Algorithms that use the Arnoldi Basis
|
|
- Griselda Preston
- 5 years ago
- Views:
Transcription
1 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 use the Arnoldi Basis FOM (usually called the Arnoldi iteration) GMRES Givens Rotations Relation between FOM and GMRES iterates Practicalities Convergence Results for GMRES The Arnoldi Basis Recall: The Arnoldi basis v 1,..., v n is obtained column by column from the relation AV = VH where H is upper Hessenberg and V T V = I. In particular, h ij = (Av j, v i ). The vectors v j can be formed by a (modified) Gram Schmidt process. For three algorithms for computing the recurrence, see Sec and Facts about the Arnoldi Basis Let s convince ourselves of the following facts: 1
2 The first m columns of V form an orthogonal basis for K m (A, v 1 ). Let V m denote the first m columns of V, and H m denote the leading m m block of H. Then AV m = V m H m + h m+1,m v m+1 e T m. The algorithm breaks down if h m+1,m = 0. In this case, we cannot get a new vector v m+1. This breakdown means that K m+1 (A, v 1 ) = K m (A, v 1 ). In other words, K m (A, v 1 ) is an invariant subspace of A of dimension m. How to use the Arnoldi Basis We want to solve Ax = b. We choose x 0 = 0 and v 1 = b/β, where β = b. Let s choose x m K m (A, v 1 ). Two choices: Projection: Make the residual orthogonal to K m (A, v 1 ). (Section 6.4.1) This is the Full Orthogonalization Method (FOM). Minimization: Minimize the norm of the residual over all choices of x in K m (A, v 1 ). (Section 6.5) This is the Generalized Minimum Residual Method (GMRES). FOM (usually called the Arnoldi iteration) The Arnoldi relation: AV m = V m H m + h m+1,m v m+1 e T m. If x m K m (A, v 1 ), then we can express it as x m = V m y m for some vector y m. The residual is r m = b Ax m = βv 1 AV m y m = βv 1 (V m H m + h m+1,m v m+1 e T m)y m. 2
3 To enforce orthogonality, we want 0 = Vmr T m = βvmv T 1 (VmV T m H m + h m+1,m Vmv T m+1 e T m)y m. = βe 1 H m y m. Therefore, given the Arnoldi relation at step m, we solve the linear system and set x m = V m y m, r m = b Ax m = b AV m y m H m y m = βe 1 = βv m e 1 (V m H m + h m+1,m v m+1 e T m)y m = V m (βe 1 H m y m ) h m+1,m v m+1 e T my m = h m+1,m v m+1 e T my m = h m+1,m (y m ) m v m+1. Therefore, the norm of the residual is just h m+1,m (y m ) m, which can be computed without forming x m. GMRES The Arnoldi relation: AV m = V m H m + h m+1,m v m+1 e T m. The iterate: x m = V m y m for some vector y m. Define The residual: H m to be the leading (m + 1) m block of H. r m = βv 1 (V m H m + h m+1,m v m+1 e T m)y m = βv 1 V m+1 H m y m = V m+1 (βe 1 H m y m ). The norm of the residual: r m = V m+1 (βe 1 H m y m ) = βe 1 H m y m. We minimize this quantity w.r.t. y m to get the GMRES iterate. 3
4 Givens Rotations The linear system involving H m is solved using m Givens rotations. The least squares problems involving H m is solved using 1 additional Givens rotation. Relation between FOM and GMRES iterates FOM: H m y F m = βe 1, or H T mh m y F m = βh T me 1. GMRES: minimize βe 1 H m ym G by solving the normal equations H mt H m ym G = βh T m e1. Note that so H T mh m and H mt [ H m = H m h m+1,m e T m ], H m differ by a matrix of rank 2. Therefore (Sec and Sec 6.5.8), If we compute the FOM iterate, we can easily get the GMRES iterate, and vice versa. It can be shown that the smallest residual computed by FOM in iterations 1,..., m is within a factor of m of the smallest for GMRES. There is a recursion for computing the GMRES iterates given the FOM iterates: x G m = s 2 mx G m 1 c 2 mx F m, where s 2 m + c 2 m = 1 and c m is the cosine used in the last step of the reduction of H m to upper triangular form. Practicalities Practicality 1: GMRES and FOM can stagnate. 4
