Rank Revealing QR factorization. F. Guyomarc h, D. Mezher and B. Philippe
|
|
- Clinton McDaniel
- 5 years ago
- Views:
Transcription
1 Rank Revealing QR factorization F. Guyomarc h, D. Mezher and B. Philippe 1
2 Outline Introduction Classical Algorithms Full matrices Sparse matrices Rank-Revealing QR Conclusion CSDA 2005, Cyprus 2
3 Situation of the problem See Bjorck SIAM96 : Numerical Methods for Least Squares Problems. Data : A R m n, b R m. Problem P : Find x S such that x minimum where S = {x R n b Ax minimum }. (.. 2 ) The solution is unique : ˆx = A + b. Property : x S iff A T (b Ax) = 0, iff A T Ax = A T b. The last equation is consistent since R(A T A) = R(A T ). CSDA 2005, Cyprus 3
4 Rank-Revealing QR factorization Theorem : A R m n with rank(a) = r (r min(m, n)). There exist Q, E, R 11 and R 12 such that Q R m m is orthogonal, E R n n is a permutation, R 11 R r r is upper-triangular with positive diagonal entries, R 12 R r (n r), and AE = Q ( R11 R ). RRQR The factorization is not unique. Let RRQR be any of them. CSDA 2005, Cyprus 4
5 Actually, if we consider a complete orthogonal decomposition ( ) T 0 A = Q V T 0 0 where Q and V are orthogonal ( and T triangular with positive diagonal entries, we have A + T = V 1 ) 0 Q 0 0 T. Such a factorization can be obtained from the previous one by performing a QR factorization of ( R T 11 R T 12 ) CSDA 2005, Cyprus 5
6 Column pivoting strategy Householder QR factorization using Householder reflections : A = H 1 A = v = A 1..12,1 + A 1..12,1 e 1 H 1 = I 12 2 vvt v t v v = A 2..12,2 + A 2..12,2 e 1 H 2 = I 11 2 vvt v t v CSDA 2005, Cyprus 6
7 Sparse QR factorization Factorizing a sparse matrix implies fill-in in the factors. The situation is worse with QR than with LU since when updating the tailing matrix : LU : the elementary transformation x y = (I αve T k )x keeps x invariant when x k = 0. QR : the elementary transformation x y = (I αvv T )x keeps x invariant when x v. Since A T A = R T R, the QR factorization A is related to the Cholesky factorization of A T A. It is known that a symmetric permutation on a sparse s.p.d. matrix changes the level of fill-in. CSDA 2005, Cyprus 7
8 Therefore, a permutation of the columns of A changes the fill-in of R. there is conflict between pivoting to minimize fill-ins and pivoting associated with numerical properties. We choose to decouple the sparse factorization phase and the rankrevealing phase For a standard QR factorization of a sparse matrix : Multi-frontal QR method [Amestoy-Duff-Puglisi 94] Routine MA49AD in Library HSL. CSDA 2005, Cyprus 8
9 Column pivoting strategy Householder QR factorization using Householder reflections : A = H 1 A = v = A 1..12,1 + A 1..12,1 e 1 H 1 = I 12 2 vvt v t v v = A 2..12,2 + A 2..12,2 e 1 H 2 = I 11 2 vvt v t v Householder QR with Column Pivoting (Businger and Golub) : At step k, the column j k k defining the Householder transformation is chosen so that A(k : m, j k ) A(k : m, j) for j k. CSDA 2005, Cyprus 9
10 Some properties on R 11 At step k : Any column a of A (k) H k H 1 AE(k) = R(k) 12 (k,:) A (k) 22 R(k) 11 R (k) 12 0 A (k) 22. satisfies R (k) 11 (k, k) a. Moreover A R (k) 11 R(k) 11 (1,1) R (k) 11 (k, k) σ min(r (k) 11 ) σ min(a), This implies that : cond(a) cond(r (k) 11 ) R(k) 11 (1,1) R (k) 11 (k,k). CSDA 2005, Cyprus 10
11 Bad new The quantity R(k) 11 (1,1) R (k) cannot be considered as an estimator of cond(r(k) 11 (k,k) since it can be of different order of magnitude : 11 ) Example (Kahan s matrix of order n) : for θ (0, π 2 ), c = cos θ and s = arcsin θ: MATLAB : d=c.ˆ [0:n-1] ; M=diag(d)*(eye(n)-s*triu(ones(n),1)); n=100 ; θ = arcsin(0.2) : σ min (R 11 ) = 3.7e 9 R 11 (100,100) = 1.3e 1 Therefore a better estimate of cond(r (k) 11 ) is needed. CSDA 2005, Cyprus 11
