7 Gaussian Elimination and LU Factorization


 Valentine Norris
 2 years ago
 Views:
Transcription
1 7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method for solving systems of linear equations) The basic idea is to use leftmultiplication of A C m m by (elementary) lower triangular matrices, L, L,, L m to convert A to upper triangular form, ie, L m L m L L } {{ } A = U = e L Note that the product of lower triangular matrices is a lower triangular matrix, and the inverse of a lower triangular matrix is also lower triangular Therefore, LA = U A = LU, where L = L This approach can be viewed as triangular triangularization 7 Why Would We Want to Do This? Consider the system Ax = b with LU factorization A = LU Then we have L }{{} Ux = b =y Therefore we can perform (a now familiar) step solution procedure: Solve the lower triangular system Ly = b for y by forward substitution Solve the upper triangular system Ux = y for x by back substitution Moreover, consider the problem AX = B (ie, many different righthand sides that are associated with the same system matrix) In this case we need to compute the factorization A = LU only once, and then and we proceed as before: AX = B LUX = B, Solve LY = B by many forward substitutions (in parallel) Solve UX = Y by many back substitutions (in parallel) In order to appreciate the usefulness of this approach note that the operations count for the matrix factorization is O( 3 m3 ), while that for forward and back substitution is O(m ) Example Take the matrix A =
2 and compute its LU factorization by applying elementary lower triangular transformation matrices We choose L such that leftmultiplication corresponds to subtracting multiples of row from the rows below such that the entries in the first column of A are zeroed out (cf the first homework assignment) Thus L A = = Next, we repeat this operation analogously for L (in order to zero what is left in column of the matrix on the righthand side above): 0 0 L (L A) = = 0 3 = U Now L = (L L ) = L so that L = L with L = and L = Remark Note that L always is a unit lower triangular matrix, ie, it has ones on the diagonal Moreover, L is always obtained as above, ie, the multipliers are accumulated into the lower triangular part with a change of sign The claims made above can be verified as follows First, we note that the multipliers in L k are of the form l jk = a jk, j = k +,, m, a kk so that Now, let L k = l k+,k l m,k l k = 0 0 l k+,k 60 l m,k,
3 Then L k = I l k e k, and therefore (I l k e k } {{ } ) (I + l k ek ) = I l } {{ } k e k l ke k = I, =L k L k since the inner product e k l k = 0 because the only nonzero entry in e k (the in the kth position) does not hit any nonzero entries in l k which start in the k + st position So, for any k we have L k = l k+,k l m,k as claimed In addition, L k L k+ = (I + l k e k ) ( I + l k+ e ) k+ = I + l k e k + l k+e k+ + l k e k l k+ } {{ } =0 and in general we have = L = L L m = l k+,k We can summarize the factorization in Algorithm (LU Factorization) Initialize U = A, L = I for k = : m for j = k + : m e k+ l k+,k+ l m,k l m,k+ l, l 3, l m, l m, l m,m, 6
4 end L(j, k) = U(j, k)/u(k, k) U(j, k : m) = U(j, k : m) L(j, k)u(k, k : m) end Remark In practice one can actually store both L and U in the original matrix A since it is known that the diagonal of L consists of all ones The LU factorization is the cheapest factorization algorithm Its operations count can be verified to be O( 3 m3 ) However, LU factorization cannot be guaranteed to be stable The following examples illustrate this fact Example A fundamental problem is given if we encounter a zero pivot as in A = 5 = L A = Now the (,) position contains a zero and the algorithm will break down since it will attempt to divide by zero Example A more subtle example is the following backward instability Take A = + ε 5 with small ε If ε = then we have the initial example in this chapter, and for ε = 0 we get the previous example LU factorization will result in and The multipliers were L A = L L A = L = 0 ε ε ε ε Now we assume that a righthand side b is given as b = 0 0 and we attempt to solve Ax = b via 6 = U
