Solving Sets of Equations. 150 B.C.E., 九章算術 Carl Friedrich Gauss,
|
|
- Noel Nicholson
- 7 years ago
- Views:
Transcription
1 Solving Sets of Equations 5 B.C.E., 九章算術 Carl Friedrich Gauss,
2 Gaussian-Jordan Elimination In Gauss-Jordan elimination, matrix is reduced to diagonal rather than triangular form Row combinations are used to eliminate entries above as well as below diagonal Numerical Methods Wen-Chieh Lin
3 Numerical Methods Wen-Chieh Lin Gaussian-Jordan Elimination (cont.) Elimination matrix used for given column vector a is of form n i a a m a a a a a a m m m m k i i k n k k k n k k,,, where M k a
4 Gaussian-Jordan Elimination (cont.) Gauss-Jordan elimination requires about n / multiplications and similar number of additions, 5% more expensive than LU factorization During elimination phase, same row operations are also applied to right-hand-side vector (vectors) of system of linear equations Once matrix is in diagonal form, components of solution are computed by dividing each entry of transformed right-hand side by corresponding diagonal entry of matrix Numerical Methods Wen-Chieh Lin 4
5 Using LU for multiple right-hand sides If LU factorization of a matrix A is given, we can solve Ax = b for different b vectors as follows: Ax = b LUx = b Solve Ly = b using forward substitution Then solve Ux = y using backward substitution Numerical Methods Wen-Chieh Lin 5
6 Row Interchanges Gaussian elimination breaks down if leading diagonal entry of remaining unreduced matrix is zero at any stage Easy fix: if diagonal entry in column k is zero, then interchange row k with some subsequent row having nonzero entry in column k and then proceed as usual If there is no nonzero on or below diagonal in column k, then there is nothing to do at this stage, so skip to next column Numerical Methods Wen-Chieh Lin 6
7 Row Interchanges (cont.) Zero on diagonal causes resulting upper triangular matrix to be singular, but LU factorization can still be completed Subsequent back-substitution will fail, however, as it should for singular matrix Numerical Methods Wen-Chieh Lin 7
8 Partial Pivoting In principle, any nonzero value will do as pivot, but in practice pivot should be chosen to minimize error propagation To avoid amplifying previous rounding errors when multiplying remaining portion of matrix by elementary elimination matrix, multipliers should not exceed in magnitude This can be accomplished by choosing entry of largest magnitude on or below diagonal as pivot at each stage Numerical Methods Wen-Chieh Lin 8
9 Partial Pivoting (cont.) Partial pivoting is necessary in practice for numerically stable implementation of Gaussian elimination for general linear system Numerical Methods Wen-Chieh Lin 9
10 LU Factorization with Partial Pivoting With partial pivoting, each M k is preceded by permutation P k to interchange rows to bring entry of largest magnitude into diagonal pivot position Still obtain MA = U, with U upper triangular, but now M = M n- P n- M P L=M - is not a triangular due to permutations L M ( M P T T nn MP MP ) P LP L P Numerical Methods Wen-Chieh Lin T n L n
11 Numerical Methods Wen-Chieh Lin Example: Pivoting Ax b x x x ] [ b A P M ] [ b A M P
12 Numerical Methods Wen-Chieh Lin Example: Pivoting (cont.) ] [ b A P M P ] [ b A M P M P
13 Numerical Methods Wen-Chieh Lin Example: Pivoting (cont.) ) ( L L P P M P M P M L T T U is still upper triangular but L is not lower triangular due to permutations
14 Complete Pivoting Complete pivoting is more exhaustive strategy in which largest entry in entire remaining unreduced submatrix is permuted into diagonal pivot position Requires interchanging columns as well as rows leading to factorization PAQ = LU with L unit lower triangular, U upper triangular, and P, Q permutations Numerical Methods Wen-Chieh Lin 4
15 Complete Pivoting (cont.) Numerical stability of complete pivoting is theoretically superior, but pivot search is more expensive than for partial pivoting Numerical stability of partial pivoting is more than adequate in practice, so it is almost always used in solving linear system by Gaussian elimination Numerical Methods Wen-Chieh Lin 5
16 Example: Pivoting and Precision Consider Without pivoting x ( E y x By Dx Ey C BD F E ) y C F x By CD BD B ( C C F CD CD BD B) C C C B Numerical Methods Wen-Chieh Lin 6
17 Numerical Methods Wen-Chieh Lin 7 Example: Pivoting and Precision With pivoting C By x F Ey Dx D F C y D E B F Ey Dx ) ( B C D E B D F C y BD CE BF D B C E F x ) (
18 Scaling Linear Systems In principle, solution to linear system is unaffected by diagonal scaling of matrix and right-hand-side vector Example: row scaling premultiplying both sides of system by nonsingular diagonal matrix D, the solution is unchanged DAx Db x ( DA) Db A b Numerical Methods Wen-Chieh Lin 8
19 Scaling Linear Systems (cont.) In practice, scaling affects both conditioning of matrix and selection of pivots in Gaussian elimination, which in turn affect numerical accuracy in finite-precision arithmetic It is usually best if all entries of matrix have about same size Numerical Methods Wen-Chieh Lin 9
20 Scaling Linear Systems (cont.) Sometimes it may be obvious how to accomplish this by choice of measurement units for variables, but there is no foolproof method for doing so in general Scaling can introduce error if not done carefully! Numerical Methods Wen-Chieh Lin