12 Incremental Condition Estimator (ICE) [Bischof 90] which is implemented in LAPACK : it estimates cond(r (k) 11 ) from an estimation of cond(r(k 1) 11 ). However, the strategy which consists in stopping the factorization when cond(r (k) 11 ) < 1 ǫ cond(r(k+1) 11 ) may fail in very special situations : Counterexample : M = Kahan s matrix of order 30 with θ = arccos(0.2) cond(r (16) 11 ) < 1 ǫ < cond(r(17) 11 ) indicates a numerical rank equal to 16 although the numerical rank of M computed by SVD is 22. CSDA 2005, Cyprus 12
13 Pivoting strategies Convert R to reveal the rank by pushing the singularities towards the right end Apply an Incremental Condition Estimator to evaluate σ max and σ min of R 1 i,1 i, if σ max /σ min > τ, move column i towards the right end and reorthogonalise R using Givens rotations CSDA 2005, Cyprus 13
14 Pivoting strategies RRQR uses a set of thresholds to overcome numerical singularities, A predifined set of thresholds might fail A = rank=2 if τ = {10 7 }, rank=3 if τ = {10 4,10 7 } RRQR adapts the thresholds to avoid failure Upon completion, check if R 22 < σ max τ σ min On failure, insert new thresholds and restart algorithm CSDA 2005, Cyprus 14
15 Numerical results smax/smin cond([q,r,p]=qr(a) RRQR, thres=1e3,1e7,1e12,1e25 RRQR, thres=1e3,1e9,1e condition number kahan(50,acos(0.2)), cond=2.4721e+49, rank=20, size=50x50, req=0 CSDA 2005, Cyprus 15
16 Numerical results smax/smin cond([q,r,p]=qr(a) RRQR, thres=1e30 RRQR, thres=1e20 RRQR, thres=1e25, req=2 condition number kahan(50,acos(0.2)), cond=2.4721e+49, rank=20, size=50x50 CSDA 2005, Cyprus 16
17 Second strategy The k + 1 column is not the worst usually for the condition number. Can we select the worst one for the conditionement of R 11? Then we reject this worst column to the end. k+1 j k+1 We need to recompute the singular values and vectors estimates. Reverse ICE CSDA 2005, Cyprus 17
18 Second strategy Random matrix, n= Exact Estimated tol=1e 12 Estimated tol=1e CSDA 2005, Cyprus 18
19 Conclusion Work is still on progress : Case where R is sparse ICE might fail, so does the Reverse ICE. Using the elimination tree, we can try to keep R as sparse as possible. CSDA 2005, Cyprus 19
14.2 QR Factorization with Column Pivoting
page 531 Chapter 14 Special Topics Background Material Needed Vector and Matrix Norms (Section 25) Rounding Errors in Basic Floating Point Operations (Section 33 37) Forward Elimination and Back Substitution
More informationNumerical Linear Algebra
Numerical Linear Algebra Direct Methods Philippe B. Laval KSU Fall 2017 Philippe B. Laval (KSU) Linear Systems: Direct Solution Methods Fall 2017 1 / 14 Introduction The solution of linear systems is one
More informationLinear Least squares
Linear Least squares Method of least squares Measurement errors are inevitable in observational and experimental sciences Errors can be smoothed out by averaging over more measurements than necessary to
More informationMatrix decompositions
Matrix decompositions Zdeněk Dvořák May 19, 2015 Lemma 1 (Schur decomposition). If A is a symmetric real matrix, then there exists an orthogonal matrix Q and a diagonal matrix D such that A = QDQ T. The
More informationGram-Schmidt Orthogonalization: 100 Years and More
Gram-Schmidt Orthogonalization: 100 Years and More September 12, 2008 Outline of Talk Early History (1795 1907) Middle History 1. The work of Åke Björck Least squares, Stability, Loss of orthogonality
More informationMS&E 318 (CME 338) Large-Scale Numerical Optimization
Stanford University, Management Science & Engineering (and ICME MS&E 38 (CME 338 Large-Scale Numerical Optimization Course description Instructor: Michael Saunders Spring 28 Notes : Review The course teaches
More informationNumerical Methods. Elena loli Piccolomini. Civil Engeneering. piccolom. Metodi Numerici M p. 1/??