5 It is possible (but unlikely) that GMRES and FOM make no progress at all until the last iteration. Example: Let A = [ e T 1 I 0 ], x true = e n, b = e 1. (e is the vector with every entry equal to 1.) Then K m (A, b) = span{e 1,..., e m }, so the solution vector is orthogonal to K m (A, b) for m < n. This is called complete stagnation. Partial stagnation can also occur. The usual remedy is preconditioning (discussed later). Practicality 2: m must be kept relatively small. As m gets big, The work per iteration becomes overwhelming, since the upper Hessenberg part of H m is generally dense. In fact, the work for the orthogonalization grows as O(m 2 n) The orthogonality relations are increasingly contaminated by round-off error, and this can cause the computation to compute bad approximations to the true GMRES/FOM iterates. Two remedies: restarting (Sec and Sec 6.5.5) truncating the orthogonalization (Sec and Sec 6.5.6) Restarting In the restarting algorithm, we run the Arnoldi iteration for a fixed number of steps, perhaps m = 100. If the resulting solution is not good enough, we repeat, by setting v 1 to be the normalized residual. This limits the work per iteration, but removes the finite-termination property. In fact, due to stagnation, it is possible (but unlikely) that the iteration never makes any progress at all! 5
6 Truncating the orthogonalization In this algorithm, we modify the Arnoldi iteration so that we only orthogonalize against the latest k vectors, rather than orthogonalizing against all of the old vectors. H approx m The matrix then has only k 1 nonzeros above the main diagonal, and the work for orthogonalization is reduced from O(m 2 n) to O(kmn). We are still working with the same Krylov subspace, but The basis vectors v 1,..., v m are not exactly orthogonal. We are approximating the matrix H m, so we don t minimize the residual (exactly) or make the residual (exactly) orthogonal to the Krylov subspace. All is not lost, though. In Section 6.5.6, Saad works out the relation between the GMRES residual norm and the one produced from the truncated method (quasi-gmres). Note: Quasi-GMRES is not often used. Restarted-GMRES is the standard algorithm. Practicality 3: Breakdown of the Arnoldi iteration is a good thing. If the Arnoldi iteration breaks down, with h m+1,m = 0, so that there is no new vector v m+1, then H m = H m, so both GMRES and FOM compute the same iterate. More importantly, Arnoldi breaks down if and only if x G m = x F m = x true. This follows from the fact that the norm of the GMRES residual is h m+1,m (y m ) m. Practicality 4: These algorithms also work in complex arithmetic. If A or b is complex rather than real, the algorithms still work, as long as we remember to use conjugate transpose instead of transpose. 6
7 use complex Givens rotations (Section 6.5.9): [ ] [ ] [ ] c s x γ = s c y 0 if s = y x 2 + y, c = x 2 x 2 + y. 2 Convergence Results for GMRES (Sec ) Two definitions: We say that a matrix A is positive definite if x T Ax > 0. for all x 0. If A is symmetric, this means that all of its eigenvalues are positive. For a nonsymmetric matrix, this means that the symmetric part of A, which is 1 2 (A + AT ), is positive definite. The algorithm that restarts GMRES every m iterations is called GMRES(m). GMRES(m) converges for positive definite matrices. Proof: The iteration can t stagnate, because it makes progress at the first iteration. In particular, if we seek x = αv 1 to minimize r 2 = b αav 1 2 where b = βv 1 and β > 0 then so the residual norm is reduced. = b T b 2αb T Av 1 + α 2 v T 1 A T Av 1, α = bt Av 1 v T 1 AT Av 1 > 0, By working a little harder, we can show that it reduced enough to guarantee convergence. [] GMRES(m) is an optimal polynomial method. The iterates take the form x m = x 0 + q m 1 (A)r 0, where q m 1 is a polynomial of degree at most m 1. 7
