COURSE Numerical methods for solving linear systems. Practical solving of many problems eventually leads to solving linear systems.
|
|
- Barrie Hawkins
- 5 years ago
- Views:
Transcription
1 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 unknowns up to several tens of thousands; they provide the exact solution of the system in a finite number of steps - iterative methods - with medium number of unknowns; it is obtained an approximation of the solution as the limit of a sequence
2 - semiiterative methods - with large number of unknowns; it is obtained an approximation of the solution 4 Perturbation of linear systems Consider the linear system Ax b Definition The number cond A A A is called conditioning number of the matrix A It measures the sensibility of the solution x of the system Ax b to the perturbation of A and b The system is good conditioned if cond A is small <000 or it is ill conditioned if cond A is great Remark 2 cond A 2 conda depends on the norm used
3 Consider an example with the solution x x 2 x 3 x , We perturbate the right hand side: and obtain the exact solution x + δx x 2 + δx 2 x 3 + δx 3 x 4 + δx ,
4 Consider, for example, b 2 b 2 + δb 2 b 2 b 2 δb , where δb i, i, 3 denote the perturbations of b, and x 2 x 2 + δx 2 x 2 x 2 δx Thus, a relative error of order 200 on the right hand side precision of 200 for the data in a linear system attracts a relative error of order 0 on the solution, 2000 times larger Consider the same system, and perturb the matrix A: x + δx x 2 + δx 2 x 3 + δx 3 x 4 + δx ,
5 with exact solution The matrix A seems to have good properties symmetric, with determinant, and the inverse A is also with integer numbers This example is very concerning as such orders of the errors in many experimental sciences are considered as satisfactory Remark 3 For this example cond A 2984 in euclidian norm
6 Analyze the phenomenon: In the first case, when b is perturbed, we compare de exact solutions x and x + δx of the systems and Ax b Ax + δx b + δb Let be a norm on R n and the induced matrix norm We have the systems Ax b and Ax + Aδx b + δb Aδx δb
7 From δx A δb we get δx A δb and from b Ax we get b A x x A b, so the relative error of the result is bounded by δx x A A δb b denoted cond A δb b In the second case, when the matrix A is perturbed, we compare the exact solutions of the linear systems and Ax b A + δax + δx b Ax + Aδx + δax + δaδx b Aδx δax + δx From δx A δax + δx, we get δx A δa x + δx, or δx A x + δx δa A A δa A cond A δa A 2
8 42 Direct methods for solving linear systems Why Cramer s method is not suitable for solving linear systems for n 00 and it will not be in near future? For applying Cramer s method for a n n system we need in a rough evaluation the following number of operations: n +! aditions n + 2! multiplications n divisions Consider, hypothetically, a volume V km 3 of cubic processors of each having the side l 0 8 cm radius of an atom, the time for execution of an operation is supposed to be equal to the time needed for the light to pass through an atom Light speed is km/s In this hypothetically case, the time necessary for solving the n n system, n 00, will be more than 0 94 years!
9 42 Gauss method for solving linear systems Consider the linear system Ax b, ie, a a 2 a n a 2 a 22 a 2n a n a n2 a nn x x 2 x n b b 2 b n 3 The method consists of two stages: - reducing the system 3 to an equivalent one, U x d, with U an upper triangular matrix - solving of the upper triangular linear system U x d by backward substitution At least one of the elements on the first column is nonzero, otherwise A is singular We choose one of these nonzero elements using some criterion and this will be called the first elimination pivot
10 If the case, we change the line of the pivot with the first line, both in A and in b, and next we successively make zeros under the first pivot: a a 2 a n 0 a 22 a 2n 0 a n2 a nn x x 2 x n b b 2 b n Analogously, after k steps we obtain the system a a 2 a k a,k+ a n 0 a 2 22 a2 2k a2 2,k+ a 2 2n 0 0 a k kk ak k,k+ a k kn a k k+,k+ ak k+,n a k n,k+ a k nn x x 2 x k x k+ x n b b 2 2 b k k b k k+ b k n
11 If a k kk 0, denote m ik ak ik a k kk and we get aij k+ a k ij m ika k kj, j k,, n b k+ i b k i m ikb k k, i k +,, n After n steps we obtain the system a a 2 a n 0 a 2 22 a2 2n 0 0 a 3 3n 0 0 a n nn x x 2 x 3 x n b b 2 2 b 3 3 b n n Remark 4 The total number of elementary operations is of order 2 3 n3 Example 5 Consider the system 0000 x y 2
12 Gauss algorithm yields: m a 2 a m 0 m m 9998 y Replacing in the first equation we get x x y By division with a pivot of small absolute value there could be induced errors For avoiding this there are two ways: A Partial pivoting: finding an index p {k,, n} such that: a k p,k max ik,n a k i,k
13 B Total pivoting: finding p, q {k,, n} such that: a k p,q max i,jk,n a k ij Example 6 Solve the following system of equations using partial pivoting: x x 2 x 3 x 4, The pivot is a 4 We interchange the st line and the 4 th line We have x x 2 x 3 x ,
