DOCSLIB.ORG

Course Notes: Week 1 Math 270C: Applied Numerical Linear Algebra

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Course Notes: Week 1 Math 270C: Applied Numerical Linear Algebra Course Notes: Week 1 Math 270C: Applied Numerical Linear Algebra 1 Lecture 1: Introduction We will focus on iterative methods for solving linear systems of equations (and some discussion of eigenvalues problems): n×n Ax = b; Ax = λx A 2 R where n is very large (e.g. n = 109 on your (powerful) desktop PC) and typically A is sparse. In other words, we want to obtain x from A and b (and λ for eigenvalue problems) but not necessarily A−1. We will also be focusing a great deal on the types of matrices that arise in numerical methods for partial differential equations (e.g. finite difference, finite volume methods etc.). However, I am assuming no familiarity with such techniques. Mainly they will provide examples for testing the algorithms we explore, but those algorithms will of course be applicable for all matrices (or at least a broad class such as symmetric matrices or symmetric positive definite matrices depending on the method we focus on). Iterative methods are potentially much faster than direct methods (e.g. Gaussian elimination, QR factorization etc., see Math 270B for this class of methods). Notably, when the matrix is sparse we may be able to solve in linear time: O(n) (and quadratic in general: O(n2)). This is a very appealing possibility considering the direct methods are always O(n2) and O(n3) for sparse and non-sparse. Furthermore, the direct methods typically incur O(n2) storage in both the sparse and non-sparse cases whereas iterative methods will typically only require storage of A (i.e. O(n) stor- age for sparse and O(n2) for dense). The analysis of iterative methods is in general much more involved. Also it can require great care to achieve linear performance in practice. Generally, the iterative method must be designed with some knowledge of the matrices it is designed for. This aspect is less appealing than the direct methods, but the potential for linear performance more than justifies the effort. Sparsity means that each row in the matrix A has at most a c non-zero entries where c << n. E.g. c = 3 for the matrix: 0 2 −1 0 0 ::: 0 1 B −1 2 −1 0 ::: 0 C B C B 0 −1 2 −1 ::: 0 C A = B C B .. .. .. C @ . A 0 ::: 0 −1 2 Basically, the matrix is mostly zero so we can evaluate matrix vector multiplies in linear time (since n each row can be dotted with an R vector in (2c − 1)n flops. The idea is that with (functional) iterative methods we should be able to get a very good approximation to x = A−1b by some low number of matrix vector multiplies being the dominant cost in the algorithm. This may sound too good to be true (and sometimes it is). However, this is the promise of iterative methods for sparse matrices and it is very often possible to achieve this performance. 1.1 Geometric Convergence Trefethen and Bau refer to \geometric convergence" as the optimal case for an iterative method (but exponential convergence is also often possible). In these scenarios, a constant number of iterations 1 is required in practice to reduce the residual to the system by a given factor (independent of the size of the problem n). In practice, my criterion for optimal convergence is that the residual is reduced by a constant factor each iteration. Let xk be the kth iterate in our method. Therefore, the residual at this step is rk = b − Axk. In general, the residual will be our only means of estimating th k k+1 k+2 the accuracy of the k iterate. Also, in general we hope that jjr jj2 < jjr jj2 < jjr jj2:::. When I am estimating the efficacy of a given iterative method I usually use the following formula: k+1 k jjr jj2 = ρjjr jj2 and estimate ρ asymptotically. I.e. as k ! 