8 Saying that x is in K m (A, r 0 ) is equivalent to saying x = q m 1 (A)r 0 for some polynomial of degree at most m 1. The resulting residual is r m = r 0 q m 1 (A)Ar 0 = (I q m 1 (A)A)r 0 = p m (A)r 0, where p m is a polynomial of degree m satisfying p m (0) = 1. since GMRES(m) minimizes the norm of the residual, it must pick the optimal polynomial p m. Let A = XΛX 1, where Λ is diagonal and contains the eigenvalues of A. (In other words, we are assuming that A has a complete set of linearly independent eigenvectors, the columns of X.) Then r m = min p m(0)=1 p m(a)r 0 = min p m(0)=1 Xp m(λ)x 1 r 0 X X 1 min p m(0)=1 p m(λ) r 0 X X 1 min p m(0)=1 max p m(λ j ) r 0. j=1,...,n Therefore, any polynomial that is small on the eigenvalues can give us a bound on the GMRES(m) residual norm, since GMRES(m) picks the optimal polynomial. By choosing a Chebyshev polynomial, Saad derives the bound ( a + ) k a r m 2 d 2 c + X X 1 r 0. c 2 d 2 whenever the ellipse centered at c with focal distance d and semi-major axis a contains all of the eigenvalues and excludes the origin (See the picture on p. 207). 8
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 informationConjugate gradient method. Descent method. Conjugate search direction. Conjugate Gradient Algorithm (294)
Conjugate gradient method Descent method Hestenes, Stiefel 1952 For A N N SPD In exact arithmetic, solves in N steps In real arithmetic No guaranteed stopping Often converges in many fewer than N steps
More informationSummary 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 informationSolving Sparse Linear Systems: Iterative methods
Scientific Computing with Case Studies SIAM Press, 2009 http://www.cs.umd.edu/users/oleary/sccs Lecture Notes for Unit VII Sparse Matrix Computations Part 2: Iterative Methods Dianne P. O Leary c 2008,2010
More informationSolving Sparse Linear Systems: Iterative methods
Scientific Computing with Case Studies SIAM Press, 2009 http://www.cs.umd.edu/users/oleary/sccswebpage Lecture Notes for Unit VII Sparse Matrix Computations Part 2: Iterative Methods Dianne P. O Leary
More informationTopics. The CG Algorithm Algorithmic Options CG s Two Main Convergence Theorems
Topics The CG Algorithm Algorithmic Options CG s Two Main Convergence Theorems What about non-spd systems? Methods requiring small history Methods requiring large history Summary of solvers 1 / 52 Conjugate
More informationAMS526: 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 informationSOLVING 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 informationFEM 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 informationThe amount of work to construct each new guess from the previous one should be a small multiple of the number of nonzeros in A.
AMSC/CMSC 661 Scientific Computing II Spring 2005 Solution of Sparse Linear Systems Part 2: Iterative methods Dianne P. O Leary c 2005 Solving Sparse Linear Systems: Iterative methods The plan: Iterative
More informationIterative 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 informationA short course on: Preconditioned Krylov subspace methods. Yousef Saad University of Minnesota Dept. of Computer Science and Engineering
A short course on: Preconditioned Krylov subspace methods Yousef Saad University of Minnesota Dept. of Computer Science and Engineering Universite du Littoral, Jan 19-3, 25 Outline Part 1 Introd., discretization
More informationKrylov 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 informationAMS526: 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 informationKrylov 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 informationON 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 informationSimple 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 informationAMS526: Numerical Analysis I (Numerical Linear Algebra)
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 Outline 1
More informationCourse 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 informationIDR(s) as a projection method
Delft University of Technology Faculty of Electrical Engineering, Mathematics and Computer Science Delft Institute of Applied Mathematics IDR(s) as a projection method A thesis submitted to the Delft Institute
More informationOn 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 informationM.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 information1 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 informationAMSC 600 /CMSC 760 Advanced Linear Numerical Analysis Fall 2007 Krylov Minimization and Projection (KMP) Dianne P. O Leary c 2006, 2007.
