Introduction to PDEs and Numerical Methods Lecture 7. Solving linear systems
|
|
- David Walton
- 5 years ago
- Views:
Transcription
1 Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Introduction to PDEs and Numerical Methods Lecture 7. Solving linear systems Dr. Noemi Friedman,
2 Reminder: Instationary heat equation what to solve? Stability checking from eigenvalue analysis: Method of lines Euler forward method find the eigenvalues (λ j ) and eigenvectors (v j ) of matrix A u n+ = I + ΔtA B u n u n+ = Bu n Euler backward method u n = I ΔtA B u n+ Theta method I θδta u n+ = I + θ ΔtA u n B u n+ = u n B θ u n+ = B 2θ u n Solve system of equations Gx = b solve for x B θ B 2θ Dr. Noemi Friedman PDE lecture Seite 2
3 Reminder: Stationary heat equation what to solve? Conclusion instationary heat equation with implicit FD System of linear equations: methods (Euler backward, Theta method) stationary heat equation Ax = b solve for x Where the A matrix is in general sparse, banded can get very large with refined spatial and temporal discretisation for D heat equation with three-point-stencils: tridiagonal for D heat equation with five-point-stencils: pentadiagonal for 2D heat equation: banded with sparse band Dr. Noemi Friedman PDE lecture Seite
4 Existance and uniquness of solutions to Ax=b Nullspace of A If the only element of the nullspace is the zero vector: trivial nullspace Unique solution: if and only if the nullspace is trivial Infinitive number of solutions: if nullspace is not trivial and b satisfies the compatibility condition (Fredholm alternative): Dr. Noemi Friedman PDE lecture Seite 4
5 How to solve? Direct methods If the matix A is a) Diagonal b) Lower triangular c) Upper triangular d) Orthogonal matrix A T = A e) Tridiagonal matrix Then to solve Ax = b is easy. a) a a a b) a 0 0 a 2 a 22 0 a a 2 a x x 2 x = x x 2 x = b b 2 b b b 2 b 2 x = b a x 2 = b 2 a 22 x = b a x = b a x 2 = f(x ) x = f(x, x 2 ) c) a a 2 a 0 a 22 a a x x 2 x = b b 2 b Backward substitution 2 x = f(x 2, x ) x 2 = f(x ) x = b a Forward substitution Dr. Noemi Friedman PDE lecture Seite 5
6 How to solve? Direct methods.) Solve system of equation directly Calculate inverse from Cramer-rule: x i = det(a i) det(a) Ax = b 2 n +! operations A = a a 2 a 2 a 22 a i a i2 a n a n2 a i a n a 2i a 2n a ii a in a ni a nn A i = a a 2 a 2 a 22 a i a i2 a n a n2 b i a n b 2i a 2n b ii a in b ni a nn Gauß-Jordan elimination Gauß elimination, LU decomposition (Cholesky decomp. If A is symmetric and pos.def., Thomas algorithm, if matrix is tridiagonal) Dr. Noemi Friedman PDE lecture Seite 6
7 Solve Ax=b Gauß-Jordan elimination Gauß-Jordan elimination for solving the system of equation: a x + a 2 x 2 + a x = b () a 2 x + a 22 x 2 + a 2 x = b 2 a x + a 2 x 2 + a x = b () Where a ii and b i are known x, x 2, x =? A = a a 2 a a 2 a 22 a 2 a a 2 a b = b b 2 b x = x x 2 x Ax = b Method: apply elementary transformation on matrix A (multiply both side of the equation with elementary tranformation matrices T i ) to get diagonal matrix. The elementary operations: Switch rows of A Multiply a row with a nonzero scalar Add to one row a scalar multiple of another Dr. Noemi Friedman PDE lecture Seite 7
8 Solve Ax=b Gauß-Jordan elimination I. Devide equation () by a * : x + a 2 a x 2 + a a x = b a () a 2 x + a 22 x 2 + a 2 x = b 2 a x + a 2 x 2 + a x = b () T 0 Ax =T 0 b a 2 a 0 0 a a a T 0 A = a 2 a 22 a 2 T 0 = T b = a a 2 a 0 0 * if a is nonzero, otherwise before this step rows should be switched such that first diagonal element of A is nonzero. If all elements in the first column of matrix A is zero, A= A and skip steps I and II Dr. Noemi Friedman PDE lecture Seite b a b 2 b
