Linear Algebraic Equations
|
|
- Jocelin Tabitha Brown
- 5 years ago
- Views:
Transcription
1 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 the rank of A and of A b satisfy r(a) r(a b ) (2) If r(a) r(a b ) k < n, there are n k variables that may be assigned arbitrarily. If k n, then there is a unique solution. A straightforward way of solving for x is to evaluate the inverse of A, A 1, using Cramer s method. However, this is computationally infeasible because of the evaluation of so many determinants. Instead we must develop other methods. The questions that must be considered how fast is the method for large n? how accurate is the method because of roundoff? There are two types of problems in terms of the matrix A 1) filled with n small usually solved with direct methods 2) sparse with n large usually solved with iterative methods 1.1 Ill Conditioned Matrices Consider the solution given by x A 1 b. The inverse is constructed using cofactors, A ij 1 (i+j) M ij where M ij is the ij minor. The elements a 1 ji A ij / A where A is the determinant of A. Suppose that a 1 ji has a large element. Then A ij must be large in relation to A. Now one of the terms in the expansion of the determinant is a ij A ij. Thus a small error in a ij when multiplied by A ij, which is large, will cause a large error in A and thus in a 1 ji and thus in the solution x. Similarly, a small error or change in b will also cause a large error in x. 1
2 1.2 Residuals Let us define the computed solution as x c and the residuals r by thus r b Ax c Ax Ax c A(x x c ) (3a) x x c A 1 r (3b) Consequently while small values of x x c lead to small residuals, r, small residuals do not guarantee good solutions. If A 1 has large elements, i.e., is ill-conditioned, then x c will have large errors. That is The residuals r are not a good measure of the accuracy of the solution The question is how do we measure the degree of ill-conditioning of a matrix? Consider the equations whose solution is 2x + 6y 8 2x y (4a) (4b) Now consider whose solution is x 1, y 1 2x + 6y 8 2x y (4c) (5a) (5b) x 10, y 2 (5c) The determinant of A and the inverse is ( A ) (6) So it looks like the determinant is a good measure of ill-conditioning. However, consider x y 8 (7a) x y (7b) (7c) 2
3 and the determinant of A A small relative change in a 22 and in b 2 will produce the same gross change in the answer, but now the elements of A 1 are all of the order of 1. ( ) A (8) Thus the determinant measures ill-conditioning only if the maximum element of A is of the order of 1, that is the matrix is normalized. 2 Direct Methods: Direct methods operate on the elements of the matrix to either solve for x or to find the inverse, A 1. There are several direct methods which differ slightly in the order of the operations and the number of additions, multiplications, and divisions. Several popular algorithms are 1. Gaussian Elimination 2. Gauss-Jordan 3. Crout Method 4. LU decomposition 5. Cholesky The decision of what method is often based upon the number of arithmetic operations. For Gaussian elimination, we have n, n 3 /3 + n 2 /2 n/3, n 3 /3 + n 2 5n/6 division, multiplications and additions to solve Ax b and n, n 3 1, n 3 2n 2 + n operations to find the inverse. 2.1 Gaussian Elimination Consider a 3x3 matrix, A and a right hand vector, b a 0 11 a 0 12 a 0 13 b 0 1 A a 0 21 a 0 22 a 0 23 b 0 2 (9) a 0 31 a 0 32 a 0 33 b 0 3 Multiply row 1 by a 0 21 /a0 11 and subtract from row 2. Then repeat this for row 3. This gives a 0 11 a 0 12 a 0 13 b a 1 22 a 1 23 b a 1 32 a 1 33 b 1 3 (10) 3
4 Now multiply row 2 by a 1 32/a 1 22 and subtract from row 3, a 0 11 a 0 12 a 0 13 b a 1 22 a 1 23 b 1 2 (11) 0 0 a 2 33 b 2 3 We now have a upper triangular matrix that is easy to solve. The production of this upper triangular matrix is termed forward reduction and the solving for x is termed back substitution. 2.2 Multiple Right Hand Sides To solve a new problem with the same matrix A but with a new right hand side, b, can be done easily. All that we have to do is to repeat the operations on the new b. This requires that we store the multipliers used in the reduction. That is store the multipliers, a 0 21 /a0 11, a0 31 /a0 11, a1 32 /a1 22 a 0 21/a 0 11 a 0 31/a 0 11 a 1 32/a 1 22 (12) These can be stored in the reduced matrix since we know that the terms below the diagonal are zero. a 0 11 a 0 12 a 0 13 a 0 21/a 0 11 a 1 22 a 1 23 (13) a 0 31 /a0 11 a 1 32 /a1 22 a Pivoting Now if during the reduction, one of the diagonal terms becomes 0 the reduction must stop, or if it becomes very small significant errors are likely to be generated. To avoid this we use a process known as pivoting Partial Pivoting At each stage of the reduction we compare the diagonal a k 1 kk with the entries in the column a k 1 jk, k < j n and choose the row j with the largest column element and interchange the two rows. This requires keeping track of the interchanges and interchanging the elements of b Complete Pivoting At each stage of the reduction, interchange rows and columns to put the largest element, aij k 1, i, j > k in the diagonal position. This requires time consuming interchanges and complex bookkeeping 4