14 then pivot element m m m Subtracting multiplies of the first equation from the three others gives pivot element m m Subtracting multiplies, of the second equation from the last two equations, gives pivot element m
15 Subtracting multiplies, of the third equation form the last one, gives the upper triangular system The process of the back substitution algorithm applied to the triangular system produces the solution x x x x x 3 633x x 8 x 2 7x 3 + 2x 4 3 Example 7 Solve the system: { 2x + y 3 3x 2y
16 Sol The extended matrix is { 2x + y 3 3x 2y [ and the pivot is 3 We interchange the lines: [ 3 2 ] 2 3 ] We have L L L 2 and obtain 0 [ 3 2 ] so the system becames 7 3 { 3x 2y y 7 3
17 Solution is { x y Example 8 Solve the system: x + x 2 + x 3 4 2x 2x 2 + 3x 3 5 x x 2 + 4x 3 5
18 422 Gauss-Jordan method total elimination method Consider the linear system Ax b, ie, a a 2 a n a 2 a 22 a 2n a n a n2 a nn x x 2 x n b b 2 b n 4 We make transformations, like in Gauss elimination method, to make zeroes in the lines i+, i+2,, n and then, also in the lines, 2,, i such that the system to be reducing to: a a a n nn The solution is obtained by x x 2 x 3 x n x i bi i a i, i,, n ii b b 2 2 b 3 3 b n n
19 423 Factorization methods - LU methods Definition 9 A n n matrix A is strictly diagonally dominant if a ii > n j,j i a ij, for i, 2,, n Theorem 0 If A is a strictly diagonally dominant matrix, then A is nonsingular and moreover, Gaussian elimination can be performed on any linear system Ax b without row or column interchanges, and the computations are stable with respect to the growth of rounding errors Theorem If A is strictly diagonally dominat then it can be factored into the product of a lower triangular matrix L and an upper triangular matrix U, namely A LU If conditions of Theorem 0 are fulfilled then Ax b LUx b,
20 where L l 0 0 l 2 l 22 0 l n l n2 l nn U u u 2 u n 0 u 22 u 2n 0 0 u nn We solve the systems in two stages: First stage: Solve Lz b, Second stage: Solve Ux z Methods for computing matrices L and U : Doolittle method where all diagonal elements of L have to be ; Crout method where all diagonal elements of U have to be and Choleski method where l ii u ii for i,, n Remark 2 LU factorizations are modified forms of Gauss elimination method
21 Doolittle method We consider that the hypotesis of Theorem 0 is fulfilled, so a kk 0, k, n Denote l i,k : ak i,k a k k,k t k, i k +, n 0 0 l k+,k l n,k having zeros for the first k-th lines, and, M k I n t k e k M n n R 5 where e k 0 0 is the k-unit vector of dimension n, and
22 I n is the identity matrix of order n a 0 i,k are ele- are elements of A; a i,k ments of M k M A are elements of M A; ; a k i,k Definition 3 The matrix M k is called the Gauss matrix, the components l i,k are called the Gauss multiplies and the vector t k is the Gauss vector Remark 4 If A M n n R, then the Gauss matrices M,, M n can be determined such that U M n M n 2 M 2 M A is an upper triangular matrix Moreover, if we choose L M M 2 M n
23 then A L U Example 5 Find LU factorization for the matrix 2 A 6 8 Solve the system { 2x + x 2 3 6x + 8x 2 9
24 Sol M I 2 t e We have So U M A A L M L U
25 We have L U x Ux z 3 9 and 0 z z 2 x x z x 0 5 0
26 43 Iterative methods for solving linear systems Because of round-off errors, direct methods become less efficient than iterative methods for large systems > variables An iterative scheme for linear systems consists of converting the system to the form Ax b 6 x b Bx After an initial guess for x 0, the sequence of approximations of the solution x 0, x,, x k, is generated by computing x k b Bx k, for k, 2, 3,
27 43 Jacobi iterative method Consider the n n linear system, a x + a 2 x a n x n b a 2 x + a 22 x a 2n x n b 2 a n x + a n2 x a nn x n b n, where we assume that the diagonal terms a, a 22,, a nn are all nonzero We begin our iterative scheme by solving each equation for one of the variables: x u 2 x u n x n + c x 2 u 2 x + + u 2n x n + c 2 x n u n x + + u nn x n + c n, where u ij a ij a ii, c i b i a ii, i,, n
28 Let x 0 x 0, x0 2,, x0 n be an initial approximation of the solution The k + -th approximation is: for k 0,, 2, x k+ u 2 x k u nx k n + c x k+ 2 u 2 x k + u 23x k x k+ n An algorithmic form: x k i b i u 2nx n k + c 2 u n x k + + u nn x k n + c n, n j,j i a ij x k j a ii, i, 2,, n, for k The iterative process is terminated when a convergence criterion is satisfied Stopping criterions: x k x k x k x k < ε or x k < ε, with ε > 0 - a prescribed tolerance
29 Example 6 Solve the following system using the Jacobi iterative method Use ε 0 3 and x as the starting vector 7x 2x 2 + x 3 7 x 9x 2 + 3x 3 x 4 3 2x + 0x 3 + x 4 5 x x 2 + x 3 + 6x 4 0 These equations can be rearranged to give and, for example, x 7 + 2x 2 x 3 /7 x x + 3x 3 x 4 /9 x 3 5 2x x 4 /0 x 4 0 x + x 2 x 3 /6 x 7 + 2x 0 2 x 0 3 /7 x x 0 + 3x 0 3 x 0 4 /9 x 3 5 2x 0 x 0 4 /0 x 4 0 x 0 + x 0 2 x 0 3 /6