AMSC 600 /CMSC 760 Advanced Linear Numerical Analysis Fall 2007 Krylov Minimization and Projection (KMP) Dianne P. O Leary c 2006, 2007 This unit: So far: A survey of iterative methods for solving linear
More informationOn Solving Large Algebraic. Riccati Matrix Equations
International Mathematical Forum, 5, 2010, no. 33, 1637-1644 On Solving Large Algebraic Riccati Matrix Equations Amer Kaabi Department of Basic Science Khoramshahr Marine Science and Technology University
More informationCharles 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 informationComputational Linear Algebra
Computational Linear Algebra PD Dr. rer. nat. habil. Ralf-Peter Mundani Computation in Engineering / BGU Scientific Computing in Computer Science / INF Winter Term 2018/19 Part 4: Iterative Methods PD
More information6.4 Krylov Subspaces and Conjugate Gradients
6.4 Krylov Subspaces and Conjugate Gradients Our original equation is Ax = b. The preconditioned equation is P Ax = P b. When we write P, we never intend that an inverse will be explicitly computed. P
More informationGMRES: Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
GMRES: Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems Tsung-Ming Huang Department of Mathematics National Taiwan Normal University December 4, 2011 T.-M. Huang (NTNU) GMRES
More informationLast Time. Social Network Graphs Betweenness. Graph Laplacian. Girvan-Newman Algorithm. Spectral Bisection
Eigenvalue Problems Last Time Social Network Graphs Betweenness Girvan-Newman Algorithm Graph Laplacian Spectral Bisection λ 2, w 2 Today Small deviation into eigenvalue problems Formulation Standard eigenvalue
More informationComputational Linear Algebra
Computational Linear Algebra PD Dr. rer. nat. habil. Ralf Peter Mundani Computation in Engineering / BGU Scientific Computing in Computer Science / INF Winter Term 2017/18 Part 3: Iterative Methods PD
More informationIterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems Luca Bergamaschi e-mail: berga@dmsa.unipd.it - http://www.dmsa.unipd.it/ berga Department of Mathematical Methods and Models for Scientific Applications University
More informationIterative 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 informationThe 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 informationConjugate Gradient Method
Conjugate Gradient Method Hung M Phan UMass Lowell April 13, 2017 Throughout, A R n n is symmetric and positive definite, and b R n 1 Steepest Descent Method We present the steepest descent method for
More informationApplied Mathematics 205. Unit V: Eigenvalue Problems. Lecturer: Dr. David Knezevic
Applied Mathematics 205 Unit V: Eigenvalue Problems Lecturer: Dr. David Knezevic Unit V: Eigenvalue Problems Chapter V.4: Krylov Subspace Methods 2 / 51 Krylov Subspace Methods In this chapter we give
More informationMath 504 (Fall 2011) 1. (*) Consider the matrices
Math 504 (Fall 2011) Instructor: Emre Mengi Study Guide for Weeks 11-14 This homework concerns the following topics. Basic definitions and facts about eigenvalues and eigenvectors (Trefethen&Bau, Lecture
More informationKrylov Subspaces. Lab 1. The Arnoldi Iteration
Lab 1 Krylov Subspaces Lab Objective: Discuss simple Krylov Subspace Methods for finding eigenvalues and show some interesting applications. One of the biggest difficulties in computational linear algebra