9 Solve Ax=b Gauß-Jordan elimination II. Add a 2 times equation () to equation and a times equation () to equation (): x + a 2 a x 2 + a a x = b a () 0 + a 22 a 2 a 2 a x 2 + a 2 a 2 a a x = b 2 a 2 b a 0 + a 2 a a 2 a x 2 + a a a a x = b a b a () T T 0 Ax = T T 0 b T = A T T 0 A = 0 0 a 2 0 a 0 a a 2 a a 2 a 22 a 2 = a a 2 a a 2 a a a 0 a 22 a 2 a 2 a a 2 a 2 a a 0 a 2 a a 2 a a a a a Dr. Noemi Friedman PDE lecture Seite 9
10 Solve Ax=b Gauß-Jordan elimination III. Devide equation by a 22 * T 2 T T 0 Ax = T 2 T T 0 b T 2 T T 0 A = a a 2 a 0 a 2 a 22 0 a 2 a where a = 0 or T 2 = a IV. Add a 2 times equation to equation () and a 2 times equation to equation () (transformation T ) T T 2 T T 0 Ax = T T 2 T T 0 b A T T 2 T T 0 A = a 0 a 0 a a T = a a 2 *if a 22 is zero, 2nd and rd equations should be switched before step III. If all elements in the second column is zero, skip steps III and IV. with A = A Dr. Noemi Friedman PDE lecture Seite 0
11 Solve Ax=b Gauß-Jordan elimination V. Devide equation () by a * (transformation T 4 ) T 4 T T 2 T T 0 Ax =T 4 A = T 4 T T 2 T T 0 b T 4 A = a 0 a 0 a 22 a where a and a 22 = 0 or T 4 = a VI. Add a times equation () to equation () and a 2 times equation () to equation () (transformation T 5 ) T 5 T 4 T T 2 T T 0 Ax = T 5 T 4 T T 2 T T 0 b A T 5 T 4 A = a a *if a is zero skip steps V and VI. T 5 = 0 a a 2 If in A there are zero diagonal elements, matrix A is singular. Otherwise A is the unit matrix, and T 5 T 4 T T 2 T T 0 is the inverse of matrix A ** ** if rows had to be switched in between, the corresponding tranformations has to be inserted in between the T i transformations Dr. Noemi Friedman PDE lecture Seite
12 Solve Ax=b Gauß-Jordan elimination Example Solve with Gauß-Jordan elimination 2x x 2 + 2x = 6 () 4x x 2 + 6x = 20 6x 7x 2 + x = () x, x 2, x =? A = b = 6 20 x = x x 2 x Ax = b I () () devide equation by () () T 0 = Dr. Noemi Friedman PDE lecture Seite 2
13 Solve Ax=b Gauß-Jordan elimination Example II () add -4 times () () add -6 times () () () T = IV. (Second diagonal is, step III. is not needed) () () add 0.5 times add 4 times () () T 2 = V () () Devide by () () T = /0 VI () () add -2 times () add -2 times () () () T 4 = Dr. Noemi Friedman PDE lecture Seite
14 Solve Ax=b Gauß-Jordan elimination x = () x 2 = 2 x = () And the inverse of matrix A: A = T 4 T T 2 T T 0 T 0 = T = T 2 = T = /0 T 4 = A = T 4 T T 2 T T 0 = Note: results can be calculated without the transformation matrices, but they are necessary for the inverse calculation Dr. Noemi Friedman PDE lecture Seite 4
15 Solve Ax=b Gauß elimination, inverse calculation Inverse of A can be calculated also from A Unit matrix Example () () devide equation by () () add -4 times () add -6 times () Dr. Noemi Friedman PDE lecture Seite 5
16 Solve Ax=b Gauß elimination, inverse calculation Example () () add 0.5 times add 4 times () () Devide by () () add -2 times () add -2 times () A = Dr. Noemi Friedman PDE lecture Seite 6
17 Solve Ax=b Gauß elimination Method: same as in Gauß-Jordan elimination, but A is transformed to upper triangular matrix through the elementary operations. Method is presented through the previous example. Example 2x x 2 + 2x = 6 () 4x x 2 + 6x = 20 6x 7x 2 + x = () x, x 2, x =? A = b = 6 20 x = x x 2 x Ax = b Dr. Noemi Friedman PDE lecture Seite 7