21 Given Example: Scale Partial Pivoting A, the exact solution is x = [,, ] T If only digits of precision is used we obtain a erroneous solution x = [.99,.9,.] T.67 5 b Numerical Methods Wen-Chieh Lin
22 Example: Scale Partial Pivoting (cont.) Premultiplying by a scaling matrix A S[ A b], b / S.5.5. / / Pivoting is required at the first column! Numerical Methods Wen-Chieh Lin
23 Example: Scale Partial Pivoting (cont.) In algorithm implementation, we don t scale equations explicitly Instead, we store the scale vector and row interchange information and only use them for pivot selection s partial pivoting no pivoting is required Numerical Methods Wen-Chieh Lin
24 Complexity of Solving Linear System LU factorization requires about n / floatingpoint multiplications and similar number of additions Forward and backward substitution for single right-hand side vector together require about n multiplications and similar number of additions Numerical Methods Wen-Chieh Lin 4
25 Complexity of Solving Linear System Can also solve linear system by matrix inversion: x = A - b Computing A - is equivalent to solve n linear systems, requiring LU factorization of A followed by n forward and backward substitutions, one for each column of identity matrix Operation count for inversion is about n, three times expensive as LU factorization Numerical Methods Wen-Chieh Lin 5
26 Inversion vs. Factorization x=a - b Needs to solve Ax = I LU factorization n forward and backward substitutions Multiplication of matrix and vector LUx = b LU factorization One forward and backward substitution Numerical Methods Wen-Chieh Lin 6
27 Inversion vs. Factorization (cont.) Inversion gives less accuracy answer; e.g., solving x = 8 by division gives x = 8/ = 6, but inversion gives x = - 8 =. 8 = 5.99 (using -digit arithmetic) Numerical Methods Wen-Chieh Lin 7
28 Inversion vs. Factorization (cont.) Matrix inverses often occurs as convenient notation in formulas, but explicit inverse is rarely required to implement such formulas For example, product A - B should be computed by LU factorization of A, followed by forward and backward substitution using each column of B Use factorization instead of inversion Numerical Methods Wen-Chieh Lin 8
29 Ill-Conditioned Systems Recall that A system is ill-conditioned if the solution is very sensitive to changes in the input Example: a near-singular coefficient matrix x.. y. x. y.. b. 98 x. y. 98 b. x y. We cannot test the accuracy of the computed solution merely by substituting the solution into equation to see whether the right-hand sides are reproduced Numerical Methods Wen-Chieh Lin 9
30 Condition Numbers and Norms The condition number of a matrix is defined in terms of norms We ll define the condition number of a matrix after introducing vector and matrix norms Numerical Methods Wen-Chieh Lin
31 Vector Norms Magnitude, modulus, or absolute value for scalars generalizes to norm for vectors We will use only p-norm, defined by x p i i for integer p > and n-vector x Important special cases x x n p n n x x x x max x i i i i i i -norm -norm p -norm Numerical Methods Wen-Chieh Lin
32 Properties of Vector Norms For any vector norm x and x if and only if x kx k x for any scalar k x y x y (triangular inequality) The definition of a vector norm needs to satisfies the above properties Numerical Methods Wen-Chieh Lin
33 Matrix Norms Matrix norm corresponding to given vector norm is defined by A max x Ax x Norm of a matrix measures maximum stretching that the matrix does to any vector in given vector norm Numerical Methods Wen-Chieh Lin
34 Matrix Norms Matrix norm corresponding to vector -nom is maximum absolute column sum A max Matrix norm corresponding to vector -norm is maximum absolute row sum A Handy way to remember these is that matrix norms agree with corresponding vector norms for n by matrix j max i n i n a j a ij ij Numerical Methods Wen-Chieh Lin 4
35 Properties of Matrix Norms Matrix norms we have defined satisfies A ka A B AB and k A A A A B for if B and only if any scalar Above are actually the required properties when a matrix norm is defined! k A Numerical Methods Wen-Chieh Lin 5
36 Condition Number Condition number of square nonsingular matrix A is defined by cond( A) A A By convention, cond(a) = if A is singular Large cond(a) means A is singular Since A x Ax A A max max x x x x condition number measures ratio of maximum stretching to maximum shrinking does to any nonzero vectors Numerical Methods Wen-Chieh Lin 6
37 Properties of Condition Number For any matrix A, cond(a) For identity matrix, cond(i) = For any matrix A and scalar k, cond(ka) = cond(a) Numerical Methods Wen-Chieh Lin 7
38 Computing Condition Number Definition of condition number involves matrix inverse, so it is nontrivial to compute Computing condition number from definition would require much more work than computing solution whose accuracy is to be assessed In practice, condition number is estimated inexpensively as byproduct of solution process Numerical Methods Wen-Chieh Lin 8
39 Computing Condition Number Matrix norm A is easily computed as maximum column sum (or row sum, depending on norm used) Estimating challenging A at low cost is more From properties of norms, if Az = y, then z A y A y A and bound is achieved for optimally chosen y z y Numerical Methods Wen-Chieh Lin 9