Metodi Numerici M p. 1/?? Numerical Methods Elena loli Piccolomini Civil Engeneering http://www.dm.unibo.it/ piccolom elena.loli@unibo.it Metodi Numerici M p. 2/?? Least Squares Data Fitting Measurement
More informationThis can be accomplished by left matrix multiplication as follows: I
1 Numerical Linear Algebra 11 The LU Factorization Recall from linear algebra that Gaussian elimination is a method for solving linear systems of the form Ax = b, where A R m n and bran(a) In this method
More informationLinear Algebra, part 3 QR and SVD
Linear Algebra, part 3 QR and SVD Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2012 Going back to least squares (Section 1.4 from Strang, now also see section 5.2). We
More informationbe a Householder matrix. Then prove the followings H = I 2 uut Hu = (I 2 uu u T u )u = u 2 uut u
MATH 434/534 Theoretical Assignment 7 Solution Chapter 7 (71) Let H = I 2uuT Hu = u (ii) Hv = v if = 0 be a Householder matrix Then prove the followings H = I 2 uut Hu = (I 2 uu )u = u 2 uut u = u 2u =
More informationReview Questions REVIEW QUESTIONS 71
REVIEW QUESTIONS 71 MATLAB, is [42]. For a comprehensive treatment of error analysis and perturbation theory for linear systems and many other problems in linear algebra, see [126, 241]. An overview of
More informationRank revealing factorizations, and low rank approximations
Rank revealing factorizations, and low rank approximations L. Grigori Inria Paris, UPMC January 2018 Plan Low rank matrix approximation Rank revealing QR factorization LU CRTP: Truncated LU factorization
More informationLinear Algebra, part 3. Going back to least squares. Mathematical Models, Analysis and Simulation = 0. a T 1 e. a T n e. Anna-Karin Tornberg
Linear Algebra, part 3 Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2010 Going back to least squares (Sections 1.7 and 2.3 from Strang). We know from before: The vector
More informationAlgebra C Numerical Linear Algebra Sample Exam Problems
Algebra C Numerical Linear Algebra Sample Exam Problems Notation. Denote by V a finite-dimensional Hilbert space with inner product (, ) and corresponding norm. The abbreviation SPD is used for symmetric
More informationMatrix Multiplication Chapter IV Special Linear Systems
Matrix Multiplication Chapter IV Special Linear Systems By Gokturk Poyrazoglu The State University of New York at Buffalo BEST Group Winter Lecture Series Outline 1. Diagonal Dominance and Symmetry a.