More informationNotes on Some Methods for Solving Linear Systems
Notes on Some Methods for Solving Linear Systems Dianne P. O Leary, 1983 and 1999 and 2007 September 25, 2007 When the matrix A is symmetric and positive definite, we have a whole new class of algorithms
More informationA DISSERTATION. Extensions of the Conjugate Residual Method. by Tomohiro Sogabe. Presented to
A DISSERTATION Extensions of the Conjugate Residual Method ( ) by Tomohiro Sogabe Presented to Department of Applied Physics, The University of Tokyo Contents 1 Introduction 1 2 Krylov subspace methods
More informationArnoldi 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 informationOn 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 informationI = i 0,
Special Types of Matrices Certain matrices, such as the identity matrix 0 0 0 0 0 0 I = 0 0 0, 0 0 0 have a special shape, which endows the matrix with helpful properties The identity matrix is an example
More informationDELFT UNIVERSITY OF TECHNOLOGY
DELFT UNIVERSITY OF TECHNOLOGY REPORT 14-1 Nested Krylov methods for shifted linear systems M. Baumann and M. B. van Gizen ISSN 1389-652 Reports of the Delft Institute of Applied Mathematics Delft 214
More informationPROJECTED GMRES AND ITS VARIANTS
PROJECTED GMRES AND ITS VARIANTS Reinaldo Astudillo Brígida Molina rastudillo@kuaimare.ciens.ucv.ve bmolina@kuaimare.ciens.ucv.ve Centro de Cálculo Científico y Tecnológico (CCCT), Facultad de Ciencias,
More informationThe Conjugate Gradient Method
The Conjugate Gradient Method Classical Iterations We have a problem, We assume that the matrix comes from a discretization of a PDE. The best and most popular model problem is, The matrix will be as large
More informationInstitute for Advanced Computer Studies. Department of Computer Science. Iterative methods for solving Ax = b. GMRES/FOM versus QMR/BiCG
University of Maryland Institute for Advanced Computer Studies Department of Computer Science College Park TR{96{2 TR{3587 Iterative methods for solving Ax = b GMRES/FOM versus QMR/BiCG Jane K. Cullum
More informationAlternative 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 information9.1 Preconditioned Krylov Subspace Methods
Chapter 9 PRECONDITIONING 9.1 Preconditioned Krylov Subspace Methods 9.2 Preconditioned Conjugate Gradient 9.3 Preconditioned Generalized Minimal Residual 9.4 Relaxation Method Preconditioners 9.5 Incomplete
More informationLecture 11: CMSC 878R/AMSC698R. Iterative Methods An introduction. Outline. Inverse, LU decomposition, Cholesky, SVD, etc.
Lecture 11: CMSC 878R/AMSC698R Iterative Methods An introduction Outline Direct Solution of Linear Systems Inverse, LU decomposition, Cholesky, SVD, etc. Iterative methods for linear systems Why? Matrix
More informationMath 5630: Conjugate Gradient Method Hung M. Phan, UMass Lowell March 29, 2019
Math 563: Conjugate Gradient Method Hung M. Phan, UMass Lowell March 29, 219 hroughout, A R n n is symmetric and positive definite, and b R n. 1 Steepest Descent Method We present the steepest descent
More informationMATH 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 informationComputation of eigenvalues and singular values Recall that your solutions to these questions will not be collected or evaluated.