40 Computing Condition Number Efficient condition estimators heuristically pick y with large ratio z y, yielding good estimator for A Good software packages for linear systems provide efficient and reliable condition estimator Numerical Methods Wen-Chieh Lin 4
41 Error Bounds Condition number yields error bound for computed solution to linear system Let x be solution to Ax = b, and x approximate solution, r is residual be an r b Ax Ax Ax Ae AB A B r A e e A AB A B r e A r r A e A r Numerical Methods Wen-Chieh Lin 4
42 Error Bounds (cont.) Similarly, from b = Ax and x = A - b we obtain b x A A Combined with previous result r e A A We have r e A A b x b r A A r b Numerical Methods Wen-Chieh Lin 4
43 Error Bounds (cont.) Similarly, from b = Ax and x = A - b we obtain b x A A b Combined with previous result r e A r A condition number of A! We have r e r A A A A b x b Numerical Methods Wen-Chieh Lin 4
44 Error Bounds (cont.) cond( A) r b e x cond( A) r b The relative error in the computed solution vector is bounded by the relative residual divided/multiplied by the condition number When the condition number is large, the residual gives little information about the accuracy Numerical Methods Wen-Chieh Lin 44
45 Error Bounds Illustration In two dimensions, uncertainty in intersection point of two lines depends on whether lines are nearly parallel well-conditioned ill-conditioned Numerical Methods Wen-Chieh Lin 45
46 Residual Residual vector of approximate solution to linear system Ax = b is defined by r b Ax In theory, if A is nonsingular, then x x if, and only if, r but they are not necessarily small simultaneously Since e r cond( A) x b small relative residual implies small relative error in approximate solution only if A is wellconditioned Numerical Methods Wen-Chieh Lin 46
47 Iterative Refinement Given approximate solution x to linear system Ax = b, compute residual r b Ax Now solve linear system Az = r and take x x z as new and better approximate solution, since Ax A ( x z ) Ax Az (b r ) r b Numerical Methods Wen-Chieh Lin 47
48 Iterative Refinement (cont.) Process can be repeated to refine solution successively until convergence, potentially producing solution accurate to full machine precision Numerical Methods Wen-Chieh Lin 48
49 Error in Coefficients of Matrix Let A A matrix and system Using x A x x Ax b A A be E the perturbed coefficient x the solution to the perturbed b A and ( Ax) ( A Ax ( Ax) A A ) x b A ( Ax) Ax x A x Ex x x x A Ex Numerical Methods Wen-Chieh Lin 49
50 Error in Coefficients of Matrix (cont.) x x x x A A Ex E x A x x E cond( A) x A A E A x Error of the solution relative to the norm of the computed solution can be as large as the relative error in the coefficients of A multiplied by the condition number Numerical Methods Wen-Chieh Lin 5
Direct 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 Gaussian Elimination and LU Factorization
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
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 information7. LU factorization. factor-solve method. LU factorization. solving Ax = b with A nonsingular. the inverse of a nonsingular matrix
7. LU factorization EE103 (Fall 2011-12) factor-solve method LU factorization solving Ax = b with A nonsingular the inverse of a nonsingular matrix LU factorization algorithm effect of rounding error sparse
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 information10.2 ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS. The Jacobi Method
578 CHAPTER 1 NUMERICAL METHODS 1. ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS As a numerical technique, Gaussian elimination is rather unusual because it is direct. That is, a solution is obtained after
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 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 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 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 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 non-empty
More information6. Cholesky factorization
6. Cholesky factorization EE103 (Fall 2011-12) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix
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 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 informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional 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 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column
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 floating-point
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 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 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 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 informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation
More informationSYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89. by Joseph Collison
SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections
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 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 informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationLinear Equations ! 25 30 35$ & " 350 150% & " 11,750 12,750 13,750% MATHEMATICS LEARNING SERVICE Centre for Learning and Professional Development
MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) Topic 4 Module 9 Introduction Systems of to Matrices Linear Equations Income = Tickets!