More information3 QR factorization revisited
LINEAR ALGEBRA: NUMERICAL METHODS. Version: August 2, 2000 30 3 QR factorization revisited Now we can explain why A = QR factorization is much better when using it to solve Ax = b than the A = LU factorization
More informationSTAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 13
STAT 309: MATHEMATICAL COMPUTATIONS I FALL 208 LECTURE 3 need for pivoting we saw that under proper circumstances, we can write A LU where 0 0 0 u u 2 u n l 2 0 0 0 u 22 u 2n L l 3 l 32, U 0 0 0 l n l
More informationIntroduction to Mathematical Programming
Introduction to Mathematical Programming Ming Zhong Lecture 6 September 12, 2018 Ming Zhong (JHU) AMS Fall 2018 1 / 20 Table of Contents 1 Ming Zhong (JHU) AMS Fall 2018 2 / 20 Solving Linear Systems A
More informationMatrix Factorization and Analysis
Chapter 7 Matrix Factorization and Analysis Matrix factorizations are an important part of the practice and analysis of signal processing. They are at the heart of many signal-processing algorithms. Their
More informationLAPACK-Style Codes for Pivoted Cholesky and QR Updating
LAPACK-Style Codes for Pivoted Cholesky and QR Updating Sven Hammarling 1, Nicholas J. Higham 2, and Craig Lucas 3 1 NAG Ltd.,Wilkinson House, Jordan Hill Road, Oxford, OX2 8DR, England, sven@nag.co.uk,
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 informationNumerical Linear Algebra
Chapter 3 Numerical Linear Algebra We review some techniques used to solve Ax = b where A is an n n matrix, and x and b are n 1 vectors (column vectors). We then review eigenvalues and eigenvectors and
More informationLecture 13 Stability of LU Factorization; Cholesky Factorization. Songting Luo. Department of Mathematics Iowa State University
Lecture 13 Stability of LU Factorization; Cholesky Factorization Songting Luo Department of Mathematics Iowa State University MATH 562 Numerical Analysis II ongting Luo ( Department of Mathematics Iowa
More informationMatrix decompositions
Matrix decompositions How can we solve Ax = b? 1 Linear algebra Typical linear system of equations : x 1 x +x = x 1 +x +9x = 0 x 1 +x x = The variables x 1, x, and x only appear as linear terms (no powers
More informationLU factorization with Panel Rank Revealing Pivoting and its Communication Avoiding version
1 LU factorization with Panel Rank Revealing Pivoting and its Communication Avoiding version Amal Khabou Advisor: Laura Grigori Université Paris Sud 11, INRIA Saclay France SIAMPP12 February 17, 2012 2
More informationSince the determinant of a diagonal matrix is the product of its diagonal elements it is trivial to see that det(a) = α 2. = max. A 1 x.
APPM 4720/5720 Problem Set 2 Solutions This assignment is due at the start of class on Wednesday, February 9th. Minimal credit will be given for incomplete solutions or solutions that do not provide details
More informationComputational Methods. Least Squares Approximation/Optimization
Computational Methods Least Squares Approximation/Optimization Manfred Huber 2011 1 Least Squares Least squares methods are aimed at finding approximate solutions when no precise solution exists Find the
More informationETNA Kent State University
Electronic Transactions on Numerical Analysis. Volume 17, pp. 76-92, 2004. Copyright 2004,. ISSN 1068-9613. ETNA STRONG RANK REVEALING CHOLESKY FACTORIZATION M. GU AND L. MIRANIAN Ý Abstract. For any symmetric
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 informationAMS526: 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 informationSolving linear equations with Gaussian Elimination (I)
Term Projects Solving linear equations with Gaussian Elimination The QR Algorithm for Symmetric Eigenvalue Problem The QR Algorithm for The SVD Quasi-Newton Methods Solving linear equations with Gaussian
More informationETNA Kent State University
C 8 Electronic Transactions on Numerical Analysis. Volume 17, pp. 76-2, 2004. Copyright 2004,. ISSN 1068-613. etnamcs.kent.edu STRONG RANK REVEALING CHOLESKY FACTORIZATION M. GU AND L. MIRANIAN Abstract.
More informationBindel, Fall 2016 Matrix Computations (CS 6210) Notes for
1 A cautionary tale Notes for 2016-10-05 You have been dropped on a desert island with a laptop with a magic battery of infinite life, a MATLAB license, and a complete lack of knowledge of basic geometry.
