AMS526: Numerical Analysis I (Numerical Linear Algebra)

Size: px
Start display at page:

Download "AMS526: Numerical Analysis I (Numerical Linear Algebra)"

Transcription

1 AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 19: More on Arnoldi Iteration; Lanczos Iteration Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis I 1 / 17

2 Outline 1 More on Arnoldi Iteration 2 Lanczos Iterations Xiangmin Jiao Numerical Analysis I 2 / 17

3 Review: Arnoldi Iteration Arnoldi iteration reduces a general, nonsymmetric matrix A to Hessenberg form by similarity transformation A = QHQ Start with a random q 1. Let Q k = [q 1 q 2 q k ] be m k matrix with first k columns of Q and H k be (k + 1) k upper-left section of H, i.e., H k = H 1:k+1,1:k kth columns of AQ k = Q k+1 H k can be written as where h ik = q i Aq k. Aq k = h 1k q h kk q k + h k+1,k q k+1 Xiangmin Jiao Numerical Analysis I 3 / 17

4 Review: Arnoldi Algorithm Algorithm: Arnoldi Iteration given random nonzero b, let q 1 = b/ b for k =1, 2, 3,... v = Aq k for j = 1 to k h jk = q j v v = v h jk q j h k+1,k = v q k+1 = v/h k+1,k Xiangmin Jiao Numerical Analysis I 4 / 17

5 QR Factorization of Krylov Matrix Vectors q j from Arnoldi iteration are orthonormal bases of successive Krylov subspaces K k = b, Ab,..., A k 1 b = q 1, q 2,..., q k C m assuming h k+1,k 0 Q k is reduced QR factorization K k = Q k R k of Krylov matrix K k = b Ab A k 1 b However, K k and R k are not formed explicitly; forming them explicitly would be unstable and can suffer from overflow and underflow Xiangmin Jiao Numerical Analysis I 5 / 17

6 Comparison Against QR Algorithm QR factorization of Krylov subspace gives intuitive explanation of Arnoldi iteration This is analogous to using simultaneous iteration to explain QR algorithm for computing eigenvalues quasi-direct iterative intuitive but unstable simultaneous iteration QR of Krylov subspace subtle but stable QR algorithm Arnoldi Xiangmin Jiao Numerical Analysis I 6 / 17

7 Projection onto Krylov Subspaces Arnoldi process computes projections of A onto successive Krylov subspaces H k = Q k AQ k because AQ k = Q k+1 Hk, H k = Q k+1 AQ k, and H k = H 1:k,1:k H k can be interpreted as orthogonal projection of A onto K k in the basis {q 1, q 2,..., q k }, restricting mapping A : C m C m to H k : K k K k. This kind of projection is known as Rayleigh-Ritz procedure Arnoldi iteration is useful as 1 basis for iterative algorithms (such as GMRES, to be discussed later) 2 technique for estimating eigenvalues of nonhermitian matrices Caution: eigenvalues of nonmormal matrices may have little to do with physical system, since eigenvalues of such equations are ill-conditioned. When such problems arise, the original problem is mostly likely posed improperly Xiangmin Jiao Numerical Analysis I 7 / 17

8 Estimating Eigenvalues by Arnoldi Iteration Diagonal entries of H k are Rayleigh quotients of A w.r.t. vectors q i H k is generalized Rayleigh quotient w.r.t Q k, whose eigenvalues {θ j } are called Arnoldi estimates or Ritz values w.r.t. K k of A Ritz vectors corresponds to θ j are Q k y j, where H k y j = θ j y j To use Arnoldi iteration to estimate eigenvalues, compute eigenvalues of H k at kth step When k = m, Ritz values are eigenvalues In general, k m, so we can only estimate only a few eigenvalues Which eigenvalues? Typically, it finds extreme eigenvalues first In many applications, extreme eigenvalues are of main interests Stability analysis typically requires estimating spectral radius Principal component analysis requires estimating largest eigenvalues and corresponding eigenvectors of A T A Xiangmin Jiao Numerical Analysis I 8 / 17