5 Solve Ly = b Solve Ux = y If ε is on the order of machine accuracy, then the 4 in the entry 4 6 ε insignificant Therefore, we have Ũ = 0 ε 3 and L = L, ε in U is which leads to LŨ = + ε A In fact, the product is significantly different from A Thus, using L and Ũ we are not able to solve a nearby problem, and thus the LU factorization method is not backward stable If we use the factorization based on L and Ũ with the above righthand side b, then we obtain 3 ε x = 3 ε 3 3 Whereas if we were to use the exact factorization A = LU, then we get the exact answer 7 3 x = 3 3 4ε 7 ε 3 ε 3 ε ε 3 Remark Even though L and Ũ are close to L and U, the product LŨ is not close to LU = A and the computed solution x is worthless 7 Pivoting Example The breakdown of the algorithm in our earlier example with L A = can be prevented by simply swapping rows, ie, instead of trying to apply L to L A we first create 0 0 P L A = 0 0 L A = and are done 63
6 More generally, stability problems can be avoided by swapping rows before applying L k, ie, we perform L m P m L P L P A = U The strategy we use for swapping rows in step k is to find the largest element in column k below (and including) the diagonal the socalled pivot element and swap its row with row k This process is referred to as partial (row) pivoting Partial column pivoting and complete (row and column) pivoting are also possible, but not very popular Example Consider again the matrix A = + ε 5 The largest element in the first column is the 4 in the (3, ) position This is our first pivot, and we swap rows and 3 Therefore 0 0 P A = ε 5 = + ε 5, 0 0 and then L P A = ε 5 = 0 ε 0 Now we need to pick the second pivot element For sufficiently small ε (in fact, unless < ε < 3 ), we pick ε as the largest element in the second column below the first row Therefore, the second permutation matrix is just the identity, and we have 0 0 P L P A = ε = 0 ε To complete the elimination phase, we need to perform the elimination in the second column: 0 0 L P L P A = ε = 0 ε = U 0 (ε ) 0 3 ε 0 0 (ε ) The lower triangular matrix L is given by L = L L = and assuming that ε we get 0 0 L = 0 and Ũ = 4 64 (ε ),
7 If we now check the computed factorization LŨ, then we see LŨ = 5 = P Ã, which is just a permuted version of the original matrix A with permutation matrix 0 0 P = P P = Thus, this approach was backward stable Finally, since we have the factorization P A = LU, we can solve the linear system Ax = b as P Ax = P b LUx = P b, and apply the usual twostep procedure Solve the lower triangular system Ly = P b for y Solve the upper triangular system Ux = y for x This yields x = 7+4ε 3+ε ε 3 ε ε 3 If we use the rounded factors L and Ũ instead, then the computed solution is 7 3 x = 3, 3 which is the exact answer to the problem (see also the Maple worksheet 473 LUmws) In general, LU factorization with pivoting results in P A = LU, where P = P m P m P P, and L = (L m L m L L ) with L k = P m P k+ L k P k+ P m, ie, L k is the same as L k except that the entries below the diagonal are appropriately permuted In particular, L k is still lower triangular Remark Since the permutation matrices used here involve only a single row swap each we have P k = P k (while in general, of course, P = P T ) In the example above L = L, and L = P L P = L since P = I 65
8 Remark Due to the pivoting strategy the multipliers will always satisfy l ij A possible interpretation of the pivoting strategy is that the matrix P is determined so that it would yield a permuted matrix A whose standard LU factorization is backward stable Of course, we do not know how to do this in advance, and so the P is determined as the algorithm progresses An algorithm for the factorization of an m m matrix A is given by Algorithm (LU Factorization with Partial Pivoting) Initialize U = A, L = I, P = I for k = : m end find i k to maximize U(i, k) U(k, k : m) U(i, k : m) L(k, : k ) L(i, : k ) P (k, :) P (i, :) for j = k + : m end L(j, k) = U(j, k)/u(k, k) U(j, k : m) = U(j, k : m) L(j, k)u(k, k : m) The operations count for this algorithm is also O( 3 m ) However, while the swaps for partial pivoting require O(m ) operations, they would require O(m 3 ) operations in the case of complete pivoting Remark The algorithm above is not really practical since one would usually not physically swap rows Instead one would use pointers to the swapped rows and store the permutation operations instead 73 Stability We saw earlier that Gaussian elimination without pivoting is can be unstable According to our previous example the algorithm with pivoting seems to be stable What can be proven theoretically? Since the entries of L are at most in absolute value, the LU factorization becomes unstable if the entries of U are unbounded relative to those of A (we need L U = O( P A ) Therefore we define a growth factor One can show ρ = max i,j U(i, j) max i,j A(i, j) Theorem 7 Let A C m m Then LU factorization with partial pivoting guarantees that ρ m 66