Math 504, Homework 5 Computation of eigenvalues and singular values Recall that your solutions to these questions will not be collected or evaluated 1 Find the eigenvalues and the associated eigenspaces
More informationParallel Numerics, WT 2016/ Iterative Methods for Sparse Linear Systems of Equations. page 1 of 1
Parallel Numerics, WT 2016/2017 5 Iterative Methods for Sparse Linear Systems of Equations page 1 of 1 Contents 1 Introduction 1.1 Computer Science Aspects 1.2 Numerical Problems 1.3 Graphs 1.4 Loop Manipulations
More informationEECS 275 Matrix Computation
EECS 275 Matrix Computation Ming-Hsuan Yang Electrical Engineering and Computer Science University of California at Merced Merced, CA 95344 http://faculty.ucmerced.edu/mhyang Lecture 20 1 / 20 Overview
More informationBlock Bidiagonal Decomposition and Least Squares Problems
Block Bidiagonal Decomposition and Least Squares Problems Åke Björck Department of Mathematics Linköping University Perspectives in Numerical Analysis, Helsinki, May 27 29, 2008 Outline Bidiagonal Decomposition
More informationIterative Methods for Linear Systems of Equations
Iterative Methods for Linear Systems of Equations Projection methods (3) ITMAN PhD-course DTU 20-10-08 till 24-10-08 Martin van Gijzen 1 Delft University of Technology Overview day 4 Bi-Lanczos method
More informationCourse Notes: Week 4
Course Notes: Week 4 Math 270C: Applied Numerical Linear Algebra 1 Lecture 9: Steepest Descent (4/18/11) The connection with Lanczos iteration and the CG was not originally known. CG was originally derived
More informationLab 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 informationWHEN studying distributed simulations of power systems,
1096 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL 21, NO 3, AUGUST 2006 A Jacobian-Free Newton-GMRES(m) Method with Adaptive Preconditioner and Its Application for Power Flow Calculations Ying Chen and Chen
More informationCommunication-avoiding Krylov subspace methods
Motivation Communication-avoiding Krylov subspace methods Mark mhoemmen@cs.berkeley.edu University of California Berkeley EECS MS Numerical Libraries Group visit: 28 April 2008 Overview Motivation Current
More informationUNIT 6: The singular value decomposition.
UNIT 6: The singular value decomposition. María Barbero Liñán Universidad Carlos III de Madrid Bachelor in Statistics and Business Mathematical methods II 2011-2012 A square matrix is symmetric if A T
More informationAgenda: Understand the action of A by seeing how it acts on eigenvectors.
Eigenvalues and Eigenvectors If Av=λv with v nonzero, then λ is called an eigenvalue of A and v is called an eigenvector of A corresponding to eigenvalue λ. Agenda: Understand the action of A by seeing
More informationLinear Algebra Review
Chapter 1 Linear Algebra Review It is assumed that you have had a course in linear algebra, and are familiar with matrix multiplication, eigenvectors, etc. I will review some of these terms here, but quite
More informationITERATIVE METHODS FOR SPARSE LINEAR SYSTEMS
ITERATIVE METHODS FOR SPARSE LINEAR SYSTEMS YOUSEF SAAD University of Minnesota PWS PUBLISHING COMPANY I(T)P An International Thomson Publishing Company BOSTON ALBANY BONN CINCINNATI DETROIT LONDON MADRID
More informationScientific Computing with Case Studies SIAM Press, Lecture Notes for Unit VII Sparse Matrix
Scientific Computing with Case Studies SIAM Press, 2009 http://www.cs.umd.edu/users/oleary/sccswebpage Lecture Notes for Unit VII Sparse Matrix Computations Part 1: Direct Methods Dianne P. O Leary c 2008
More informationABSTRACT OF DISSERTATION. Ping Zhang
ABSTRACT OF DISSERTATION Ping Zhang The Graduate School University of Kentucky 2009 Iterative Methods for Computing Eigenvalues and Exponentials of Large Matrices ABSTRACT OF DISSERTATION A dissertation
More informationThe 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 informationLARGE SPARSE EIGENVALUE PROBLEMS. General Tools for Solving Large Eigen-Problems
LARGE SPARSE EIGENVALUE PROBLEMS Projection methods The subspace iteration Krylov subspace methods: Arnoldi and Lanczos Golub-Kahan-Lanczos bidiagonalization General Tools for Solving Large Eigen-Problems
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J Olver 8 Numerical Computation of Eigenvalues In this part, we discuss some practical methods for computing eigenvalues and eigenvectors of matrices Needless to
More informationLinear 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 informationIterative Methods for Solving A x = b
Iterative Methods for Solving A x = b A good (free) online source for iterative methods for solving A x = b is given in the description of a set of iterative solvers called templates found at netlib: http
More informationMATRICES ARE SIMILAR TO TRIANGULAR MATRICES
MATRICES ARE SIMILAR TO TRIANGULAR MATRICES 1 Complex matrices Recall that the complex numbers are given by a + ib where a and b are real and i is the imaginary unity, ie, i 2 = 1 In what we describe below,
More informationConceptual Questions for Review
Conceptual Questions for Review Chapter 1 1.1 Which vectors are linear combinations of v = (3, 1) and w = (4, 3)? 1.2 Compare the dot product of v = (3, 1) and w = (4, 3) to the product of their lengths.