18 Solve Ax=b Gauß elimination Example () () I. devide equation by 2 II III. IV () add -4 times () () add -6 times () () () () () add 4 times Devide by () () () () () () () () T 0 = T = 2 T 2 = T = Dr. Noemi Friedman PDE lecture Seite /0
19 Solve Ax=b Gauß elimination, LU decomposition Example x 0.5x 2 + x = () x 2 + 2x = x = () If equation Ax=b has to be solved for several b: Using back-substitution, each x i can be solved for () x = x 2 = 2x = 2 = 2 () x = + 0.5x 2 x = = T 0 - T - T 2 - T - T T 2 T T 0 A = A = LU L: lower triangular matrix U: upper triangular matrix.) Calculate L and U 2.) Solve Ax=LUx=b in two steps: solve y from Ly = b where y = Ux (equation can be solved with forward substitution) solve x from Ux= y (equation can be solved with backward substitution) Note: LU decomposition can be calculated from different algorithms Dr. Noemi Friedman PDE lecture Seite 9
20 Stationary and instationary heat equation how to solve?.) Solve system of equation directly Ax = b Calculate inverse from Cramer-rule: x i = det(a i) det(a) 2 n +! operations Gauß-Jordan elimination Gauß elimination, LU decomposition (Cholesky decomp. If G is symmetric and pos.def., Thomas algorithm, if matrix is tridiagonal) A = L U 2n 2 operations Ax = L Ux y Ly = b Ux = y = b forward substitution back substitution Dr. Noemi Friedman PDE lecture Seite 20
21 Stationary and instationary heat equation how to solve? Direct solve A = LU LU factorisation with Gauß method a a 2 a 2 a 22 = l 0 l 2 l 22 u u 2 0 u 22 a = l u g 2 = l u 2 a 2 = l 2 u g 22 = l 2 u 2 + l 22 u 22 4 equations 6 unknowns 0 l 2 u u 2 0 u 22 a = u a 2 = u 2 0 l 2 a a 2 0 u 22 a 2 = l 2 a a 22 = l 2 a 2 + u 22 0 a 2 a a a 2 0 a 22 l 2 a Dr. Noemi Friedman PDE lecture Seite 2
22 Stationary and instationary heat equation how to solve? Direct solve Gauß factorization A = LU in general n 2 equations n 2 l + n unknowns ii = General algorithm for i = k + n l ik = a ik (k) a kk (k) for j = k +.. n a ij (k+) = a ij (k) l ik a kj (k) ~ 2n operations But even if the matrix is nonsingular the elements g kk (k) (pivot elements) can be zero Pivoting (flip rows or columns) can be also important for reducing roundoff errors g kk (k) won t be zero if the matrix is positivive definit or if it is diagonally dominant Dr. Noemi Friedman PDE lecture Seite 22
23 Stationary and instationary heat equation how to solve? Direct solve If A is positive definit+symmetric Tridiagonal system: A = A = LU = HH T a c e 2 a 2 c 2 e n a n L = LU factorisation with Cholesky decomposition Thomas algorithm β 2 β n U = ~ n operations (half of the Gauß method) α c α 2 c 2 α n α = a β i = e i α α i = a i β i c i i = 2 n What happens with the roundoff errors in A = LU = A + δa A Ax = b A + δa x = b + δb x x x λ max λ min δb b O(n) operations λ max λ min = K(A) Dr. Noemi Friedman PDE lecture Seite 2
24 Iterative methods illustrative example the Jacobi method x y z = x + y = 0 x + 4y + z = 2 y + 4z = 0 x = (0 y) 4 y = (2 x z) 4 z = (0 y) 4 x (0) y (0) z (0) : = x () y () z () : = The true solution: x () = 4 0 y 0 = 2.5 y () = 4 2 x 0 z 0 = z () = 4 0 y 0 = 2.5 x y z : = x y z = x () y () z () : = x (k+) = 4 0 y k = 2.5 y (k+) = 4 2 x k z k = z (k+) = 4 0 y k = 2.5 x i (k+) = a ii b i j i a ij x j (k) Dr. Noemi Friedman PDE lecture Seite 24