9 Invariance Properties of Arnoldi Iteration Theorem Let the Arnoldi iteration be applied to matrix A C m m as described above. Translation invariance. If A is changed to A + σi for some σ C, and b is unchanged, then Ritz values {θ j } change to {θ j + σ}. Scale invariance. If A is changed to σa for some σ C, and b is unchanged, then {θ j } change to {σθ j }. Invariance under unitary similarity transformation. If A is changed to UAU for some unitary matrix U, and b is changed to Ub, then {θ j } do not change. In all three cases, the Ritz vectors, namely Q k y k corresponding to eigenvectors y j of H k do not change under indicated transformation. Xiangmin Jiao Numerical Analysis I 9 / 17

10 Convergence of Arnoldi Iteration If A has n distinct eigenvalues, Arnoldi iteration finds them all in n steps Under certain circumstances, convergence of some Arnoldi estimates is geometric (i.e., linear), and it accelerates in later iterations However, these matters are not yet fully understood Example convergence of extreme Arnoldi eigenvalue estimation. Xiangmin Jiao Numerical Analysis I 10 / 17

11 Outline 1 More on Arnoldi Iteration 2 Lanczos Iterations Xiangmin Jiao Numerical Analysis I 11 / 17

12 Lanczos Iteration for Symmetric Matrices For symmetric A, H k and H k in Arnoldi iteration are tridiagonal We denote them by T k and T k, respectively. Let α k = h kk and β k = h k+1,k = h k,k+1 AQ k = Q k+1 H k can then be written as three-term recurrence Aq k = β k 1 q k 1 + α k q k + β k q k+1 where α i are diagonal entries and β i are sub-diagonal entries of T k α 1 β 1 β 1 α 2 β 2 T n =. β 2 α βn 1 β n 1 Arnoldi iteration for symmetric matrices is known as Lanczos iteration α n Xiangmin Jiao Numerical Analysis I 12 / 17

13 Algorithm of Lanczos Iteration Algorithm: Lanczos Iteration β 0 = 0, q 0 = 0 given random b, let q 1 = b/ b for k = 1, 2, 3,... v = Aq k α k = q k v v = v β k 1 q k 1 α k q k β k = v q k+1 = v/β k Each step consists of matrix-vector multiplication, an inner product, and a couple of vector operations This is particularly efficient for sparse matrices. In practice, Lanczos iteration is used to compute eigenvalues of large symmetric matrices Like Arnoldi iteration, Lanczos iteration is useful as 1 basis for other iterative algorithms (such as conjugate gradient) 2 technique for estimating eigenvalues of Hermitian matrices Xiangmin Jiao Numerical Analysis I 13 / 17

14 Estimating Eigenvalues by Lanczos Iterations For symmetric matrices with evenly spaced eigenvalues, Ritz values tend to first convert to extreme eigenvalue Ritz values for first 20 steps for Lanczos iteration applied to example matrix. Convergence of extreme eigenvalues is geometric. Xiangmin Jiao Numerical Analysis I 14 / 17

15 Effect of Rounding Errors Rounding errors have complex effects on Lanczos iteration and all iterations based on three-term recurrence Rounding errors cause loss of orthogonality of q 1, q 2,..., q k In Arnoldi iteration, vectors q 1, q 2,..., q k are enforced to be orthogonal by explicit modified Gram-Schmidt orthogonalization, which suffer some but not as much loss of orthogonality In Lanczos iteration, orthogonality of qj, q j 1 and q j 2 are enforced, but orthogonality of q j with q j 3,..., q 1 are automatic, based on mathematical identities In practice, such mathematical identities are not accurately preserved in the presence of rounding errors In practice, periodic re-orthogonalization of Q k is sometimes used to alleviate effect of rounding errors Xiangmin Jiao Numerical Analysis I 15 / 17

16 Rounding Errors and Ghost Eigenvalues With rounding errors, Lanczos iteration can suffer from loss of orthogonality and can in turn lead to spurious ghost eigenvalues Continuation to 120 steps of Lanczos iteration. Numbers indicate multiplicities of Ritz values. 4 ghost copies of 3.0 and 2 ghost copies of 2.5 appear. Xiangmin Jiao Numerical Analysis I 16 / 17