9 This bound is unacceptably high, and indicates that the algorithm (with pivoting) can be unstable The following (contrived) example illustrates this Example Take A = Then LU factorization produces the following sequence of matrices: and we see that the largest element in U is 6 = m = U, Remark BUT in practice it turns out that matrices which such large growth factors almost never arise For most practical cases a realistic bound on ρ seems to be m This is still an active area of research and some more details can be found in the [Trefethen/Bau] book In summary we can say that for most practical purposes LU factorization with partial pivoting is considered to be a stable algorithm 74 Cholesky Factorization In the case when the matrix A is Hermitian (or symmetric) positive definite we can devise a faster (and more stable) algorithm Recall that A C m m is Hermitian if A = A This implies x Ay = y Ax for any x, y C m (see homework for details) If we let y = x then we see that x Ax is real Now, if x Ax > 0 for any x 0 then A is called Hermitian positive definite Remark Symmetric positive definite matrices are defined analogously with replaced by T 74 Some Properties of Positive Definite Matrices Assume A C m m is positive definite If X C m n has full rank, then X AX C n n is positive definite 67
10 Any principal submatrix (ie, the intersection of a set of rows and the corresponding columns) of A is positive definite 3 All diagonal entries of A are positive and real 4 The maximum element of A is on the diagonal 5 All eigenvalues of A are positive 74 How Does Cholesky Factorization Work? Consider the Hermitian positive definite A C m m in block form [ ] w A = w K and apply the first step of the LU factorization algorithm Then [ ] [ ] 0 T w A = w I 0 K ww The main idea of the Cholesky factorization algorithms is to take advantage of the fact that A is Hermitian and perform operations symmetrically, ie, also zero the first row Thus [ ] [ ] [ ] 0 T 0 T w A = w I 0 K ww 0 I Note that the three matrices are lower triangular, blockdiagonal, and upper triangular (in fact the adjoint of the lower triangular factor) Now we continue iteratively However, in general A will not have a in the upper lefthand corner We will have to be able to deal with an arbitrary (albeit positive) entry in the (,) position Therefore we reconsider the first step with slightly more general notation, ie, [ a w A = ] w K so that A = [ a 0 T a w I ] [ 0 T 0 K a ww ] [ a a w 0 I ] = R A R Now we can iterate on the inner matrix A Note that the (,) entry of K ww /a has to be positive since A was positive definite and R is nonsingular This guarantees that A = R AR is positive definite and therefore has positive diagonal entries The iteration yields A = RA R so that and eventually A = R R A R R, A = RR Rm R } {{ } m R R, } {{ } =R =R 68
11 the Cholesky factorization of A Note that R is upper triangular and its diagonal is positive (because of the square roots) Thus we have proved Theorem 7 Every Hermitian positive definite matrix A has a unique Cholesky factorization A = R R with R an upper triangular matrix with positive diagonal entries Example Consider A = Cholesky factorization as explained above yields the sequence A = = Distributing copies of the identity matrix in the middle we arrive at 0 0 R = Algorithm (Cholesky Factorization) initialize R = upper triangular part of A (including the diagonal) for k = : m end for j = k + : m end R(j, j : m) = R(j, j : m) R(k, j : m) R(k,j) R(k,k) R(k, k : m) = R(k, k : m)/ R(k, k) The operations count for the Cholesky algorithm can be computed as m m (m k + ) + ((m j + ) + ) = 3 m3 + m m k= j=k+ Thus the count is O( 3 m3 ), which is half the amount needed for the LU factorization Another nice feature of Cholesky factorization is that it is always stable even without pivoting 69
12 Remark The simplest (and cheapest) way to test whether a matrix is positive definite is to run Cholesky factorization it If the algorithm works it is, if not, it isn t The solution of Ax = b with Hermitian positive definite A is given by: Compute Cholesky factorization A = R R Solve the lower triangular system R y = b for y 3 Solve the upper triangular system Rx = y for x 70
Linear Systems. Singular and Nonsingular Matrices. Find x 1, x 2, x 3 such that the following three equations hold:
Linear Systems Example: Find x, x, x such that the following three equations hold: x + x + x = 4x + x + x = x + x + x = 6 We can write this using matrixvector notation as 4 {{ A x x x {{ x = 6 {{ b General
More informationAbstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix multiplication).