25 Iterative methods illustrative example the Gauß-Seidel method x y z = x + y = 0 x + 4y + z = 2 y + 4z = 0 x = (0 y) 4 y = (2 x z) 4 z = (0 y) 4 x (0) y (0) z (0) : = x () y () z () : = The true solution: x () = 4 0 y 0 = 2.5 y () = 4 2 x z 0 = 2.25 z () = 4 0 y =.975 x y z = x i (k+) = a ii x (k+) = 4 0 y k = 2.5 y (k+) = 4 2 x k+ z k = z (k+) = 4 0 y k+ = 2.5 (k) b i a ij x j(k+) a ij x j j<i j>i Dr. Noemi Friedman PDE lecture Seite 25
26 Iterative methods illustrative example the Jacobi method A = a a 2 a 2 a 22 a i a i2 a n a n2 a i a n a 2i a 2n a ii a in a ni a nn = D + E + F x i (k+) = a ii b i j i a ij x j (k) a a 22 a i a n a 2i a 2n a ii a nn x k+ x 2 k+ x i k+ x n k+ = b b 2 b i b n 0 a 2 a 2 0 a i a i2 a n a n2 a i a n a 2i a 2n 0 a in a ni 0 x k x 2 k x i k x n k D E + F x (k+) = D b E + F x (k) Dr. Noemi Friedman PDE lecture Seite 26
27 Iterative methods illustrative example the Jacobi method A = a a 2 a 2 a 22 a i a i2 a n a n2 a i a n a 2i a 2n a ii a in a ni a nn = D + E + F x i (k+) = a ii j<i b i a ij x j (k+) + aii x i (k+) = j<i a ij x j(k+) j>i bi j>i a ij x j (k) a ij x j (k) Ex (k+) + Dx (k+) = b Fx (k) E + D x (k+) = b Fx (k) x (k+) = E + D b Fx (k) Dr. Noemi Friedman PDE lecture Seite 27
Introduction to PDEs and Numerical Methods Tutorial 5. Finite difference methods equilibrium equation and iterative solvers
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Introduction to PDEs and Numerical Methods Tutorial 5. Finite difference methods equilibrium equation and iterative solvers Dr. Noemi
More informationIntroduction to PDEs and Numerical Methods Tutorial 1: Overview of essential linear algebra and analysis
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Introduction to PDEs and Numerical Methods Tutorial 1: Overview of essential linear algebra and analysis Dr. Noemi Friedman, 25.10.201.
More informationPlatzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Introduction to PDEs and Numerical Methods Lecture 6: Numerical solution of the heat equation with FD method: method of lines, Euler
More informationIntroduction to PDEs and Numerical Methods Tutorial 4. Finite difference methods stability, concsistency, convergence
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Introduction to PDEs and Numerical Methods Tutorial 4. Finite difference methods stability, concsistency, convergence Dr. Noemi Friedman,
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 informationThe Solution of Linear Systems AX = B
Chapter 2 The Solution of Linear Systems AX = B 21 Upper-triangular Linear Systems We will now develop the back-substitution algorithm, which is useful for solving a linear system of equations that has
More informationDepartment of Mathematics California State University, Los Angeles Master s Degree Comprehensive Examination in. NUMERICAL ANALYSIS Spring 2015
Department of Mathematics California State University, Los Angeles Master s Degree Comprehensive Examination in NUMERICAL ANALYSIS Spring 2015 Instructions: Do exactly two problems from Part A AND two
More informationLU Factorization. LU Decomposition. LU Decomposition. LU Decomposition: Motivation A = LU
LU Factorization To further improve the efficiency of solving linear systems Factorizations of matrix A : LU and QR LU Factorization Methods: Using basic Gaussian Elimination (GE) Factorization of Tridiagonal
More informationCOURSE Numerical methods for solving linear systems. Practical solving of many problems eventually leads to solving linear systems.
