴 if there is a nonsingular matrix 푇, such that 퐵 = 푇−1퐴푇 Then, 퐵푦 = 휆푦 ⇒ 푇−1퐴푇푦 = 휆푦 ⇒ 퐴 푇푦 = 휆 푇푦 Similarity transformations preserve eigenvalues and eigenvectors are easily recovered Diagonal form Eigenvalues
    [Show full text]
  • Lecture 33. the Arnoldi Iteration
    Lecture 33. The Arnoldi Iteration Despite the many names and acronyms that have proliferated in the field of Krylov subspaee rnatrix iterations, these algorithrIls are built upon a eOIIlIIlon foundation of a few fuudarnental ideas. One can take various approaehes to describing this foundation. Ours will be to consider the Arnoldi proeess, a. Gram-Schmidt-style iteration for transforming a matrix to Hessenberg form. The Arnoldi/Gram-Schmidt Analogy Suppose, to pass the time while marooned on a desert island, you challenged yourself t.o devise an algoritlnll to reduce a nOllhermitian Illatrix to Hes.senberg form by orthogonal similarity transformations, proceeding column by column from a prescribed first column q,. To your surprise, you would probably find you could solve this problem in an hour and still have time to gather coconuts for dinner. The method you would come up with goes by the name of the Arnoldi iteration. If A is hermitian, the Hessenberg matrix becomes tridiagonal, an n-tcrm recurrence relation becomes a three-term recurrence relation, and the name changes to the Lanc:zos iteration, to be discussed in Lecture 36. Here is an analogy. For computing the QR factorization A = (JR of a llWtrix A, \ve have discussed t\'w methods in this book: Householder reflec- tions, which triangularhe A by a succession of orthogonal operations, and GraIn-Schmidt orthogonalization, which A by a succession of triangular operations. Though Householder reflections lead to a more nearly 250 LECTURE 33. THE ARNOLDI ITERATION 251 orthogonal matrix Q in the presence of rounding errors, the Gram Schmidt process has the advantage that it can be stopped part-way, leaving one with a reduced QR factorization of the first.
    [Show full text]
  • CALCULATION of PSEUDOSPECTRA by the ARNOLDI ITERATION* KIM-CHUAN Toht and I,LOYD N
    SlAM J. ScI. COMPUT. () 1996 Society for Industrial and Applied Mathematics Vol. 17, No. 1, pp. 1-15, January 1996 001 CALCULATION OF PSEUDOSPECTRA BY THE ARNOLDI ITERATION* KIM-CHUAN TOHt AND I,LOYD N. TREFETHEN Abstract. The Amoldi iteration, usually viewed as a method for calculating eigenvalues, can also be used to estimate pseudospectra. This possibility may be of practical importance, because in applications involving highly nonnormal matrices or operators, such as hydrodynamic stability, pseudospectra may be physically more significant than spectra. Key words. Amoldi, Lanczos, pseudospectra, numerical range, hydrodynamic stability AMS subject classifications. 65F15, 47A12 1. Introduction. Large-scale nonsymmetric matrix eigenvalue problems, which typi- cally arise via discretization of non-self-adjoint differential or integral operators, are com- monly solved numerically by the Arnoldi iteration and its variants [1], [8], [18], [25]. In this paper we explore the possibility that the Arnoldi iteration can also be used for the esti- mation of pseudospectra. Such an idea was first proposed by Nachtigal, Reichel, and Tre- fethen [17] and Freund [7], and methods much closer to those of the present paper have been presented by Ruhe in talks at the Householder (1993) and St. Girons (1994) symposia on linear algebra [23]. Recent developments indicate that in some applications, the pseu- dospectra of a matrix or operator may be more significant physically than its spectrum (see 7). Since calculation of pseudospectra is much more expensive than calculation of spec- tra, this suggests that it may be desirable to develop methods for determining them itera- tively. The idea investigated here is that the pseudospectra of a matrix A can be approximated by those of the Hessenberg matrices constructed by an Arnoldi iteration.
    [Show full text]