1 I estimate ρ (numerically). In this case our asymptotic k γk γ k performance is jjr jj2 = Ce with ρ = e and γ < 0. In practice you can plot log(jjr jj2) vs. k and estimate the slope of the line. 1.2 Examples The two most simplistic iterative methods are probably Jacobi and Gauss-Seidel. These methods are very simple to describe, but typically do not yield optimal performance. I.e. they usually require many many iterations and the the asymptotic ρ discussed above is 1. I.e. the method nearly stagnates as you run it longer and longer (typically without bringing you close enough to the solution). These methods are still very useful, just not typically on their own. For example, both of these techniques can be shown to \smooth" the error very efficiently for many matrices. That is, they reduce the error of the high frequency modes very quickly but reduce the error of the low fre- quency modes very slowly. This fact is leveraged in multigrid methods for discrete elliptic equations. Not that in both examples below, all operations can be made much more efficient in the case of sparse matrices. Matrix vector multiplies and residual updates can all be done with sparsity in mind to make each iteration of both methods linear in computational cost (linear in the size of the problem n). 1.2.1 Jacobi Jacobi iteration is a very simple algorithm for approximating x = A−1b. In practice, I would describe Jacobi iteration as: for k = 1 to max it do // At this point, x is xk r = b − Ax for i = 1 to n do δ ri Aii xi xi + δ //Note that each xi update can be done independently end for // At this point, x is now equal to xk+1 end for This can also be described as: MJ xk+1 = NJ xk + b 2 where MJ is the diagonal part of A (D) and NJ = − (L + U) and L and U are the lower and upper triangular parts of the matrix A. E.g. for 0 2 −1 0 0 ::: 0 1 B −1 2 −1 0 ::: 0 C B C B 0 −1 2 −1 ::: 0 C A = B C B .. .. .. C @ . A 0 ::: 0 −1 2 0 2 0 0 0 ::: 0 1 0 0 0 0 0 ::: 0 1 0 0 −1 0 0 ::: 0 1 B 0 2 0 0 ::: 0 C B −1 0 0 0 ::: 0 C B 0 0 −1 0 ::: 0 C B C B C B C J B 0 0 2 0 ::: 0 C B 0 −1 0 0 ::: 0 C B 0 0 0 −1 ::: 0 C M = B C ; L = B C ; U = B C : B .. .. .. C B .. .. .. C B .. .. .. C @ . A @ . A @ . A 0 ::: 0 0 2 0 ::: 0 −1 0 0 ::: 0 0 0 1.2.2 Gauss-Seidel Gauss-Seidel usually performs better than Jacobi and is a little more expensive than Jacobi. Also, the order in which you apply a correction matters for Gauss-Seidel and individual updates are not independent. Jacobi is usually considered more parallel friendly than Gauss-Seidel for this reason. The Gauss-Seidel iteration can be described as follows: for k = 1 to max it do // At this point, x is xk for i = 1 to n do δ ri Aii xi xi + δ r = b − Ax //Update the residual to reflect our change to xi end for // At this point, x is now equal to xk+1 end for This can also be described as: MGxk+1 = NGxk + b where MG = D + L and NG = −U. 2 Lecture 2: Krylov Subspaces and Arnoldi Iteration Krylov subspace methods are very powerful techniques that often achieve near optimal performance. That is, these often have the potential of requiring just a few iterations independent of the size of the problem n. Furthermore, they may be optimized for different classes of matrices. E.g. if the matrix is symmetric positive definite, the Kryolv method is conjugate gradients (CG). If is just symmetric the method is MINRES. In general it is GMRES. CG is the most efficient in terms of memory and computation, MINRES second most so and GMRES least so. We will initially discuss the general case before we cover the specializations. These methods are based on the Krylov subspaces Kk = spanfb; Ab; A2b; :::; Ak−1bg. These 3 spaces have the property Kk ⊂ Kk^ 8k^ > k and 9k with A−1b 2 Kk. The first property is trivial, the second property follows because (assuming A is full rank) we can write: k−1 −1 X i A = ciA : i=0 To see this, we can construct a polynomial for which A is a root. For example, consider the case of a diagonalizeable A. Let A = QΛQ−1 where Λ is a diagonal matrix with eigenvalues λ1; λ2; :::; λn. Let λ1; λ2; :::; λk be all of the eigenvalues in Λ except for any repeats (i.e. just the distinct eigenvalues). Then A is a root of the following kth degree polynomial: k−1 X i k p(λ) = (λ − λ1)(λ − λ2) ::: (λ − λk) = biλ + λ i=0 where k is then the number of distinct eigenvalues. Since A is a root of this polynomial k−1 ! 1 X A−1 = − b Ai−1 + Ak−1 : b i 0 i=1 Here d0 is the product of the distinct eigenvalues and thus will not be zero in the full rank case. Therefore, when A is full rank, the solution x = A−1b will be in the kth Krylov subspace Kk where k is the number of distinct eigenvalues.