17 Explanation of Ghost Eigenvalues Intuitive explanation of ghost eigenvalues Convergence of Ritz value annihilates corresponding eigenvector components in the vector being operated upon With rounding errors, random noise re-introduce and excite those components again We cannot trust multiplicities of Ritz values as those of eigenvalues Nevertheless, Lanczos iteration can still be very useful in practice E.g., in PCA for dimension reduction in data analysis, one needs to find leading singular values and corresponding singular vectors of A. One standard approach is to apply Lanczos iteration to A T A or AA T without forming the product explicitly, and then use Ritz vectors to approximate singular vectors Xiangmin Jiao Numerical Analysis I 17 / 17

4.8 Arnoldi Iteration, Krylov Subspaces and GMRES

4.8 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 information

AMS526: Numerical Analysis I (Numerical Linear Algebra)

AMS526: Numerical Analysis I (Numerical Linear Algebra) AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 23: GMRES and Other Krylov Subspace Methods Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 9 Minimizing Residual CG

More information

AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 23: GMRES and Other Krylov Subspace Methods; Preconditioning

AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 23: GMRES and Other Krylov Subspace Methods; Preconditioning AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 23: GMRES and Other Krylov Subspace Methods; Preconditioning Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 18 Outline

More information

Last Time. Social Network Graphs Betweenness. Graph Laplacian. Girvan-Newman Algorithm. Spectral Bisection

Last 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 information

Eigenvalue 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 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 information

Scientific Computing: An Introductory Survey

Scientific 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 information

AMS526: Numerical Analysis I (Numerical Linear Algebra)

AMS526: 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 information

LARGE SPARSE EIGENVALUE PROBLEMS. General Tools for Solving Large Eigen-Problems

LARGE 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 information

LARGE SPARSE EIGENVALUE PROBLEMS

LARGE SPARSE EIGENVALUE PROBLEMS LARGE SPARSE EIGENVALUE PROBLEMS Projection methods The subspace iteration Krylov subspace methods: Arnoldi and Lanczos Golub-Kahan-Lanczos bidiagonalization 14-1 General Tools for Solving Large Eigen-Problems

More information

AMS526: Numerical Analysis I (Numerical Linear Algebra)

AMS526: Numerical Analysis I (Numerical Linear Algebra) AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 1: Course Overview & Matrix-Vector Multiplication Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 20 Outline 1 Course

More information

Applied Mathematics 205. Unit V: Eigenvalue Problems. Lecturer: Dr. David Knezevic

Applied 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 information

Krylov Space Methods. Nonstationary sounds good. Radu Trîmbiţaş ( Babeş-Bolyai University) Krylov Space Methods 1 / 17

Krylov Space Methods. Nonstationary sounds good. Radu Trîmbiţaş ( Babeş-Bolyai University) Krylov Space Methods 1 / 17 Krylov Space Methods Nonstationary sounds good Radu Trîmbiţaş Babeş-Bolyai University Radu Trîmbiţaş ( Babeş-Bolyai University) Krylov Space Methods 1 / 17 Introduction These methods are used both to solve

More information

Computation of eigenvalues and singular values Recall that your solutions to these questions will not be collected or evaluated.

Computation 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 information

Numerical Methods in Matrix Computations

Numerical Methods in Matrix Computations Ake Bjorck Numerical Methods in Matrix Computations Springer Contents 1 Direct Methods for Linear Systems 1 1.1 Elements of Matrix Theory 1 1.1.1 Matrix Algebra 2 1.1.2 Vector Spaces 6 1.1.3 Submatrices

More information

AMS526: Numerical Analysis I (Numerical Linear Algebra)

AMS526: Numerical Analysis I (Numerical Linear Algebra) AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 7: More on Householder Reflectors; Least Squares Problems Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 15 Outline

More information

AMS526: Numerical Analysis I (Numerical Linear Algebra)

AMS526: Numerical Analysis I (Numerical Linear Algebra) AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 5: Projectors and QR Factorization Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 14 Outline 1 Projectors 2 QR Factorization

More information

Index. for generalized eigenvalue problem, butterfly form, 211

Index. 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 information

6.4 Krylov Subspaces and Conjugate Gradients

6.4 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 information

Bindel, Fall 2016 Matrix Computations (CS 6210) Notes for

Bindel, Fall 2016 Matrix Computations (CS 6210) Notes for 1 Arnoldi Notes for 2016-11-16 Krylov subspaces are good spaces for approximation schemes. But the power basis (i.e. the basis A j b for j = 0,..., k 1) is not good for numerical work. The vectors in the