More informationLinear Algebra Linear Algebra : Matrix decompositions Monday, February 11th Math 365 Week #4
Linear Algebra Linear Algebra : Matrix decompositions Monday, February 11th Math Week # 1 Saturday, February 1, 1 Linear algebra Typical linear system of equations : x 1 x +x = x 1 +x +9x = 0 x 1 +x x
More informationNumerical Methods I Non-Square and Sparse Linear Systems
Numerical Methods I Non-Square and Sparse Linear Systems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fall 2014 September 25th, 2014 A. Donev (Courant
More informationApplied Numerical Linear Algebra. Lecture 8
Applied Numerical Linear Algebra. Lecture 8 1/ 45 Perturbation Theory for the Least Squares Problem When A is not square, we define its condition number with respect to the 2-norm to be k 2 (A) σ max (A)/σ
More informationLinear Algebraic Equations
Linear Algebraic Equations 1 Fundamentals Consider the set of linear algebraic equations n a ij x i b i represented by Ax b j with [A b ] [A b] and (1a) r(a) rank of A (1b) Then Axb has a solution iff
More informationLAPACK-Style Codes for Pivoted Cholesky and QR Updating. Hammarling, Sven and Higham, Nicholas J. and Lucas, Craig. MIMS EPrint: 2006.
LAPACK-Style Codes for Pivoted Cholesky and QR Updating Hammarling, Sven and Higham, Nicholas J. and Lucas, Craig 2007 MIMS EPrint: 2006.385 Manchester Institute for Mathematical Sciences School of Mathematics
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 2: Direct Methods PD Dr.
More informationAM205: Assignment 2. i=1
AM05: Assignment Question 1 [10 points] (a) [4 points] For p 1, the p-norm for a vector x R n is defined as: ( n ) 1/p x p x i p ( ) i=1 This definition is in fact meaningful for p < 1 as well, although
More informationMTH 464: Computational Linear Algebra
MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University February 6, 2018 Linear Algebra (MTH
More informationLinear Algebra Review
Linear Algebra Review CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Doug James (and Justin Solomon) CS 205A: Mathematical Methods Linear Algebra Review 1 / 16 Midterm Exam Tuesday Feb
More informationMain matrix factorizations
Main matrix factorizations A P L U P permutation matrix, L lower triangular, U upper triangular Key use: Solve square linear system Ax b. A Q R Q unitary, R upper triangular Key use: Solve square or overdetrmined
More informationGaussian Elimination without/with Pivoting and Cholesky Decomposition
Gaussian Elimination without/with Pivoting and Cholesky Decomposition Gaussian Elimination WITHOUT pivoting Notation: For a matrix A R n n we define for k {,,n} the leading principal submatrix a a k A
More informationforms Christopher Engström November 14, 2014 MAA704: Matrix factorization and canonical forms Matrix properties Matrix factorization Canonical forms
Christopher Engström November 14, 2014 Hermitian LU QR echelon Contents of todays lecture Some interesting / useful / important of matrices Hermitian LU QR echelon Rewriting a as a product of several matrices.
More informationNumerical 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 informationScientific Computing
Scientific Computing Direct solution methods Martin van Gijzen Delft University of Technology October 3, 2018 1 Program October 3 Matrix norms LU decomposition Basic algorithm Cost Stability Pivoting Pivoting
More information5.6. PSEUDOINVERSES 101. A H w.
5.6. PSEUDOINVERSES 0 Corollary 5.6.4. If A is a matrix such that A H A is invertible, then the least-squares solution to Av = w is v = A H A ) A H w. The matrix A H A ) A H is the left inverse of A and
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 informationProblem set 5: SVD, Orthogonal projections, etc.