5 2.4 Gauss-Jordan Method After completing the forward reduction, do a backward reduction to zero all of the non diagonal elements, a diagonal matrix to solve. If we consider the forward reduction as the operaton of a matrix F on A and the backward reduction as B we can write FA U, BFA D, D 1 BFA I (14) Then clearly D 1 BF A 1. If we operate on the matrix [A I] then when we are through, [A] will be [I] and [I] will be [A 1 ]. 2.5 Gaussian Elimination again It is easier to show the Gaussian Elimination method using a different approach. Let E 1 be the operation that produces zeroes in the first column of A, E 2 that does it for the 2nd columns, etc. We then have E 1 and then with E a 21 /a a 31 /a a 1 32/a Since E E 2 E 1 A U we can write E 1 A we get E 2 E 1 A a 11 a 12 a a 1 22 a 1 32 a 1 23 a 1 33 a 11 a 12 a 13 0 a 1 22 a a 2 33 (15) U (16) EAx Eb, or Ux Eb c (17) If we have a new right hand side, b, then simply multiply it by E and solve U x c. But we don t want to have to store the matrix E! LU Decomposition Since EA U, A E 1 U. What is E 1? E E 2 E 1, therefore E 1 E1 1 E 1 2. Now E 1 1 E 1A A and since E 1 subtracts multiples of row 1 from the other rows we we see that E1 1 must add multiples of row 1, that is and thus E a 21 /a a 31 /a and E a 1 32/a (18) 5
6 E 1 E 1 1 E a 21 /a a 31 /a 11 a 1 32/a L (19) L is easy to form since it is nothing but the multipliers used in the forward reduction and since so we have A E 1 U LU Eb c b E 1 C Lc (20a) (20b) (20c) Lc b Ux c (20d) (20e) Because L is a lower triangular matrix, it is easy to solve for c, and likewise because U is an upper triangular matrix it is easy to solve for x Useful Theorems a) L and U are unique when based on E. If instead of using E, we write LU A, since A has n 2 entries and L and U have a total of n 2 + n elements, n of these can be chosen arbitrarily. It is common to choose either L or U to have ones on the diagonal. b) det(a) det(u) u 11 u 22...u nn. Thus if det(a) 0 at least one diagonal element of U must be zero if row interchanges are necessary, A LU. We can still find a U, but not L 2.6 Pivoting and Roundoff Let [ x1 1 x 2 2 { x x (21) We have [ Lc ]{ { c1 1 c 2 2 { c c (22) 6
7 [ Ux 0 1 x1 x { x x (23) which is correct. Now suppose that the base β 10 and the precision, n 3. Then 9999 cannot be represented, it will be rounded to and we have Lc [ c1 1 c 2 2 { x1 0 and after rounding c gives x 2 1 (24) and while x 2 is correct to 3 decimal places, x 1 is 100% in error Complete Pivoting This amounts to reordering the equations and for this problem we write the matrix, Eq. (25), as [ x1 2 x 2 1 { x x (25) which gives Lc, U, and c as [ ]{ { 1 0 c1 2 Lc c 2 1 and we obtain the correct answer. { c1 2. c (26) Ux [ x1 x { x x (27) Now if the precision is n 3 we have [ Lc c1 2 c 2 1 { x1 1 and after rounding c gives x 2 1 (28) and the answers are correct to 3 decimal places. 7
8 2.6.2 Partial Pivoting Partial pivoting means that we exchange rows of the matrix and of the right hand side as the reduction proceeds. Doing so gives the correct answer with [ LU ] [ ] (29) However, with a precision of 3, the value 9999 will be rounded to and we find { x1 0 x 2 1 (30) 3 Condition Number Let us suppose that b is known exaclty, but that the matrix A has some small errors due to rounding. This will produce small errors in the result, x. x + δx (A + δa) 1 b δx (A + δa) 1 b A 1 b ((A + δa) 1 A 1 )b (31a) (31b) from matrix theory and letting B A + δa we have B 1 A 1 A 1 (A B)B 1 (32) δx A 1 ( δa)(a + δa) 1 b (33a) A 1 δa(x + δx) (33b) or in terms of the norms δx A 1 1 δa δa x + δx A A x + δx A δx x + δx cond(a) δa A (34a) (34b) where we have defined A A 1 as the condition number of A. If A is known exactly, but b has uncertainty δx x cond(a) δb b (35) Noer that if b is known to d decimal places, δb / b is around d or d 1 decimal places and δx / x will be known to 10 d cond(a). 8