MAT 2 (Badger, Spring 202) LU Factorization Selected Notes September 2, 202 Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of 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 information7. LU factorization. factorsolve method. LU factorization. solving Ax = b with A nonsingular. the inverse of a nonsingular matrix
7. LU factorization EE103 (Fall 201112) factorsolve method LU factorization solving Ax = b with A nonsingular the inverse of a nonsingular matrix LU factorization algorithm effect of rounding error sparse
More informationSolving Sets of Equations. 150 B.C.E., 九章算術 Carl Friedrich Gauss,
Solving Sets of Equations 5 B.C.E., 九章算術 Carl Friedrich Gauss, 777855 GaussianJordan Elimination In GaussJordan elimination, matrix is reduced to diagonal rather than triangular form Row combinations
More informationPractical Numerical Training UKNum
Practical Numerical Training UKNum 7: Systems of linear equations C. Mordasini Max Planck Institute for Astronomy, Heidelberg Program: 1) Introduction 2) Gauss Elimination 3) Gauss with Pivoting 4) Determinants
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
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 informationSolution of Linear Systems
Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start
More informationSOLVING LINEAR SYSTEMS
SOLVING LINEAR SYSTEMS Linear systems Ax = b occur widely in applied mathematics They occur as direct formulations of real world problems; but more often, they occur as a part of the numerical analysis
More informationFactorization Theorems
Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization
More information2.5 Gaussian Elimination
page 150 150 CHAPTER 2 Matrices and Systems of Linear Equations 37 10 the linear algebra package of Maple, the three elementary 20 23 1 row operations are 12 1 swaprow(a,i,j): permute rows i and j 3 3
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationDETERMINANTS. b 2. x 2
DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in
More informationWe seek a factorization of a square matrix A into the product of two matrices which yields an
LU Decompositions We seek a factorization of a square matrix A into the product of two matrices which yields an efficient method for solving the system where A is the coefficient matrix, x is our variable
More information1.5 Elementary Matrices and a Method for Finding the Inverse
.5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:
More information(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.
Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product
More informationSolving Linear Systems, Continued and The Inverse of a Matrix
, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing
More informationElementary Matrices and The LU Factorization
lementary Matrices and The LU Factorization Definition: ny matrix obtained by performing a single elementary row operation (RO) on the identity (unit) matrix is called an elementary matrix. There are three
More information6 Gaussian Elimination
G1BINM Introduction to Numerical Methods 6 1 6 Gaussian Elimination 61 Simultaneous linear equations Consider the system of linear equations a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 +
More information6. Cholesky factorization
6. Cholesky factorization EE103 (Fall 201112) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix
More informationLecture 3: Finding integer solutions to systems of linear equations
Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture
More informationLecture 5  Triangular Factorizations & Operation Counts
LU Factorization Lecture 5  Triangular Factorizations & Operation Counts We have seen that the process of GE essentially factors a matrix A into LU Now we want to see how this factorization allows us
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationDiagonal, Symmetric and Triangular Matrices
Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by
More informationMatrix Inverse and Determinants
DM554 Linear and Integer Programming Lecture 5 and Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1 2 3 4 and Cramer s rule 2 Outline 1 2 3 4 and
More informationDirect Methods for Solving Linear Systems. Linear Systems of Equations
Direct Methods for Solving Linear Systems Linear Systems of Equations Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University
More informationMATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.
MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted
More informationLinear Dependence Tests
Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks
More informationSection Notes 3. The Simplex Algorithm. Applied Math 121. Week of February 14, 2011
Section Notes 3 The Simplex Algorithm Applied Math 121 Week of February 14, 2011 Goals for the week understand how to get from an LP to a simplex tableau. be familiar with reduced costs, optimal solutions,
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationLinear Systems COS 323
Linear Systems COS 323 Last time: Constrained Optimization Linear constrained optimization Linear programming (LP) Simplex method for LP General optimization With equality constraints: Lagrange multipliers
More information1 Solving LPs: The Simplex Algorithm of George Dantzig
Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.
More informationMAT Solving Linear Systems Using Matrices and Row Operations
MAT 171 8.5 Solving Linear Systems Using Matrices and Row Operations A. Introduction to Matrices Identifying the Size and Entries of a Matrix B. The Augmented Matrix of a System of Equations Forming Augmented
More informationSolving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
More informationLecture 6. Inverse of Matrix
Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that
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 informationSCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self Study Course
SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self Study Course MODULE 17 MATRICES II Module Topics 1. Inverse of matrix using cofactors 2. Sets of linear equations 3. Solution of sets of linear
More information10.3 POWER METHOD FOR APPROXIMATING EIGENVALUES
55 CHAPTER NUMERICAL METHODS. POWER METHOD FOR APPROXIMATING EIGENVALUES In Chapter 7 we saw that the eigenvalues of an n n matrix A are obtained by solving its characteristic equation n c n n c n n...
More information(57) (58) (59) (60) (61)
Module 4 : Solving Linear Algebraic Equations Section 5 : Iterative Solution Techniques 5 Iterative Solution Techniques By this approach, we start with some initial guess solution, say for solution and
More information1 Gaussian Elimination
Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 GaussJordan reduction and the Reduced
More informationSolving Linear Systems of Equations. Gerald Recktenwald Portland State University Mechanical Engineering Department gerry@me.pdx.
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 informationeissn: pissn: X Page 67
International Journal of Computer & Communication Engineering Research (IJCCER) Volume 2  Issue 2 March 2014 Performance Comparison of Gauss and GaussJordan Yadanar Mon 1, Lai Lai Win Kyi 2 1,2 Information
More informationCofactor Expansion: Cramer s Rule
Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationDeterminants. Dr. Doreen De Leon Math 152, Fall 2015
Determinants Dr. Doreen De Leon Math 52, Fall 205 Determinant of a Matrix Elementary Matrices We will first discuss matrices that can be used to produce an elementary row operation on a given matrix A.
More informationLinear Least Squares
Linear Least Squares Suppose we are given a set of data points {(x i,f i )}, i = 1,...,n. These could be measurements from an experiment or obtained simply by evaluating a function at some points. One
More information1 Determinants. Definition 1
Determinants The determinant of a square matrix is a value in R assigned to the matrix, it characterizes matrices which are invertible (det 0) and is related to the volume of a parallelpiped described
More informationmax cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x
Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study
More informationMATH 2030: SYSTEMS OF LINEAR EQUATIONS. ax + by + cz = d. )z = e. while these equations are not linear: xy z = 2, x x = 0,
MATH 23: SYSTEMS OF LINEAR EQUATIONS Systems of Linear Equations In the plane R 2 the general form of the equation of a line is ax + by = c and that the general equation of a plane in R 3 will be we call
More informationIntroduction to Matrix Algebra I
Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model
More informationSystems of Linear Equations
A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler Systems of Linear Equations Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION AttributionNonCommercialShareAlike (CC
More informationNUMERICALLY EFFICIENT METHODS FOR SOLVING LEAST SQUARES PROBLEMS
NUMERICALLY EFFICIENT METHODS FOR SOLVING LEAST SQUARES PROBLEMS DO Q LEE Abstract. Computing the solution to Least Squares Problems is of great importance in a wide range of fields ranging from numerical
More information13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.