Problem set 5: SVD, Orthogonal projections, etc. February 21, 2017 1 SVD 1. Work out again the SVD theorem done in the class: If A is a real m n matrix then here exist orthogonal matrices such that where
More informationSparse BLAS-3 Reduction
Sparse BLAS-3 Reduction to Banded Upper Triangular (Spar3Bnd) Gary Howell, HPC/OIT NC State University gary howell@ncsu.edu Sparse BLAS-3 Reduction p.1/27 Acknowledgements James Demmel, Gene Golub, Franc
More informationAM 205: lecture 8. Last time: Cholesky factorization, QR factorization Today: how to compute the QR factorization, the Singular Value Decomposition
AM 205: lecture 8 Last time: Cholesky factorization, QR factorization Today: how to compute the QR factorization, the Singular Value Decomposition QR Factorization A matrix A R m n, m n, can be factorized
More informationIterative Methods. Splitting Methods
Iterative Methods Splitting Methods 1 Direct Methods Solving Ax = b using direct methods. Gaussian elimination (using LU decomposition) Variants of LU, including Crout and Doolittle Other decomposition
More informationMatrix decompositions
Matrix decompositions How can we solve Ax = b? 1 Linear algebra Typical linear system of equations : x 1 x +x = x 1 +x +9x = 0 x 1 +x x = The variables x 1, x, and x only appear as linear terms (no powers
More informationHomework 2 Foundations of Computational Math 2 Spring 2019
Homework 2 Foundations of Computational Math 2 Spring 2019 Problem 2.1 (2.1.a) Suppose (v 1,λ 1 )and(v 2,λ 2 ) are eigenpairs for a matrix A C n n. Show that if λ 1 λ 2 then v 1 and v 2 are linearly independent.
More informationSubset Selection. Deterministic vs. Randomized. Ilse Ipsen. North Carolina State University. Joint work with: Stan Eisenstat, Yale
Subset Selection Deterministic vs. Randomized Ilse Ipsen North Carolina State University Joint work with: Stan Eisenstat, Yale Mary Beth Broadbent, Martin Brown, Kevin Penner Subset Selection Given: real
More informationMATH 350: Introduction to Computational Mathematics
MATH 350: Introduction to Computational Mathematics Chapter V: Least Squares Problems Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Spring 2011 fasshauer@iit.edu MATH
More informationLinear Algebra Section 2.6 : LU Decomposition Section 2.7 : Permutations and transposes Wednesday, February 13th Math 301 Week #4
Linear Algebra Section. : LU Decomposition Section. : Permutations and transposes Wednesday, February 1th Math 01 Week # 1 The LU Decomposition We learned last time that we can factor a invertible matrix
More informationParallel Numerical Algorithms
Parallel Numerical Algorithms Chapter 6 Matrix Models Section 6.2 Low Rank Approximation Edgar Solomonik Department of Computer Science University of Illinois at Urbana-Champaign CS 554 / CSE 512 Edgar
More informationBlockMatrixComputations and the Singular Value Decomposition. ATaleofTwoIdeas
BlockMatrixComputations and the Singular Value Decomposition ATaleofTwoIdeas Charles F. Van Loan Department of Computer Science Cornell University Supported in part by the NSF contract CCR-9901988. Block
More informationLecture # 11 The Power Method for Eigenvalues Part II. The power method find the largest (in magnitude) eigenvalue of. A R n n.
Lecture # 11 The Power Method for Eigenvalues Part II The power method find the largest (in magnitude) eigenvalue of It makes two assumptions. 1. A is diagonalizable. That is, A R n n. A = XΛX 1 for some
More informationLU Factorization with Panel Rank Revealing Pivoting and Its Communication Avoiding Version
LU Factorization with Panel Rank Revealing Pivoting and Its Communication Avoiding Version Amal Khabou James Demmel Laura Grigori Ming Gu Electrical Engineering and Computer Sciences University of California
More informationLecture 6, Sci. Comp. for DPhil Students
Lecture 6, Sci. Comp. for DPhil Students Nick Trefethen, Thursday 1.11.18 Today II.3 QR factorization II.4 Computation of the QR factorization II.5 Linear least-squares Handouts Quiz 4 Householder s 4-page
More informationHouseholder reflectors are matrices of the form. P = I 2ww T, where w is a unit vector (a vector of 2-norm unity)
Householder QR Householder reflectors are matrices of the form P = I 2ww T, where w is a unit vector (a vector of 2-norm unity) w Px x Geometrically, P x represents a mirror image of x with respect to
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 informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 6
CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 6 GENE H GOLUB Issues with Floating-point Arithmetic We conclude our discussion of floating-point arithmetic by highlighting two issues that frequently
More informationMath 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination
Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column