COURSE 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 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 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 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 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 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 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 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 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 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 informationLinear Systems of n equations for n unknowns
Linear Systems of n equations for n unknowns In many application problems we want to find n unknowns, and we have n linear equations Example: Find x,x,x such that the following three equations hold: x
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 informationLecture 9. Errors in solving Linear Systems. J. Chaudhry (Zeb) Department of Mathematics and Statistics University of New Mexico
Lecture 9 Errors in solving Linear Systems J. Chaudhry (Zeb) Department of Mathematics and Statistics University of New Mexico J. Chaudhry (Zeb) (UNM) Math/CS 375 1 / 23 What we ll do: Norms and condition
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 informationIntroduction to PDEs and Numerical Methods Lecture 7. Solving linear systems
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Introduction to PDEs and Numerical Methods Lecture 7. Solving linear systems Dr. Noemi Friedman, 09.2.205. Reminder: Instationary heat
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 informationFormula for the inverse matrix. Cramer s rule. Review: 3 3 determinants can be computed expanding by any row or column
Math 20F Linear Algebra Lecture 18 1 Determinants, n n Review: The 3 3 case Slide 1 Determinants n n (Expansions by rows and columns Relation with Gauss elimination matrices: Properties) Formula for the
More informationLinear System of Equations
Linear System of Equations Linear systems are perhaps the most widely applied numerical procedures when real-world situation are to be simulated. Example: computing the forces in a TRUSS. F F 5. 77F F.
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 informationLecture 7. Gaussian Elimination with Pivoting. David Semeraro. University of Illinois at Urbana-Champaign. February 11, 2014
Lecture 7 Gaussian Elimination with Pivoting David Semeraro University of Illinois at Urbana-Champaign February 11, 2014 David Semeraro (NCSA) CS 357 February 11, 2014 1 / 41 Naive Gaussian Elimination
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 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 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 information5.7 Cramer's Rule 1. Using Determinants to Solve Systems Assumes the system of two equations in two unknowns
5.7 Cramer's Rule 1. Using Determinants to Solve Systems Assumes the system of two equations in two unknowns (1) possesses the solution and provided that.. The numerators and denominators are recognized
More informationChapter 2 - Linear Equations
Chapter 2 - Linear Equations 2. Solving Linear Equations One of the most common problems in scientific computing is the solution of linear equations. It is a problem in its own right, but it also occurs
More informationSystems of Linear Equations
Systems of Linear Equations Last time, we found that solving equations such as Poisson s equation or Laplace s equation on a grid is equivalent to solving a system of linear equations. There are many other
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 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 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 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 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 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 informationSolving Linear Systems of Equations
Solving Linear Systems of Equations Gerald Recktenwald Portland State University Mechanical Engineering Department gerry@me.pdx.edu These slides are a supplement to the book Numerical Methods with Matlab:
More informationMethods for Solving Linear Systems Part 2
Methods for Solving Linear Systems Part 2 We have studied the properties of matrices and found out that there are more ways that we can solve Linear Systems. In Section 7.3, we learned that we can use
More informationMatrix notation. A nm : n m : size of the matrix. m : no of columns, n: no of rows. Row matrix n=1 [b 1, b 2, b 3,. b m ] Column matrix m=1
Matrix notation A nm : n m : size of the matrix m : no of columns, n: no of rows Row matrix n=1 [b 1, b 2, b 3,. b m ] Column matrix m=1 n = m square matrix Symmetric matrix Upper triangular matrix: matrix
More informationMATH 2050 Assignment 8 Fall [10] 1. Find the determinant by reducing to triangular form for the following matrices.