3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in threespace, we write a vector in terms
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 4. Gaussian Elimination In this part, our focus will be on the most basic method for solving linear algebraic systems, known as Gaussian Elimination in honor
More information4. MATRICES Matrices
4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:
More informationJones and Bartlett Publishers, LLC. NOT FOR SALE OR DISTRIBUTION
8498_CH08_WilliamsA.qxd 11/13/09 10:35 AM Page 347 Jones and Bartlett Publishers, LLC. NOT FOR SALE OR DISTRIBUTION C H A P T E R Numerical Methods 8 I n this chapter we look at numerical techniques for
More information( % . This matrix consists of $ 4 5 " 5' the coefficients of the variables as they appear in the original system. The augmented 3 " 2 2 # 2 " 3 4&
Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important
More information2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors
2.1: Determinants by Cofactor Expansion Math 214 Chapter 2 Notes and Homework Determinants The minor M ij of the entry a ij is the determinant of the submatrix obtained from deleting the i th row and the
More information2. Perform elementary row operations to get zeros below the diagonal.
Gaussian Elimination We list the basic steps of Gaussian Elimination, a method to solve a system of linear equations. Except for certain special cases, Gaussian Elimination is still state of the art. After
More informationDefinition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that
0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
More informationCS3220 Lecture Notes: QR factorization and orthogonal transformations
CS3220 Lecture Notes: QR factorization and orthogonal transformations Steve Marschner Cornell University 11 March 2009 In this lecture I ll talk about orthogonal matrices and their properties, discuss
More informationChapter 8. Matrices II: inverses. 8.1 What is an inverse?
Chapter 8 Matrices II: inverses We have learnt how to add subtract and multiply matrices but we have not defined division. The reason is that in general it cannot always be defined. In this chapter, we
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 3 Linear Least Squares Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign Copyright c 2002. Reproduction
More information1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form
1. LINEAR EQUATIONS A linear 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, b are given real numbers. For example, with x and
More information10.3 POWER METHOD FOR APPROXIMATING EIGENVALUES
58 CHAPTER NUMERICAL METHODS. POWER METHOD FOR APPROXIMATING EIGENVALUES In Chapter 7 you saw that the eigenvalues of an n n matrix A are obtained by solving its characteristic equation n c nn c nn...
More informationSimilar matrices and Jordan form
Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive
More informationCalculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants
Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants Hartmut Führ fuehr@matha.rwthaachen.de Lehrstuhl A für Mathematik, RWTH Aachen October 30, 2008 Overview
More informationChap 4 The Simplex Method
The Essence of the Simplex Method Recall the Wyndor problem Max Z = 3x 1 + 5x 2 S.T. x 1 4 2x 2 12 3x 1 + 2x 2 18 x 1, x 2 0 Chap 4 The Simplex Method 8 corner point solutions. 5 out of them are CPF solutions.
More informationLecture 11. Shuanglin Shao. October 2nd and 7th, 2013
Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous
More informationSolving Systems of Linear Equations; Row Reduction
Harvey Mudd College Math Tutorial: Solving Systems of Linear Equations; Row Reduction Systems of linear equations arise in all sorts of applications in many different fields of study The method reviewed
More informationInverses and powers: Rules of Matrix Arithmetic
Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3
More informationBy W.E. Diewert. July, Linear programming problems are important for a number of reasons:
APPLIED ECONOMICS By W.E. Diewert. July, 3. Chapter : Linear Programming. Introduction The theory of linear programming provides a good introduction to the study of constrained maximization (and minimization)
More informationLecture 10: Invertible matrices. Finding the inverse of a matrix
Lecture 10: Invertible matrices. Finding the inverse of a matrix Danny W. Crytser April 11, 2014 Today s lecture Today we will Today s lecture Today we will 1 Single out a class of especially nice matrices
More informationMATH 240 Fall, Chapter 1: Linear Equations and Matrices
MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a nonempty
More informationLinear Programming. March 14, 2014
Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1
More informationSolving Systems of Linear Equations. Substitution
Solving Systems of Linear Equations There are two basic methods we will use to solve systems of linear equations: Substitution Elimination We will describe each for a system of two equations in two unknowns,
More informationLecture 1: Schur s Unitary Triangularization Theorem
Lecture 1: Schur s Unitary Triangularization Theorem This lecture introduces the notion of unitary equivalence and presents Schur s theorem and some of its consequences It roughly corresponds to Sections
More information1.2 Solving a System of Linear Equations
1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems  Basic De nitions As noticed above, the general form of a linear system of m equations in n variables
More informationThe Inverse of a Matrix
The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a ( 0) has a reciprocal b written as a or such that a ba = ab =. Some, but not all, square matrices have inverses. If a square
More informationAu = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.
Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationLINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LUdecomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
More informationLECTURE 1 I. Inverse matrices We return now to the problem of solving linear equations. Recall that we are trying to find x such that IA = A
LECTURE I. Inverse matrices We return now to the problem of solving linear equations. Recall that we are trying to find such that A = y. Recall: there is a matri I such that for all R n. It follows that
More informationChapter 4  Systems of Equations and Inequalities
Math 233  Spring 2009 Chapter 4  Systems of Equations and Inequalities 4.1 Solving Systems of equations in Two Variables Definition 1. A system of linear equations is two or more linear equations to
More informationElementary Row Operations and Matrix Multiplication
Contents 1 Elementary Row Operations and Matrix Multiplication 1.1 Theorem (Row Operations using Matrix Multiplication) 2 Inverses of Elementary Row Operation Matrices 2.1 Theorem (Inverses of Elementary
More informationLinear Systems and Gaussian Elimination
Eivind Eriksen Linear Systems and Gaussian Elimination September 2, 2011 BI Norwegian Business School Contents 1 Linear Systems................................................ 1 1.1 Linear Equations...........................................
More informationSolution to Homework 2
Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if
More informationLecture 1: Systems of Linear Equations
MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables
More informationMatrices 2. Solving Square Systems of Linear Equations; Inverse Matrices
Matrices 2. Solving Square Systems of Linear Equations; Inverse Matrices Solving square systems of linear equations; inverse matrices. Linear algebra is essentially about solving systems of linear equations,
More informationUnit 18 Determinants
Unit 18 Determinants Every square matrix has a number associated with it, called its determinant. In this section, we determine how to calculate this number, and also look at some of the properties of
More informationThe basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23
(copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, highdimensional
More informationOperation Count; Numerical Linear Algebra
10 Operation Count; Numerical Linear Algebra 10.1 Introduction Many computations are limited simply by the sheer number of required additions, multiplications, or function evaluations. If floatingpoint
More informationMATRICES WITH DISPLACEMENT STRUCTURE A SURVEY
MATRICES WITH DISPLACEMENT STRUCTURE A SURVEY PLAMEN KOEV Abstract In the following survey we look at structured matrices with what is referred to as low displacement rank Matrices like Cauchy Vandermonde
More informationHelpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:
Helpsheet Giblin Eunson Library ATRIX ALGEBRA Use this sheet to help you: Understand the basic concepts and definitions of matrix algebra Express a set of linear equations in matrix notation Evaluate determinants
More informationLecture 23: The Inverse of a Matrix
Lecture 23: The Inverse of a Matrix Winfried Just, Ohio University March 9, 2016 The definition of the matrix inverse Let A be an n n square matrix. The inverse of A is an n n matrix A 1 such that A 1
More informationNUMERICAL METHODS C. Carl Gustav Jacob Jacobi 10.1 GAUSSIAN ELIMINATION WITH PARTIAL PIVOTING
0. Gaussian Elimination with Partial Pivoting 0.2 Iterative Methods for Solving Linear Systems 0.3 Power Method for Approximating Eigenvalues 0.4 Applications of Numerical Methods Carl Gustav Jacob Jacobi
More informationIterative Methods for Computing Eigenvalues and Eigenvectors
The Waterloo Mathematics Review 9 Iterative Methods for Computing Eigenvalues and Eigenvectors Maysum Panju University of Waterloo mhpanju@math.uwaterloo.ca Abstract: We examine some numerical iterative
More information