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]
  • Accelerating the LOBPCG Method on Gpus Using a Blocked Sparse Matrix Vector Product
    Accelerating the LOBPCG method on GPUs using a blocked Sparse Matrix Vector Product Hartwig Anzt and Stanimire Tomov and Jack Dongarra Innovative Computing Lab University of Tennessee Knoxville, USA Email: [email protected], [email protected], [email protected] Abstract— the computing power of today’s supercomputers, often accel- erated by coprocessors like graphics processing units (GPUs), This paper presents a heterogeneous CPU-GPU algorithm design and optimized implementation for an entire sparse iter- becomes challenging. ative eigensolver – the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) – starting from low-level GPU While there exist numerous efforts to adapt iterative lin- data structures and kernels to the higher-level algorithmic choices ear solvers like Krylov subspace methods to coprocessor and overall heterogeneous design. Most notably, the eigensolver technology, sparse eigensolvers have so far remained out- leverages the high-performance of a new GPU kernel developed side the main focus. A possible explanation is that many for the simultaneous multiplication of a sparse matrix and a of those combine sparse and dense linear algebra routines, set of vectors (SpMM). This is a building block that serves which makes porting them to accelerators more difficult. Aside as a backbone for not only block-Krylov, but also for other from the power method, algorithms based on the Krylov methods relying on blocking for acceleration in general. The subspace idea are among the most commonly used general heterogeneous LOBPCG developed here reveals the potential of eigensolvers [1]. When targeting symmetric positive definite this type of eigensolver by highly optimizing all of its components, eigenvalue problems, the recently developed Locally Optimal and can be viewed as a benchmark for other SpMM-dependent applications.
    [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]
  • Limited Memory Block Krylov Subspace Optimization for Computing Dominant Singular Value Decompositions
    Limited Memory Block Krylov Subspace Optimization for Computing Dominant Singular Value Decompositions Xin Liu∗ Zaiwen Weny Yin Zhangz March 22, 2012 Abstract In many data-intensive applications, the use of principal component analysis (PCA) and other related techniques is ubiquitous for dimension reduction, data mining or other transformational purposes. Such transformations often require efficiently, reliably and accurately computing dominant singular value de- compositions (SVDs) of large unstructured matrices. In this paper, we propose and study a subspace optimization technique to significantly accelerate the classic simultaneous iteration method. We analyze the convergence of the proposed algorithm, and numerically compare it with several state-of-the-art SVD solvers under the MATLAB environment. Extensive computational results show that on a wide range of large unstructured matrices, the proposed algorithm can often provide improved efficiency or robustness over existing algorithms. Keywords. subspace optimization, dominant singular value decomposition, Krylov subspace, eigen- value decomposition 1 Introduction Singular value decomposition (SVD) is a fundamental and enormously useful tool in matrix computations, such as determining the pseudo-inverse, the range or null space, or the rank of a matrix, solving regular or total least squares data fitting problems, or computing low-rank approximations to a matrix, just to mention a few. The need for computing SVDs also frequently arises from diverse fields in statistics, signal processing, data mining or compression, and from various dimension-reduction models of large-scale dynamic systems. Usually, instead of acquiring all the singular values and vectors of a matrix, it suffices to compute a set of dominant (i.e., the largest) singular values and their corresponding singular vectors in order to obtain the most valuable and relevant information about the underlying dataset or system.
    [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]
  • Section 5.3: Other Krylov Subspace Methods
    Section 5.3: Other Krylov Subspace Methods Jim Lambers March 15, 2021 • So far, we have only seen Krylov subspace methods for symmetric positive definite matrices. • We now generalize these methods to arbitrary square invertible matrices. Minimum Residual Methods • In the derivation of the Conjugate Gradient Method, we made use of the Krylov subspace 2 k−1 K(r1; A; k) = spanfr1;Ar1;A r1;:::;A r1g to obtain a solution of Ax = b, where A was a symmetric positive definite matrix and (0) (0) r1 = b − Ax was the residual associated with the initial guess x . • Specifically, the Conjugate Gradient Method generated a sequence of approximate solutions x(k), k = 1; 2;:::; where each iterate had the form k (k) (0) (0) X x = x + Qkyk = x + qjyjk; k = 1; 2;:::; (1) j=1 and the columns q1; q2;:::; qk of Qk formed an orthonormal basis of the Krylov subspace K(b; A; k). • In this section, we examine whether a similar approach can be used for solving Ax = b even when the matrix A is not symmetric positive definite. (k) • That is, we seek a solution x of the form of equation (??), where again the columns of Qk form an orthonormal basis of the Krylov subspace K(b; A; k). • To generate this basis, we cannot simply generate a sequence of orthogonal polynomials using a three-term recurrence relation, as before, because the bilinear form T hf; gi = r1 f(A)g(A)r1 does not satisfy the requirements for a valid inner product when A is not symmetric positive definite.
    [Show 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.
    [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]