MATH 2050 Assignment 8 Fall 2016 [10] 1. Find the determinant by reducing to triangular form for the following matrices. 0 1 2 (a) A = 2 1 4. ANS: We perform the Gaussian Elimination on A by the following
More informationCSE 160 Lecture 13. Numerical Linear Algebra
CSE 16 Lecture 13 Numerical Linear Algebra Announcements Section will be held on Friday as announced on Moodle Midterm Return 213 Scott B Baden / CSE 16 / Fall 213 2 Today s lecture Gaussian Elimination
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 informationProcess Model Formulation and Solution, 3E4
Process Model Formulation and Solution, 3E4 Section B: Linear Algebraic Equations Instructor: Kevin Dunn dunnkg@mcmasterca Department of Chemical Engineering Course notes: Dr Benoît Chachuat 06 October
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 2 Systems of Linear Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction
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 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 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 informationlecture 2 and 3: algorithms for linear algebra
lecture 2 and 3: algorithms for linear algebra STAT 545: Introduction to computational statistics Vinayak Rao Department of Statistics, Purdue University August 27, 2018 Solving a system of linear equations
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 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 information1 GSW Sets of Systems
1 Often, we have to solve a whole series of sets of simultaneous equations of the form y Ax, all of which have the same matrix A, but each of which has a different known vector y, and a different unknown
More informationLinear System of Equations
Linear System of Equations Linear systems are perhaps the most widely applied numerical procedures when real-world situation are to be simulated. Example: computing the forces in a TRUSS. F F 5. 77F F.
More informationElementary Linear Algebra
Matrices J MUSCAT Elementary Linear Algebra Matrices Definition Dr J Muscat 2002 A matrix is a rectangular array of numbers, arranged in rows and columns a a 2 a 3 a n a 2 a 22 a 23 a 2n A = a m a mn We
More informationLecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 2. Systems of Linear Equations
Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 2 Systems of Linear Equations Copyright c 2001. Reproduction permitted only for noncommercial,
More information(17) (18)
Module 4 : Solving Linear Algebraic Equations Section 3 : Direct Solution Techniques 3 Direct Solution Techniques Methods for solving linear algebraic equations can be categorized as direct and iterative
More informationAMS 147 Computational Methods and Applications Lecture 17 Copyright by Hongyun Wang, UCSC
Lecture 17 Copyright by Hongyun Wang, UCSC Recap: Solving linear system A x = b Suppose we are given the decomposition, A = L U. We solve (LU) x = b in 2 steps: *) Solve L y = b using the forward substitution
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 information4 Elementary matrices, continued
4 Elementary matrices, continued We have identified 3 types of row operations and their corresponding elementary matrices. To repeat the recipe: These matrices are constructed by performing the given row
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 information1.5 Gaussian Elimination With Partial Pivoting.
Gaussian Elimination With Partial Pivoting In the previous section we discussed Gaussian elimination In that discussion we used equation to eliminate x from equations through n Then we used equation to
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 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 informationSOLVING LINEAR SYSTEMS
SOLVING LINEAR SYSTEMS We want to solve the linear system a, x + + a,n x n = b a n, x + + a n,n x n = b n This will be done by the method used in beginning algebra, by successively eliminating unknowns
More informationlecture 3 and 4: algorithms for linear algebra
lecture 3 and 4: algorithms for linear algebra STAT 545: Introduction to computational statistics Vinayak Rao Department of Statistics, Purdue University August 30, 2016 Solving a system of linear equations
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 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 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 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 informationLecture Note 2: The Gaussian Elimination and LU Decomposition
MATH 5330: Computational Methods of Linear Algebra Lecture Note 2: The Gaussian Elimination and LU Decomposition The Gaussian elimination Xianyi Zeng Department of Mathematical Sciences, UTEP The method
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 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 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 informationGaussian Elimination for Linear Systems
Gaussian Elimination for Linear Systems Tsung-Ming Huang Department of Mathematics National Taiwan Normal University October 3, 2011 1/56 Outline 1 Elementary matrices 2 LR-factorization 3 Gaussian elimination
More informationy b where U. matrix inverse A 1 ( L. 1 U 1. L 1 U 13 U 23 U 33 U 13 2 U 12 1