30 Substitute x 0 0, 0, 0, 0 into the right-hand side of each of these equations to get x / x / x /0 5 x / and so x , , 5, The Jacobi iterative process: x k+ x k+ 2 x k+ 3 x k x k 3 + x k 2 xk 3 /7 + 3xk 3 xk 4 5 2x k xk 4 /0 0 x k + xk 2 xk 3 We obtain a sequence that converges to /9 /6, k x , , ,
Linear 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 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 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 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 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 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 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 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 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 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 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 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 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 informationCOURSE Iterative methods for solving linear systems
COURSE 0 4.3. Iterative methods for solving linear systems Because of round-off errors, direct methods become less efficient than iterative methods for large systems (>00 000 variables). An iterative scheme
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 informationNumerical methods for solving linear systems
Chapter 2 Numerical methods for solving linear systems Let A C n n be a nonsingular matrix We want to solve the linear system Ax = b by (a) Direct methods (finite steps); Iterative methods (convergence)
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 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 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 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 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 informationLecture 12 (Tue, Mar 5) Gaussian elimination and LU factorization (II)
Math 59 Lecture 2 (Tue Mar 5) Gaussian elimination and LU factorization (II) 2 Gaussian elimination - LU factorization For a general n n matrix A the Gaussian elimination produces an LU factorization if
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 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 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 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 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 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 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 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 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 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 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 informationChapter 4 No. 4.0 Answer True or False to the following. Give reasons for your answers.
MATH 434/534 Theoretical Assignment 3 Solution Chapter 4 No 40 Answer True or False to the following Give reasons for your answers If a backward stable algorithm is applied to a computational problem,
More informationNumerical Analysis: Solutions of System of. Linear Equation. Natasha S. Sharma, PhD
Mathematical Question we are interested in answering numerically How to solve the following linear system for x Ax = b? where A is an n n invertible matrix and b is vector of length n. Notation: x denote
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 informationLECTURE NOTES ELEMENTARY NUMERICAL METHODS. Eusebius Doedel
LECTURE NOTES on ELEMENTARY NUMERICAL METHODS Eusebius Doedel TABLE OF CONTENTS Vector and Matrix Norms 1 Banach Lemma 20 The Numerical Solution of Linear Systems 25 Gauss Elimination 25 Operation Count
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 informationIntroduction to Numerical Analysis
Université de Liège Faculté des Sciences Appliquées Introduction to Numerical Analysis Edition 2015 Professor Q. Louveaux Department of Electrical Engineering and Computer Science Montefiore Institute
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 informationGAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511)
GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511) D. ARAPURA Gaussian elimination is the go to method for all basic linear classes including this one. We go summarize the main ideas. 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 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 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 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 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 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 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 informationParallel Scientific Computing
IV-1 Parallel Scientific Computing Matrix-vector multiplication. Matrix-matrix multiplication. Direct method for solving a linear equation. Gaussian Elimination. Iterative method for solving a linear equation.