More informationAMS526: 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 informationAlgorithm 853: an Efficient Algorithm for Solving Rank-Deficient Least Squares Problems
Algorithm 853: an Efficient Algorithm for Solving Rank-Deficient Least Squares Problems LESLIE FOSTER and RAJESH KOMMU San Jose State University Existing routines, such as xgelsy or xgelsd in LAPACK, for
More informationDense LU factorization and its error analysis
Dense LU factorization and its error analysis Laura Grigori INRIA and LJLL, UPMC February 2016 Plan Basis of floating point arithmetic and stability analysis Notation, results, proofs taken from [N.J.Higham,
More informationLecture 4 Orthonormal vectors and QR factorization
Orthonormal vectors and QR factorization 4 1 Lecture 4 Orthonormal vectors and QR factorization EE263 Autumn 2004 orthonormal vectors Gram-Schmidt procedure, QR factorization orthogonal decomposition induced
More informationCommunication Avoiding LU Factorization using Complete Pivoting
Communication Avoiding LU Factorization using Complete Pivoting Implementation and Analysis Avinash Bhardwaj Department of Industrial Engineering and Operations Research University of California, Berkeley
More informationNumerical Methods I Solving Square Linear Systems: GEM and LU factorization
Numerical Methods I Solving Square Linear Systems: GEM and LU factorization Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fall 2014 September 18th,
More informationENGG5781 Matrix Analysis and Computations Lecture 8: QR Decomposition
ENGG5781 Matrix Analysis and Computations Lecture 8: QR Decomposition Wing-Kin (Ken) Ma 2017 2018 Term 2 Department of Electronic Engineering The Chinese University of Hong Kong Lecture 8: QR Decomposition
More informationComputing least squares condition numbers on hybrid multicore/gpu systems
Computing least squares condition numbers on hybrid multicore/gpu systems M. Baboulin and J. Dongarra and R. Lacroix Abstract This paper presents an efficient computation for least squares conditioning
More informationMath 407: Linear Optimization
Math 407: Linear Optimization Lecture 16: The Linear Least Squares Problem II Math Dept, University of Washington February 28, 2018 Lecture 16: The Linear Least Squares Problem II (Math Dept, University
More informationMODULE 7. where A is an m n real (or complex) matrix. 2) Let K(t, s) be a function of two variables which is continuous on the square [0, 1] [0, 1].
Topics: Linear operators MODULE 7 We are going to discuss functions = mappings = transformations = operators from one vector space V 1 into another vector space V 2. However, we shall restrict our sights
More information6 Linear Systems of Equations
6 Linear Systems of Equations Read sections 2.1 2.3, 2.4.1 2.4.5, 2.4.7, 2.7 Review questions 2.1 2.37, 2.43 2.67 6.1 Introduction When numerically solving two-point boundary value problems, the differential
More informationApplied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in
More informationNumerical Linear Algebra
Numerical Linear Algebra Decompositions, numerical aspects Gerard Sleijpen and Martin van Gijzen September 27, 2017 1 Delft University of Technology Program Lecture 2 LU-decomposition Basic algorithm Cost
More informationProgram Lecture 2. Numerical Linear Algebra. Gaussian elimination (2) Gaussian elimination. Decompositions, numerical aspects
Numerical Linear Algebra Decompositions, numerical aspects Program Lecture 2 LU-decomposition Basic algorithm Cost Stability Pivoting Cholesky decomposition Sparse matrices and reorderings Gerard Sleijpen
More informationSection 4.5 Eigenvalues of Symmetric Tridiagonal Matrices
Section 4.5 Eigenvalues of Symmetric Tridiagonal Matrices Key Terms Symmetric matrix Tridiagonal matrix Orthogonal matrix QR-factorization Rotation matrices (plane rotations) Eigenvalues We will now complete
More informationLecture 3: QR-Factorization
Lecture 3: QR-Factorization This lecture introduces the Gram Schmidt orthonormalization process and the associated QR-factorization of matrices It also outlines some applications of this factorization
More informationSome Notes on Least Squares, QR-factorization, SVD and Fitting
Department of Engineering Sciences and Mathematics January 3, 013 Ove Edlund C000M - Numerical Analysis Some Notes on Least Squares, QR-factorization, SVD and Fitting Contents 1 Introduction 1 The Least
More informationImportant Matrix Factorizations
LU Factorization Choleski Factorization The QR Factorization LU Factorization: Gaussian Elimination Matrices Gaussian elimination transforms vectors of the form a α, b where a R k, 0 α R, and b R n k 1,
More informationPivoting. Reading: GV96 Section 3.4, Stew98 Chapter 3: 1.3
Pivoting Reading: GV96 Section 3.4, Stew98 Chapter 3: 1.3 In the previous discussions we have assumed that the LU factorization of A existed and the various versions could compute it in a stable manner.