More information

AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences)

AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences) AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences) Lecture 19: Computing the SVD; Sparse Linear Systems Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical

More information

AMS526: Numerical Analysis I (Numerical Linear Algebra)

AMS526: Numerical Analysis I (Numerical Linear Algebra) AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 16: Rayleigh Quotient Iteration Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 10 Solving Eigenvalue Problems All

More information

Simple iteration procedure

Simple iteration procedure Simple iteration procedure Solve Known approximate solution Preconditionning: Jacobi Gauss-Seidel Lower triangle residue use of pre-conditionner correction residue use of pre-conditionner Convergence Spectral

More information

Krylov Subspace Methods that Are Based on the Minimization of the Residual

Krylov Subspace Methods that Are Based on the Minimization of the Residual Chapter 5 Krylov Subspace Methods that Are Based on the Minimization of the Residual Remark 51 Goal he goal of these methods consists in determining x k x 0 +K k r 0,A such that the corresponding Euclidean

More information

Krylov Subspaces. Lab 1. The Arnoldi Iteration

Krylov 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 information

On prescribing Ritz values and GMRES residual norms generated by Arnoldi processes

On prescribing Ritz values and GMRES residual norms generated by Arnoldi processes On prescribing Ritz values and GMRES residual norms generated by Arnoldi processes Jurjen Duintjer Tebbens Institute of Computer Science Academy of Sciences of the Czech Republic joint work with Gérard

More information

The Lanczos and conjugate gradient algorithms

The Lanczos and conjugate gradient algorithms The Lanczos and conjugate gradient algorithms Gérard MEURANT October, 2008 1 The Lanczos algorithm 2 The Lanczos algorithm in finite precision 3 The nonsymmetric Lanczos algorithm 4 The Golub Kahan bidiagonalization

More information

Computational Methods. Eigenvalues and Singular Values

Computational Methods. Eigenvalues and Singular Values Computational Methods Eigenvalues and Singular Values Manfred Huber 2010 1 Eigenvalues and Singular Values Eigenvalues and singular values describe important aspects of transformations and of data relations

More information

AMS526: Numerical Analysis I (Numerical Linear Algebra)

AMS526: Numerical Analysis I (Numerical Linear Algebra) AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 21: Sensitivity of Eigenvalues and Eigenvectors; Conjugate Gradient Method Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis

More information

AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences)

AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences) AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences) Lecture 1: Course Overview; Matrix Multiplication Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical

More information

AMS526: Numerical Analysis I (Numerical Linear Algebra)

AMS526: Numerical Analysis I (Numerical Linear Algebra) AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 2: Orthogonal Vectors and Matrices; Vector Norms Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 11 Outline 1 Orthogonal

More information

Course Notes: Week 1

Course Notes: Week 1 Course Notes: Week 1 Math 270C: Applied Numerical Linear Algebra 1 Lecture 1: Introduction (3/28/11) We will focus on iterative methods for solving linear systems of equations (and some discussion of eigenvalues

More information

Eigenvalues and eigenvectors

Eigenvalues 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 information

Math 504 (Fall 2011) 1. (*) Consider the matrices

Math 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 information

AMS526: Numerical Analysis I (Numerical Linear Algebra)

AMS526: Numerical Analysis I (Numerical Linear Algebra) AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 16: Eigenvalue Problems; Similarity Transformations Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis I 1 / 18 Eigenvalue

More information

Numerical Methods - Numerical Linear Algebra

Numerical 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 information

Iterative methods for symmetric eigenvalue problems

Iterative methods for symmetric eigenvalue problems s Iterative s for symmetric eigenvalue problems, PhD McMaster University School of Computational Engineering and Science February 11, 2008 s 1 The power and its variants Inverse power Rayleigh quotient

More information

FEM and sparse linear system solving

FEM and sparse linear system solving FEM & sparse linear system solving, Lecture 9, Nov 19, 2017 1/36 Lecture 9, Nov 17, 2017: Krylov space methods http://people.inf.ethz.ch/arbenz/fem17 Peter Arbenz Computer Science Department, ETH Zürich