More information4.2 Floating-Point Numbers
101 Approximation 4.2 Floating-Point Numbers 4.2 Floating-Point Numbers The number 3.1416 in scientific notation is 0.31416 10 1 or (as computer output) -0.31416E01..31416 10 1 exponent sign mantissa base
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 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 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 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 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 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 informationScientific Computing WS 2018/2019. Lecture 9. Jürgen Fuhrmann Lecture 9 Slide 1
Scientific Computing WS 2018/2019 Lecture 9 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 9 Slide 1 Lecture 9 Slide 2 Simple iteration with preconditioning Idea: Aû = b iterative scheme û = û
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 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 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 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 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 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 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 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 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 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 informationJACOBI S ITERATION METHOD
ITERATION METHODS These are methods which compute a sequence of progressively accurate iterates to approximate the solution of Ax = b. We need such methods for solving many large linear systems. Sometimes
More informationDense LU factorization and its error analysis
Dense LU factorization and its error analysis Laura Grigori INRIA and LJLL, UPMC February 2016 Plan Basis of floating point arithmetic and stability analysis Notation, results, proofs taken from [N.J.Higham,
More 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 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 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 informationIterative techniques in matrix algebra
Iterative techniques in matrix algebra Tsung-Ming Huang Department of Mathematics National Taiwan Normal University, Taiwan September 12, 2015 Outline 1 Norms of vectors and matrices 2 Eigenvalues and
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 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 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 informationThe matrix will only be consistent if the last entry of row three is 0, meaning 2b 3 + b 2 b 1 = 0.
) Find all solutions of the linear system. Express the answer in vector form. x + 2x + x + x 5 = 2 2x 2 + 2x + 2x + x 5 = 8 x + 2x + x + 9x 5 = 2 2 Solution: Reduce the augmented matrix [ 2 2 2 8 ] to
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 informationComputational Economics and Finance
Computational Economics and Finance Part II: Linear Equations Spring 2016 Outline Back Substitution, LU and other decomposi- Direct methods: tions Error analysis and condition numbers Iterative methods:
More informationMath 471 (Numerical methods) Chapter 3 (second half). System of equations
Math 47 (Numerical methods) Chapter 3 (second half). System of equations Overlap 3.5 3.8 of Bradie 3.5 LU factorization w/o pivoting. Motivation: ( ) A I Gaussian Elimination (U L ) where U is upper triangular
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 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 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 informationGaussian Elimination -(3.1) b 1. b 2., b. b n
Gaussian Elimination -() Consider solving a given system of n linear equations in n unknowns: (*) a x a x a n x n b where a ij and b i are constants and x i are unknowns Let a n x a n x a nn x n a a a
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 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 informationA LINEAR SYSTEMS OF EQUATIONS. By : Dewi Rachmatin
A LINEAR SYSTEMS OF EQUATIONS By : Dewi Rachmatin Back Substitution We will now develop the backsubstitution algorithm, which is useful for solving a linear system of equations that has an upper-triangular
More informationMAC1105-College Algebra. Chapter 5-Systems of Equations & Matrices
MAC05-College Algebra Chapter 5-Systems of Equations & Matrices 5. Systems of Equations in Two Variables Solving Systems of Two Linear Equations/ Two-Variable Linear Equations A system of equations is
More information22A-2 SUMMER 2014 LECTURE 5
A- SUMMER 0 LECTURE 5 NATHANIEL GALLUP Agenda Elimination to the identity matrix Inverse matrices LU factorization Elimination to the identity matrix Previously, we have used elimination to get a system
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 informationOutline. Math Numerical Analysis. Errors. Lecture Notes Linear Algebra: Part B. Joseph M. Mahaffy,
Math 54 - Numerical Analysis Lecture Notes Linear Algebra: Part B Outline Joseph M. Mahaffy, jmahaffy@mail.sdsu.edu Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences
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 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 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 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 informationLU Factorization. LU factorization is the most common way of solving linear systems! Ax = b LUx = b
AM 205: lecture 7 Last time: LU factorization Today s lecture: Cholesky factorization, timing, QR factorization Reminder: assignment 1 due at 5 PM on Friday September 22 LU Factorization LU factorization
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 informationNumerical methods, midterm test I (2018/19 autumn, group A) Solutions
Numerical methods, midterm test I (2018/19 autumn, group A Solutions x Problem 1 (6p We are going to approximate the limit 3/2 x lim x 1 x 1 by substituting x = 099 into the fraction in the present form
More information