More informationGMRES ON (NEARLY) SINGULAR SYSTEMS
SIAM J. MATRIX ANAL. APPL. c 1997 Society for Industrial and Applied Mathematics Vol. 18, No. 1, pp. 37 51, January 1997 004 GMRES ON (NEARLY) SINGULAR SYSTEMS PETER N. BROWN AND HOMER F. WALKER Abstract.
More informationAugmented GMRES-type methods
Augmented GMRES-type methods James Baglama 1 and Lothar Reichel 2, 1 Department of Mathematics, University of Rhode Island, Kingston, RI 02881. E-mail: jbaglama@math.uri.edu. Home page: http://hypatia.math.uri.edu/
More informationLARGE 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 informationDELFT UNIVERSITY OF TECHNOLOGY
DELFT UNIVERSITY OF TECHNOLOGY REPORT 06-05 Solution of the incompressible Navier Stokes equations with preconditioned Krylov subspace methods M. ur Rehman, C. Vuik G. Segal ISSN 1389-6520 Reports of the
More informationANY FINITE CONVERGENCE CURVE IS POSSIBLE IN THE INITIAL ITERATIONS OF RESTARTED FOM
Electronic Transactions on Numerical Analysis. Volume 45, pp. 133 145, 2016. Copyright c 2016,. ISSN 1068 9613. ETNA ANY FINITE CONVERGENCE CURVE IS POSSIBLE IN THE INITIAL ITERATIONS OF RESTARTED FOM
More informationMATH 235. Final ANSWERS May 5, 2015
MATH 235 Final ANSWERS May 5, 25. ( points) Fix positive integers m, n and consider the vector space V of all m n matrices with entries in the real numbers R. (a) Find the dimension of V and prove your
More informationLINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS
LINEAR ALGEBRA, -I PARTIAL EXAM SOLUTIONS TO PRACTICE PROBLEMS Problem (a) For each of the two matrices below, (i) determine whether it is diagonalizable, (ii) determine whether it is orthogonally diagonalizable,
More informationLecture 17 Methods for System of Linear Equations: Part 2. Songting Luo. Department of Mathematics Iowa State University
Lecture 17 Methods for System of Linear Equations: Part 2 Songting Luo Department of Mathematics Iowa State University MATH 481 Numerical Methods for Differential Equations Songting Luo ( Department of
More informationCOMP 558 lecture 18 Nov. 15, 2010
Least squares We have seen several least squares problems thus far, and we will see more in the upcoming lectures. For this reason it is good to have a more general picture of these problems and how to
More informationft-uiowa-math2550 Assignment NOTRequiredJustHWformatOfQuizReviewForExam3part2 due 12/31/2014 at 07:10pm CST
me me ft-uiowa-math2550 Assignment NOTRequiredJustHWformatOfQuizReviewForExam3part2 due 12/31/2014 at 07:10pm CST 1. (1 pt) local/library/ui/eigentf.pg A is n n an matrices.. There are an infinite number
More informationMATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL
MATH 3 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL MAIN TOPICS FOR THE FINAL EXAM:. Vectors. Dot product. Cross product. Geometric applications. 2. Row reduction. Null space, column space, row space, left
More informationDS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More informationLinear Solvers. Andrew Hazel
Linear Solvers Andrew Hazel Introduction Thus far we have talked about the formulation and discretisation of physical problems...... and stopped when we got to a discrete linear system of equations. Introduction
More informationThe Conjugate Gradient Method
The Conjugate Gradient Method The minimization problem We are given a symmetric positive definite matrix R n n and a right hand side vector b R n We want to solve the linear system Find u R n such that
More informationPreface to the Second Edition. Preface to the First Edition
n page v Preface to the Second Edition Preface to the First Edition xiii xvii 1 Background in Linear Algebra 1 1.1 Matrices................................. 1 1.2 Square Matrices and Eigenvalues....................
More informationPreliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012
Instructions Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012 The exam consists of four problems, each having multiple parts. You should attempt to solve all four problems. 1.
More informationNotes on Eigenvalues, Singular Values and QR
Notes on Eigenvalues, Singular Values and QR Michael Overton, Numerical Computing, Spring 2017 March 30, 2017 1 Eigenvalues Everyone who has studied linear algebra knows the definition: given a square
More informationCME342 Parallel Methods in Numerical Analysis. Matrix Computation: Iterative Methods II. Sparse Matrix-vector Multiplication.
CME342 Parallel Methods in Numerical Analysis Matrix Computation: Iterative Methods II Outline: CG & its parallelization. Sparse Matrix-vector Multiplication. 1 Basic iterative methods: Ax = b r = b Ax
More informationCheat Sheet for MATH461
Cheat Sheet for MATH46 Here is the stuff you really need to remember for the exams Linear systems Ax = b Problem: We consider a linear system of m equations for n unknowns x,,x n : For a given matrix A
More informationRational Krylov methods for linear and nonlinear eigenvalue problems
Rational Krylov methods for linear and nonlinear eigenvalue problems Mele Giampaolo mele@mail.dm.unipi.it University of Pisa 7 March 2014 Outline Arnoldi (and its variants) for linear eigenproblems Rational
More informationChapter 5. Eigenvalues and Eigenvectors
Chapter 5 Eigenvalues and Eigenvectors Section 5. Eigenvectors and Eigenvalues Motivation: Difference equations A Biology Question How to predict a population of rabbits with given dynamics:. half of the
More informationEigenvalue Problems. Eigenvalue problems occur in many areas of science and engineering, such as structural analysis
Eigenvalue Problems Eigenvalue problems occur in many areas of science and engineering, such as structural analysis Eigenvalues also important in analyzing numerical methods Theory and algorithms apply
More informationDefinition (T -invariant subspace) Example. Example
Eigenvalues, Eigenvectors, Similarity, and Diagonalization We now turn our attention to linear transformations of the form T : V V. To better understand the effect of T on the vector space V, we begin
More informationIDR(s) Master s thesis Goushani Kisoensingh. Supervisor: Gerard L.G. Sleijpen Department of Mathematics Universiteit Utrecht
IDR(s) Master s thesis Goushani Kisoensingh Supervisor: Gerard L.G. Sleijpen Department of Mathematics Universiteit Utrecht Contents 1 Introduction 2 2 The background of Bi-CGSTAB 3 3 IDR(s) 4 3.1 IDR.............................................
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 4 Eigenvalue Problems Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction
More informationApplied Mathematics 205. Unit II: Numerical Linear Algebra. Lecturer: Dr. David Knezevic
Applied Mathematics 205 Unit II: Numerical Linear Algebra Lecturer: Dr. David Knezevic Unit II: Numerical Linear Algebra Chapter II.3: QR Factorization, SVD 2 / 66 QR Factorization 3 / 66 QR Factorization
More information