More information

Iterative methods for Linear System

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

More information

ON ORTHOGONAL REDUCTION TO HESSENBERG FORM WITH SMALL BANDWIDTH

ON ORTHOGONAL REDUCTION TO HESSENBERG FORM WITH SMALL BANDWIDTH ON ORTHOGONAL REDUCTION TO HESSENBERG FORM WITH SMALL BANDWIDTH V. FABER, J. LIESEN, AND P. TICHÝ Abstract. Numerous algorithms in numerical linear algebra are based on the reduction of a given matrix

More information

Linear Algebra: Matrix Eigenvalue Problems

Linear Algebra: Matrix Eigenvalue Problems CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given

More information

Matrix Algorithms. Volume II: Eigensystems. G. W. Stewart H1HJ1L. University of Maryland College Park, Maryland

Matrix 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 information

Orthogonal iteration to QR

Orthogonal 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 information

Krylov subspace projection methods

Krylov subspace projection methods I.1.(a) Krylov subspace projection methods Orthogonal projection technique : framework Let A be an n n complex matrix and K be an m-dimensional subspace of C n. An orthogonal projection technique seeks

More information

APPLIED NUMERICAL LINEAR ALGEBRA

APPLIED 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 information

Krylov Subspaces. The order-n Krylov subspace of A generated by x is

Krylov Subspaces. The order-n Krylov subspace of A generated by x is Lab 1 Krylov Subspaces Lab Objective: matrices. Use Krylov subspaces to find eigenvalues of extremely large One of the biggest difficulties in computational linear algebra is the amount of memory needed

More information

Algorithms that use the Arnoldi Basis

Algorithms that use the Arnoldi Basis AMSC 600 /CMSC 760 Advanced Linear Numerical Analysis Fall 2007 Arnoldi Methods Dianne P. O Leary c 2006, 2007 Algorithms that use the Arnoldi Basis Reference: Chapter 6 of Saad The Arnoldi Basis How to

More information

Introduction to Arnoldi method

Introduction to Arnoldi method Introduction to Arnoldi method SF2524 - Matrix Computations for Large-scale Systems KTH Royal Institute of Technology (Elias Jarlebring) 2014-11-07 KTH Royal Institute of Technology (Elias Jarlebring)Introduction

More information

MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors.

MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. Orthogonal sets Let V be a vector space with an inner product. Definition. Nonzero vectors v 1,v

More information

1 Singular Value Decomposition and Principal Component

1 Singular Value Decomposition and Principal Component Singular Value Decomposition and Principal Component Analysis In these lectures we discuss the SVD and the PCA, two of the most widely used tools in machine learning. Principal Component Analysis (PCA)

More information

Preconditioned inverse iteration and shift-invert Arnoldi method

Preconditioned inverse iteration and shift-invert Arnoldi method Preconditioned inverse iteration and shift-invert Arnoldi method Melina Freitag Department of Mathematical Sciences University of Bath CSC Seminar Max-Planck-Institute for Dynamics of Complex Technical

More information

Eigenvalues and Eigenvectors

Eigenvalues 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 information

Approximating the matrix exponential of an advection-diffusion operator using the incomplete orthogonalization method

Approximating the matrix exponential of an advection-diffusion operator using the incomplete orthogonalization method Approximating the matrix exponential of an advection-diffusion operator using the incomplete orthogonalization method Antti Koskela KTH Royal Institute of Technology, Lindstedtvägen 25, 10044 Stockholm,

More information

Large-scale eigenvalue problems

Large-scale eigenvalue problems ELE 538B: Mathematics of High-Dimensional Data Large-scale eigenvalue problems Yuxin Chen Princeton University, Fall 208 Outline Power method Lanczos algorithm Eigenvalue problems 4-2 Eigendecomposition

More information

Math 411 Preliminaries

Math 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 information

SOLVING SPARSE LINEAR SYSTEMS OF EQUATIONS. Chao Yang Computational Research Division Lawrence Berkeley National Laboratory Berkeley, CA, USA