COURSE 9 4 Numerical methods for solving linear systems Practical solving of many problems eventually leads to solving linear systems Classification of the methods: - direct methods - with low number of
More informationCS412: Lecture #17. Mridul Aanjaneya. March 19, 2015
CS: Lecture #7 Mridul Aanjaneya March 9, 5 Solving linear systems of equations Consider a lower triangular matrix L: l l l L = l 3 l 3 l 33 l n l nn A procedure similar to that for upper triangular systems
More informationNumerical methods for PDEs FEM convergence, error estimates, piecewise polynomials
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Numerical methods for PDEs FEM convergence, error estimates, piecewise polynomials Dr. Noemi Friedman Contents of the course Fundamentals
More informationGaussian Elimination and Back Substitution
Jim Lambers MAT 610 Summer Session 2009-10 Lecture 4 Notes These notes correspond to Sections 31 and 32 in the text Gaussian Elimination and Back Substitution The basic idea behind methods for solving
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 informationReview of matrices. Let m, n IN. A rectangle of numbers written like A =
Review of matrices Let m, n IN. A rectangle of numbers written like a 11 a 12... a 1n a 21 a 22... a 2n A =...... a m1 a m2... a mn where each a ij IR is called a matrix with m rows and n columns or an
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 informationNumerical methods for PDEs FEM convergence, error estimates, piecewise polynomials
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Numerical methods for PDEs FEM convergence, error estimates, piecewise polynomials Dr. Noemi Friedman Contents of the course Fundamentals
More informationEngineering Computation
Engineering Computation Systems of Linear Equations_1 1 Learning Objectives for Lecture 1. Motivate Study of Systems of Equations and particularly Systems of Linear Equations. Review steps of Gaussian
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 informationSolving Linear Systems of Equations
November 6, 2013 Introduction The type of problems that we have to solve are: Solve the system: A x = B, where a 11 a 1N a 12 a 2N A =.. a 1N a NN x = x 1 x 2. x N B = b 1 b 2. b N To find A 1 (inverse
More informationSolving Linear Systems of Equations
1 Solving Linear Systems of Equations Many practical problems could be reduced to solving a linear system of equations formulated as Ax = b This chapter studies the computational issues about directly
More informationToday s class. Linear Algebraic Equations LU Decomposition. Numerical Methods, Fall 2011 Lecture 8. Prof. Jinbo Bi CSE, UConn
Today s class Linear Algebraic Equations LU Decomposition 1 Linear Algebraic Equations Gaussian Elimination works well for solving linear systems of the form: AX = B What if you have to solve the linear
More informationNext topics: Solving systems of linear equations
Next topics: Solving systems of linear equations 1 Gaussian elimination (today) 2 Gaussian elimination with partial pivoting (Week 9) 3 The method of LU-decomposition (Week 10) 4 Iterative techniques:
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 informationLINEAR SYSTEMS (11) Intensive Computation
LINEAR SYSTEMS () Intensive Computation 27-8 prof. Annalisa Massini Viviana Arrigoni EXACT METHODS:. GAUSSIAN ELIMINATION. 2. CHOLESKY DECOMPOSITION. ITERATIVE METHODS:. JACOBI. 2. GAUSS-SEIDEL 2 CHOLESKY
More informationPlatzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen PARAMetric UNCertainties, Budapest STOCHASTIC PROCESSES AND FIELDS Noémi Friedman Institut für Wissenschaftliches Rechnen, wire@tu-bs.de
More informationLinear Algebra March 16, 2019
Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented
More informationCS 246 Review of Linear Algebra 01/17/19
1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector
More informationDirect Methods for Solving Linear Systems. Simon Fraser University Surrey Campus MACM 316 Spring 2005 Instructor: Ha Le
Direct Methods for Solving Linear Systems Simon Fraser University Surrey Campus MACM 316 Spring 2005 Instructor: Ha Le 1 Overview General Linear Systems Gaussian Elimination Triangular Systems The LU Factorization
More informationVectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes.
Vectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes Matrices and linear equations A matrix is an m-by-n array of numbers A = a 11 a 12 a 13 a 1n a 21 a 22 a 23 a
More informationLecture 11. Linear systems: Cholesky method. Eigensystems: Terminology. Jacobi transformations QR transformation
Lecture Cholesky method QR decomposition Terminology Linear systems: Eigensystems: Jacobi transformations QR transformation Cholesky method: For a symmetric positive definite matrix, one can do an LU decomposition
More informationIntroduction to PDEs and Numerical Methods Tutorial 11. 2D elliptic equations
Platzalter für Bild, Bild auf Titelfolie inter das Logo einsetzen Introduction to PDEs and Numerical Metods Tutorial 11. 2D elliptic equations Dr. Noemi Friedman, 3. 1. 215. Overview Introduction (classification
More informationMATHEMATICS FOR COMPUTER VISION WEEK 2 LINEAR SYSTEMS. Dr Fabio Cuzzolin MSc in Computer Vision Oxford Brookes University Year
1 MATHEMATICS FOR COMPUTER VISION WEEK 2 LINEAR SYSTEMS Dr Fabio Cuzzolin MSc in Computer Vision Oxford Brookes University Year 2013-14 OUTLINE OF WEEK 2 Linear Systems and solutions Systems of linear
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 informationDEN: Linear algebra numerical view (GEM: Gauss elimination method for reducing a full rank matrix to upper-triangular
form) Given: matrix C = (c i,j ) n,m i,j=1 ODE and num math: Linear algebra (N) [lectures] c phabala 2016 DEN: Linear algebra numerical view (GEM: Gauss elimination method for reducing a full rank matrix
More informationDirect Methods for Solving Linear Systems. Matrix Factorization
Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011
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 informationMATH 3511 Lecture 1. Solving Linear Systems 1
MATH 3511 Lecture 1 Solving Linear Systems 1 Dmitriy Leykekhman Spring 2012 Goals Review of basic linear algebra Solution of simple linear systems Gaussian elimination D Leykekhman - MATH 3511 Introduction
More informationNumerical 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 information9. Numerical linear algebra background
Convex Optimization Boyd & Vandenberghe 9. Numerical linear algebra background matrix structure and algorithm complexity solving linear equations with factored matrices LU, Cholesky, LDL T factorization
More information1.Chapter Objectives
LU Factorization INDEX 1.Chapter objectives 2.Overview of LU factorization 2.1GAUSS ELIMINATION AS LU FACTORIZATION 2.2LU Factorization with Pivoting 2.3 MATLAB Function: lu 3. CHOLESKY FACTORIZATION 3.1
More information1 Multiply Eq. E i by λ 0: (λe i ) (E i ) 2 Multiply Eq. E j by λ and add to Eq. E i : (E i + λe j ) (E i )
Direct Methods for Linear Systems Chapter Direct Methods for Solving Linear Systems Per-Olof Persson persson@berkeleyedu Department of Mathematics University of California, Berkeley Math 18A Numerical
More informationNumerical Solution Techniques in Mechanical and Aerospace Engineering
Numerical Solution Techniques in Mechanical and Aerospace Engineering Chunlei Liang LECTURE 3 Solvers of linear algebraic equations 3.1. Outline of Lecture Finite-difference method for a 2D elliptic PDE
More information5 Solving Systems of Linear Equations
106 Systems of LE 5.1 Systems of Linear Equations 5 Solving Systems of Linear Equations 5.1 Systems of Linear Equations System of linear equations: a 11 x 1 + a 12 x 2 +... + a 1n x n = b 1 a 21 x 1 +
More informationSolution of Linear Equations
Solution of Linear Equations (Com S 477/577 Notes) Yan-Bin Jia Sep 7, 07 We have discussed general methods for solving arbitrary equations, and looked at the special class of polynomial equations A subclass
More informationAMS526: 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 informationFundamentals of Engineering Analysis (650163)
Philadelphia University Faculty of Engineering Communications and Electronics Engineering Fundamentals of Engineering Analysis (6563) Part Dr. Omar R Daoud Matrices: Introduction DEFINITION A matrix is
More information2.1 Gaussian Elimination
2. Gaussian Elimination A common problem encountered in numerical models is the one in which there are n equations and n unknowns. The following is a description of the Gaussian elimination method for
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 informationLECTURES IN BASIC COMPUTATIONAL NUMERICAL ANALYSIS
Ax = b Z b a " 1 f(x) dx = h 2 (f X 1 + f n )+ f i #+ O(h 2 ) n 1 i=2 LECTURES IN BASIC COMPUTATIONAL NUMERICAL ANALYSIS x (m+1) = x (m) J(x (m) ) 1 F (x (m) ) p n (x) = X n+1 i=1 " n+1 Y j=1 j6=i # (x
More informationLinear Algebra. Solving Linear Systems. Copyright 2005, W.R. Winfrey
Copyright 2005, W.R. Winfrey Topics Preliminaries Echelon Form of a Matrix Elementary Matrices; Finding A -1 Equivalent Matrices LU-Factorization Topics Preliminaries Echelon Form of a Matrix Elementary
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 informationScientific Computing: Dense Linear Systems
Scientific Computing: Dense Linear Systems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course MATH-GA.2043 or CSCI-GA.2112, Spring 2012 February 9th, 2012 A. Donev (Courant Institute)
More informationRoundoff Analysis of Gaussian Elimination
Jim Lambers MAT 60 Summer Session 2009-0 Lecture 5 Notes These notes correspond to Sections 33 and 34 in the text Roundoff Analysis of Gaussian Elimination In this section, we will perform a detailed error
More informationLinear Equations and Matrix
1/60 Chia-Ping Chen Professor Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Gaussian Elimination 2/60 Alpha Go Linear algebra begins with a system of linear
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 information5. Direct Methods for Solving Systems of Linear Equations. They are all over the place...