LU decomposition -- manual demonstration Instructor: Nam Sun Wang lu-manualmcd LU decomposition, where L is a lower-triangular matrix with as the diagonal elements and U is an upper-triangular matrix Just
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 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 informationY = ax + b. Numerical Applications Least-squares. Start with Self-test 10-1/459. Linear equation. Error function: E = D 2 = (Y - (ax+b)) 2
Ch.10 Numerical Applications 10-1 Least-squares Start with Self-test 10-1/459. Linear equation Y = ax + b Error function: E = D 2 = (Y - (ax+b)) 2 Regression Formula: Slope a = (N ΣXY - (ΣX)(ΣY)) / (N
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 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 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 information4 Elementary matrices, continued
4 Elementary matrices, continued We have identified 3 types of row operations and their corresponding elementary matrices. If you check the previous examples, you ll find that these matrices are constructed
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 informationMAC Module 3 Determinants. Learning Objectives. Upon completing this module, you should be able to:
MAC 2 Module Determinants Learning Objectives Upon completing this module, you should be able to:. Determine the minor, cofactor, and adjoint of a matrix. 2. Evaluate the determinant of a matrix by cofactor
More informationNumerical Analysis Fall. Gauss Elimination
Numerical Analysis 2015 Fall Gauss Elimination Solving systems m g g m m g x x x k k k k k k k k k 3 2 1 3 2 1 3 3 3 2 3 2 2 2 1 0 0 Graphical Method For small sets of simultaneous equations, graphing
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 informationChapter 9: Gaussian Elimination
Uchechukwu Ofoegbu Temple University Chapter 9: Gaussian Elimination Graphical Method The solution of a small set of simultaneous equations, can be obtained by graphing them and determining the location
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 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 informationSolution of Linear systems
Solution of Linear systems Direct Methods Indirect Methods -Elimination Methods -Inverse of a matrix -Cramer s Rule -LU Decomposition Iterative Methods 2 A x = y Works better for coefficient matrices with
More informationLemma 8: Suppose the N by N matrix A has the following block upper triangular form:
17 4 Determinants and the Inverse of a Square Matrix In this section, we are going to use our knowledge of determinants and their properties to derive an explicit formula for the inverse of a square matrix
More informationChapter 2. Solving Systems of Equations. 2.1 Gaussian elimination
Chapter 2 Solving Systems of Equations A large number of real life applications which are resolved through mathematical modeling will end up taking the form of the following very simple looking matrix
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 information1 GSW Gaussian Elimination
Gaussian elimination is probabl the simplest technique for solving a set of simultaneous linear equations, such as: = A x + A x + A x +... + A x,,,, n n = A x + A x + A x +... + A x,,,, n n... m = Am,x
More informationAx = b. Systems of Linear Equations. Lecture Notes to Accompany. Given m n matrix A and m-vector b, find unknown n-vector x satisfying
Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T Heath Chapter Systems of Linear Equations Systems of Linear Equations Given m n matrix A and m-vector
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 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 informationThe System of Linear Equations. Direct Methods. Xiaozhou Li.
1/16 The Direct Methods xiaozhouli@uestc.edu.cn http://xiaozhouli.com School of Mathematical Sciences University of Electronic Science and Technology of China Chengdu, China Does the LU factorization always
More informationDeterminants Chapter 3 of Lay
Determinants Chapter of Lay Dr. Doreen De Leon Math 152, Fall 201 1 Introduction to Determinants Section.1 of Lay Given a square matrix A = [a ij, the determinant of A is denoted by det A or a 11 a 1j
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 informationMatrix Factorization Reading: Lay 2.5
Matrix Factorization Reading: Lay 2.5 October, 20 You have seen that if we know the inverse A of a matrix A, we can easily solve the equation Ax = b. Solving a large number of equations Ax = b, Ax 2 =
More information1 Error analysis for linear systems
Notes for 2016-09-16 1 Error analysis for linear systems We now discuss the sensitivity of linear systems to perturbations. This is relevant for two reasons: 1. Our standard recipe for getting an error
More information4. Direct Methods for Solving Systems of Linear Equations. They are all over the place...
They are all over the place... Numerisches Programmieren, Hans-Joachim ungartz page of 27 4.. Preliminary Remarks Systems of Linear Equations Another important field of application for numerical methods
More informationSolving Linear Systems Using Gaussian Elimination. How can we solve
Solving Linear Systems Using Gaussian Elimination How can we solve? 1 Gaussian elimination Consider the general augmented system: Gaussian elimination Step 1: Eliminate first column below the main diagonal.
More information