SOLVING SPARSE LINEAR SYSTEMS OF EQUATIONS. Chao Yang Computational Research Division Lawrence Berkeley National Laboratory Berkeley, CA, USA 1 SOLVING SPARSE LINEAR SYSTEMS OF EQUATIONS Chao Yang Computational Research Division Lawrence Berkeley National Laboratory Berkeley, CA, USA 2 OUTLINE Sparse matrix storage format Basic factorization

More information

Bindel, Spring 2016 Numerical Analysis (CS 4220) Notes for

Bindel, 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 information

Numerical Methods I Eigenvalue Problems

Numerical 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 information

AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences)

AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences) AMS526: Numerical Analysis (Numerical Linear Algebra for Computational and Data Sciences) Lecture 14: Eigenvalue Problems; Eigenvalue Revealing Factorizations Xiangmin Jiao Stony Brook University Xiangmin

More information

Numerical Methods for Solving Large Scale Eigenvalue Problems

Numerical Methods for Solving Large Scale Eigenvalue Problems Peter Arbenz Computer Science Department, ETH Zürich E-mail: arbenz@inf.ethz.ch arge scale eigenvalue problems, Lecture 2, February 28, 2018 1/46 Numerical Methods for Solving Large Scale Eigenvalue Problems

More information

Charles University Faculty of Mathematics and Physics DOCTORAL THESIS. Krylov subspace approximations in linear algebraic problems

Charles University Faculty of Mathematics and Physics DOCTORAL THESIS. Krylov subspace approximations in linear algebraic problems Charles University Faculty of Mathematics and Physics DOCTORAL THESIS Iveta Hnětynková Krylov subspace approximations in linear algebraic problems Department of Numerical Mathematics Supervisor: Doc. RNDr.

More information

Solving large scale eigenvalue problems

Solving 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 information

On the influence of eigenvalues on Bi-CG residual norms

On the influence of eigenvalues on Bi-CG residual norms On the influence of eigenvalues on Bi-CG residual norms Jurjen Duintjer Tebbens Institute of Computer Science Academy of Sciences of the Czech Republic duintjertebbens@cs.cas.cz Gérard Meurant 30, rue

More information

Summary of Iterative Methods for Non-symmetric Linear Equations That Are Related to the Conjugate Gradient (CG) Method

Summary of Iterative Methods for Non-symmetric Linear Equations That Are Related to the Conjugate Gradient (CG) Method Summary of Iterative Methods for Non-symmetric Linear Equations That Are Related to the Conjugate Gradient (CG) Method Leslie Foster 11-5-2012 We will discuss the FOM (full orthogonalization method), CG,

More information

Linear Algebra Massoud Malek

Linear Algebra Massoud Malek CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product

More information

1 Extrapolation: A Hint of Things to Come

1 Extrapolation: A Hint of Things to Come Notes for 2017-03-24 1 Extrapolation: A Hint of Things to Come Stationary iterations are simple. Methods like Jacobi or Gauss-Seidel are easy to program, and it s (relatively) easy to analyze their convergence.

More information

M.A. Botchev. September 5, 2014

M.A. Botchev. September 5, 2014 Rome-Moscow school of Matrix Methods and Applied Linear Algebra 2014 A short introduction to Krylov subspaces for linear systems, matrix functions and inexact Newton methods. Plan and exercises. M.A. Botchev

More information

Math 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination

Math 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 information

Numerical Methods Orals

Numerical Methods Orals Numerical Methods Orals Travis Askham April 4, 2012 Contents 1 Floating Point Arithmetic, Conditioning, and Stability 3 1.1 Previously Asked Questions................................ 3 1.2 Numerical Methods

More information

Linear Algebra. Brigitte Bidégaray-Fesquet. MSIAM, September Univ. Grenoble Alpes, Laboratoire Jean Kuntzmann, Grenoble.