5 Direct Methods for Solving Systems of Linear Equations They are all over the place Miriam Mehl: 5 Direct Methods for Solving Systems of Linear Equations They are all over the place, December 13, 2012
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 informationPractical Linear Algebra: A Geometry Toolbox
Practical Linear Algebra: A Geometry Toolbox Third edition Chapter 12: Gauss for Linear Systems Gerald Farin & Dianne Hansford CRC Press, Taylor & Francis Group, An A K Peters Book www.farinhansford.com/books/pla
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More information1 Positive definiteness and semidefiniteness
Positive definiteness and semidefiniteness Zdeněk Dvořák May 9, 205 For integers a, b, and c, let D(a, b, c) be the diagonal matrix with + for i =,..., a, D i,i = for i = a +,..., a + b,. 0 for i = a +
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 informationMAT Linear Algebra Collection of sample exams
MAT 342 - Linear Algebra Collection of sample exams A-x. (0 pts Give the precise definition of the row echelon form. 2. ( 0 pts After performing row reductions on the augmented matrix for a certain system
More informationExample: Current in an Electrical Circuit. Solving Linear Systems:Direct Methods. Linear Systems of Equations. Solving Linear Systems: Direct Methods
Example: Current in an Electrical Circuit Solving Linear Systems:Direct Methods A number of engineering problems or models can be formulated in terms of systems of equations Examples: Electrical Circuit
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 informationNumerical Analysis: Solving Systems of Linear Equations
Numerical Analysis: Solving Systems of Linear Equations Mirko Navara http://cmpfelkcvutcz/ navara/ Center for Machine Perception, Department of Cybernetics, FEE, CTU Karlovo náměstí, building G, office
More informationMath Linear Algebra Final Exam Review Sheet
Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of
More information. =. a i1 x 1 + a i2 x 2 + a in x n = b i. a 11 a 12 a 1n a 21 a 22 a 1n. i1 a i2 a in
Vectors and Matrices Continued Remember that our goal is to write a system of algebraic equations as a matrix equation. Suppose we have the n linear algebraic equations a x + a 2 x 2 + a n x n = b a 2
More informationLinear Algebraic Equations
Linear Algebraic Equations Linear Equations: a + a + a + a +... + a = c 11 1 12 2 13 3 14 4 1n n 1 a + a + a + a +... + a = c 21 2 2 23 3 24 4 2n n 2 a + a + a + a +... + a = c 31 1 32 2 33 3 34 4 3n n
More informationBoundary Value Problems - Solving 3-D Finite-Difference problems Jacob White
Introduction to Simulation - Lecture 2 Boundary Value Problems - Solving 3-D Finite-Difference problems Jacob White Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy Outline Reminder about
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 informationCS227-Scientific Computing. Lecture 4: A Crash Course in Linear Algebra
CS227-Scientific Computing Lecture 4: A Crash Course in Linear Algebra Linear Transformation of Variables A common phenomenon: Two sets of quantities linearly related: y = 3x + x 2 4x 3 y 2 = 2.7x 2 x
More information9. Numerical linear algebra background
Convex Optimization Boyd & Vandenberghe 9. Numerical linear algebra background matrix structure and algorithm complexity solving linear equations with factored matrices LU, Cholesky, LDL T factorization
More information(Linear equations) Applied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB (Linear equations) Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots
More informationMATH2210 Notebook 2 Spring 2018
MATH2210 Notebook 2 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009 2018 by Jenny A. Baglivo. All Rights Reserved. 2 MATH2210 Notebook 2 3 2.1 Matrices and Their Operations................................
More informationMatrices and systems of linear equations
Matrices and systems of linear equations Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T.