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 informationA Backward Stable Hyperbolic QR Factorization Method for Solving Indefinite Least Squares Problem
A Backward Stable Hyperbolic QR Factorization Method for Solving Indefinite Least Suares Problem Hongguo Xu Dedicated to Professor Erxiong Jiang on the occasion of his 7th birthday. Abstract We present
More informationNumerical Methods I: Numerical linear algebra
1/3 Numerical Methods I: Numerical linear algebra Georg Stadler Courant Institute, NYU stadler@cimsnyuedu September 1, 017 /3 We study the solution of linear systems of the form Ax = b with A R n n, x,
More informationOrthogonal Transformations
Orthogonal Transformations Tom Lyche University of Oslo Norway Orthogonal Transformations p. 1/3 Applications of Qx with Q T Q = I 1. solving least squares problems (today) 2. solving linear equations
More informationFundamentals of Numerical Linear Algebra
Fundamentals of Numerical Linear Algebra Seongjai Kim Department of Mathematics and Statistics Mississippi State University Mississippi State, MS 39762 USA Email: skim@math.msstate.edu Updated: November
More informationAMS526: 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 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 informationc 2013 Society for Industrial and Applied Mathematics
SIAM J. MATRIX ANAL. APPL. Vol. 34, No. 3, pp. 1401 1429 c 2013 Society for Industrial and Applied Mathematics LU FACTORIZATION WITH PANEL RANK REVEALING PIVOTING AND ITS COMMUNICATION AVOIDING VERSION
More informationReview. Example 1. Elementary matrices in action: (a) a b c. d e f = g h i. d e f = a b c. a b c. (b) d e f. d e f.
Review Example. Elementary matrices in action: (a) 0 0 0 0 a b c d e f = g h i d e f 0 0 g h i a b c (b) 0 0 0 0 a b c d e f = a b c d e f 0 0 7 g h i 7g 7h 7i (c) 0 0 0 0 a b c a b c d e f = d e f 0 g
More informationComputational Methods. Systems of Linear Equations
Computational Methods Systems of Linear Equations Manfred Huber 2010 1 Systems of Equations Often a system model contains multiple variables (parameters) and contains multiple equations Multiple equations
More informationOrthonormal Transformations and Least Squares
Orthonormal Transformations and Least Squares Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo October 30, 2009 Applications of Qx with Q T Q = I 1. solving
More informationParallel Numerical Algorithms
Parallel Numerical Algorithms Chapter 5 Eigenvalue Problems Section 5.1 Michael T. Heath and Edgar Solomonik Department of Computer Science University of Illinois at Urbana-Champaign CS 554 / CSE 512 Michael
More informationScientific Computing: Solving Linear Systems
Scientific Computing: Solving Linear Systems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course MATH-GA.2043 or CSCI-GA.2112, Spring 2012 September 17th and 24th, 2015 A. Donev (Courant
More informationAMS 209, Fall 2015 Final Project Type A Numerical Linear Algebra: Gaussian Elimination with Pivoting for Solving Linear Systems
AMS 209, Fall 205 Final Project Type A Numerical Linear Algebra: Gaussian Elimination with Pivoting for Solving Linear Systems. Overview We are interested in solving a well-defined linear system given
More information