Linear Algebra. Brigitte Bidégaray-Fesquet. MSIAM, September Univ. Grenoble Alpes, Laboratoire Jean Kuntzmann, Grenoble. Brigitte Bidégaray-Fesquet Univ. Grenoble Alpes, Laboratoire Jean Kuntzmann, Grenoble MSIAM, 23 24 September 215 Overview 1 Elementary operations Gram Schmidt orthonormalization Matrix norm Conditioning

More information

AMS526: Numerical Analysis I (Numerical Linear Algebra)

AMS526: Numerical Analysis I (Numerical Linear Algebra) AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 3: Positive-Definite Systems; Cholesky Factorization Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis I 1 / 11 Symmetric

More information

Foundations of Matrix Analysis

Foundations of Matrix Analysis 1 Foundations of Matrix Analysis In this chapter we recall the basic elements of linear algebra which will be employed in the remainder of the text For most of the proofs as well as for the details, the

More information

AMS526: Numerical Analysis I (Numerical Linear Algebra)

AMS526: Numerical Analysis I (Numerical Linear Algebra) AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 12: Gaussian Elimination and LU Factorization Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 10 Gaussian Elimination

More information

Quantum Computing Lecture 2. Review of Linear Algebra

Quantum Computing Lecture 2. Review of Linear Algebra Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces

More information

Eigenvalue Problems CHAPTER 1 : PRELIMINARIES

Eigenvalue Problems CHAPTER 1 : PRELIMINARIES Eigenvalue Problems CHAPTER 1 : PRELIMINARIES Heinrich Voss voss@tu-harburg.de Hamburg University of Technology Institute of Mathematics TUHH Heinrich Voss Preliminaries Eigenvalue problems 2012 1 / 14

More information

Lecture 3: QR-Factorization

Lecture 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 information

1 Number Systems and Errors 1

1 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 information

Linear algebra & Numerical Analysis

Linear 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 information

Iterative methods for Linear System of Equations. Joint Advanced Student School (JASS-2009)

Iterative methods for Linear System of Equations. Joint Advanced Student School (JASS-2009) Iterative methods for Linear System of Equations Joint Advanced Student School (JASS-2009) Course #2: Numerical Simulation - from Models to Software Introduction In numerical simulation, Partial Differential

More information

Eigenvalue and Eigenvector Problems

Eigenvalue 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 information

A HARMONIC RESTARTED ARNOLDI ALGORITHM FOR CALCULATING EIGENVALUES AND DETERMINING MULTIPLICITY

A HARMONIC RESTARTED ARNOLDI ALGORITHM FOR CALCULATING EIGENVALUES AND DETERMINING MULTIPLICITY A HARMONIC RESTARTED ARNOLDI ALGORITHM FOR CALCULATING EIGENVALUES AND DETERMINING MULTIPLICITY RONALD B. MORGAN AND MIN ZENG Abstract. A restarted Arnoldi algorithm is given that computes eigenvalues

More information

Chapter 7. Iterative methods for large sparse linear systems. 7.1 Sparse matrix algebra. Large sparse matrices

Chapter 7. Iterative methods for large sparse linear systems. 7.1 Sparse matrix algebra. Large sparse matrices Chapter 7 Iterative methods for large sparse linear systems In this chapter we revisit the problem of solving linear systems of equations, but now in the context of large sparse systems. The price to pay

More information

The German word eigen is cognate with the Old English word āgen, which became owen in Middle English and own in modern English.

The German word eigen is cognate with the Old English word āgen, which became owen in Middle English and own in modern English. Chapter 4 EIGENVALUE PROBLEM The German word eigen is cognate with the Old English word āgen, which became owen in Middle English and own in modern English. 4.1 Mathematics 4.2 Reduction to Upper Hessenberg

More information

Introduction. Chapter One

Introduction. Chapter One Chapter One Introduction The aim of this book is to describe and explain the beautiful mathematical relationships between matrices, moments, orthogonal polynomials, quadrature rules and the Lanczos and

More information

Arnoldi Methods in SLEPc

Arnoldi Methods in SLEPc Scalable Library for Eigenvalue Problem Computations SLEPc Technical Report STR-4 Available at http://slepc.upv.es Arnoldi Methods in SLEPc V. Hernández J. E. Román A. Tomás V. Vidal Last update: October,

More information

Krylov Subspace Methods for Large/Sparse Eigenvalue Problems

Krylov Subspace Methods for Large/Sparse Eigenvalue Problems Krylov Subspace Methods for Large/Sparse Eigenvalue Problems Tsung-Ming Huang Department of Mathematics National Taiwan Normal University, Taiwan April 17, 2012 T.-M. Huang (Taiwan Normal University) Krylov