More informationA Review of Matrix Analysis
Matrix Notation Part Matrix Operations Matrices are simply rectangular arrays of quantities Each quantity in the array is called an element of the matrix and an element can be either a numerical value
More informationLinear Algebra and Matrix Inversion
Jim Lambers MAT 46/56 Spring Semester 29- Lecture 2 Notes These notes correspond to Section 63 in the text Linear Algebra and Matrix Inversion Vector Spaces and Linear Transformations Matrices are much
More informationLU Factorization. Marco Chiarandini. DM559 Linear and Integer Programming. Department of Mathematics & Computer Science University of Southern Denmark
DM559 Linear and Integer Programming LU Factorization Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark [Based on slides by Lieven Vandenberghe, UCLA] Outline
More informationCPE 310: Numerical Analysis for Engineers
CPE 310: Numerical Analysis for Engineers Chapter 2: Solving Sets of Equations Ahmed Tamrawi Copyright notice: care has been taken to use only those web images deemed by the instructor to be in the public
More informationCHAPTER 6. Direct Methods for Solving Linear Systems
CHAPTER 6 Direct Methods for Solving Linear Systems. Introduction A direct method for approximating the solution of a system of n linear equations in n unknowns is one that gives the exact solution to
More informationChapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations
Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2
More informationChapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in
Chapter 4 - MATRIX ALGEBRA 4.1. Matrix Operations A a 11 a 12... a 1j... a 1n a 21. a 22.... a 2j... a 2n. a i1 a i2... a ij... a in... a m1 a m2... a mj... a mn The entry in the ith row and the jth column
More informationTMA4125 Matematikk 4N Spring 2017
Norwegian University of Science and Technology Institutt for matematiske fag TMA15 Matematikk N Spring 17 Solutions to exercise set 1 1 We begin by writing the system as the augmented matrix.139.38.3 6.
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 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 informationApplied Linear Algebra
Applied Linear Algebra Gábor P. Nagy and Viktor Vígh University of Szeged Bolyai Institute Winter 2014 1 / 262 Table of contents I 1 Introduction, review Complex numbers Vectors and matrices Determinants
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 informationII. Determinant Functions
Supplemental Materials for EE203001 Students II Determinant Functions Chung-Chin Lu Department of Electrical Engineering National Tsing Hua University May 22, 2003 1 Three Axioms for a Determinant Function
More informationMH1200 Final 2014/2015
MH200 Final 204/205 November 22, 204 QUESTION. (20 marks) Let where a R. A = 2 3 4, B = 2 3 4, 3 6 a 3 6 0. For what values of a is A singular? 2. What is the minimum value of the rank of A over all a
More informationLecture 16 Methods for System of Linear Equations (Linear Systems) Songting Luo. Department of Mathematics Iowa State University
Lecture 16 Methods for System of Linear Equations (Linear Systems) Songting Luo Department of Mathematics Iowa State University MATH 481 Numerical Methods for Differential Equations Songting Luo ( Department
More informationNumerical Linear Algebra Homework Assignment - Week 2
Numerical Linear Algebra Homework Assignment - Week 2 Đoàn Trần Nguyên Tùng Student ID: 1411352 8th October 2016 Exercise 2.1: Show that if a matrix A is both triangular and unitary, then it is diagonal.
More informationChapter 7. Tridiagonal linear systems. Solving tridiagonal systems of equations. and subdiagonal. E.g. a 21 a 22 a A =
Chapter 7 Tridiagonal linear systems The solution of linear systems of equations is one of the most important areas of computational mathematics. A complete treatment is impossible here but we will discuss
More informationPowerPoints organized by Dr. Michael R. Gustafson II, Duke University
Part 3 Chapter 10 LU Factorization PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
More informationChapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1
More informationChapter 7 Iterative Techniques in Matrix Algebra
Chapter 7 Iterative Techniques in Matrix Algebra Per-Olof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 128B Numerical Analysis Vector Norms Definition
More informationLinear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.
Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the
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 informationMath 313 Chapter 1 Review
Math 313 Chapter 1 Review Howard Anton, 9th Edition May 2010 Do NOT write on me! Contents 1 1.1 Introduction to Systems of Linear Equations 2 2 1.2 Gaussian Elimination 3 3 1.3 Matrices and Matrix Operations
More information2. Linear algebra. matrices and vectors. linear equations. range and nullspace of matrices. function of vectors, gradient and Hessian
FE661 - Statistical Methods for Financial Engineering 2. Linear algebra Jitkomut Songsiri matrices and vectors linear equations range and nullspace of matrices function of vectors, gradient and Hessian
More information