More information

Solving large scale eigenvalue problems

Solving 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 information

The quadratic eigenvalue problem (QEP) is to find scalars λ and nonzero vectors u satisfying

The quadratic eigenvalue problem (QEP) is to find scalars λ and nonzero vectors u satisfying I.2 Quadratic Eigenvalue Problems 1 Introduction The quadratic eigenvalue problem QEP is to find scalars λ and nonzero vectors u satisfying where Qλx = 0, 1.1 Qλ = λ 2 M + λd + K, M, D and K are given

More information

Orthogonal iteration to QR

Orthogonal 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 information

Some minimization problems

Some minimization problems Week 13: Wednesday, Nov 14 Some minimization problems Last time, we sketched the following two-step strategy for approximating the solution to linear systems via Krylov subspaces: 1. Build a sequence of

More information

EECS 275 Matrix Computation

EECS 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 information

Alternative correction equations in the Jacobi-Davidson method

Alternative correction equations in the Jacobi-Davidson method Chapter 2 Alternative correction equations in the Jacobi-Davidson method Menno Genseberger and Gerard Sleijpen Abstract The correction equation in the Jacobi-Davidson method is effective in a subspace

More information

5.3 The Power Method Approximation of the Eigenvalue of Largest Module

5.3 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 information

GPU accelerated Arnoldi solver for small batched matrix

GPU accelerated Arnoldi solver for small batched matrix 15. 09. 22 GPU accelerated Arnoldi solver for small batched matrix Samsung Advanced Institute of Technology Hyung-Jin Kim Contents - Eigen value problems - Solution - Arnoldi Algorithm - Target - CUDA

More information

Structured Krylov Subspace Methods for Eigenproblems with Spectral Symmetries

Structured Krylov Subspace Methods for Eigenproblems with Spectral Symmetries Structured Krylov Subspace Methods for Eigenproblems with Spectral Symmetries Fakultät für Mathematik TU Chemnitz, Germany Peter Benner benner@mathematik.tu-chemnitz.de joint work with Heike Faßbender

More information

Lab 1: Iterative Methods for Solving Linear Systems

Lab 1: Iterative Methods for Solving Linear Systems Lab 1: Iterative Methods for Solving Linear Systems January 22, 2017 Introduction Many real world applications require the solution to very large and sparse linear systems where direct methods such as

More information

ECS231 Handout Subspace projection methods for Solving Large-Scale Eigenvalue Problems. Part I: Review of basic theory of eigenvalue problems

ECS231 Handout Subspace projection methods for Solving Large-Scale Eigenvalue Problems. Part I: Review of basic theory of eigenvalue problems ECS231 Handout Subspace projection methods for Solving Large-Scale Eigenvalue Problems Part I: Review of basic theory of eigenvalue problems 1. Let A C n n. (a) A scalar λ is an eigenvalue of an n n A

More information

LINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS

LINEAR 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 information

Solution of eigenvalue problems. Subspace iteration, The symmetric Lanczos algorithm. Harmonic Ritz values, Jacobi-Davidson s method

Solution of eigenvalue problems. Subspace iteration, The symmetric Lanczos algorithm. Harmonic Ritz values, Jacobi-Davidson s method Solution of eigenvalue problems Introduction motivation Projection methods for eigenvalue problems Subspace iteration, The symmetric Lanczos algorithm Nonsymmetric Lanczos procedure; Implicit restarts

More information

Model reduction of large-scale dynamical systems

Model reduction of large-scale dynamical systems Model reduction of large-scale dynamical systems Lecture III: Krylov approximation and rational interpolation Thanos Antoulas Rice University and Jacobs University email: aca@rice.edu URL: www.ece.rice.edu/

More information

arxiv: v1 [math.na] 5 May 2011

arxiv: v1 [math.na] 5 May 2011 ITERATIVE METHODS FOR COMPUTING EIGENVALUES AND EIGENVECTORS MAYSUM PANJU arxiv:1105.1185v1 [math.na] 5 May 2011 Abstract. We examine some numerical iterative methods for computing the eigenvalues and

More information