More informationLinear Algebra: Determinants, Inverses, Rank
D Linear Algebra: Determinants, Inverses, Rank D 1 Appendix D: LINEAR ALGEBRA: DETERMINANTS, INVERSES, RANK TABLE OF CONTENTS Page D.1. Introduction D 3 D.2. Determinants D 3 D.2.1. Some Properties of
More information1 Review of Least Squares Solutions to Overdetermined Systems
cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares
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 three-space, we write a vector in terms
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 information1 Determinants and the Solvability of Linear Systems
1 Determinants and the Solvability of Linear Systems In the last section we learned how to use Gaussian elimination to solve linear systems of n equations in n unknowns The section completely side-stepped
More information8 Square matrices continued: Determinants
8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You
More informationGeneral Framework for an Iterative Solution of Ax b. Jacobi s Method
2.6 Iterative Solutions of Linear Systems 143 2.6 Iterative Solutions of Linear Systems Consistent linear systems in real life are solved in one of two ways: by direct calculation (using a matrix factorization,
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 6. Eigenvalues and Singular Values In this section, we collect together the basic facts about eigenvalues and eigenvectors. From a geometrical viewpoint,
More informationQuestion 2: How do you solve a matrix equation using the matrix inverse?
Question : How do you solve a matrix equation using the matrix inverse? In the previous question, we wrote systems of equations as a matrix equation AX B. In this format, the matrix A contains the coefficients
More information2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables
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 informationLINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
More informationSolving Systems of Linear Equations Using Matrices
Solving Systems of Linear Equations Using Matrices What is a Matrix? A matrix is a compact grid or array of numbers. It can be created from a system of equations and used to solve the system of equations.
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationLS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
More informationALGEBRAIC EIGENVALUE PROBLEM
ALGEBRAIC EIGENVALUE PROBLEM BY J. H. WILKINSON, M.A. (Cantab.), Sc.D. Technische Universes! Dsrmstedt FACHBEREICH (NFORMATiK BIBL1OTHEK Sachgebieto:. Standort: CLARENDON PRESS OXFORD 1965 Contents 1.
More informationLecture 5: Singular Value Decomposition SVD (1)
EEM3L1: Numerical and Analytical Techniques Lecture 5: Singular Value Decomposition SVD (1) EE3L1, slide 1, Version 4: 25-Sep-02 Motivation for SVD (1) SVD = Singular Value Decomposition Consider the system
More informationMAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =
MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the
More informationSpecial Situations in the Simplex Algorithm
Special Situations in the Simplex Algorithm Degeneracy Consider the linear program: Maximize 2x 1 +x 2 Subject to: 4x 1 +3x 2 12 (1) 4x 1 +x 2 8 (2) 4x 1 +2x 2 8 (3) x 1, x 2 0. We will first apply the
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 informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationMAT188H1S Lec0101 Burbulla
Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u
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 informationLinear Programming in Matrix Form
Linear Programming in Matrix Form Appendix B We first introduce matrix concepts in linear programming by developing a variation of the simplex method called the revised simplex method. This algorithm,
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationThe Determinant: a Means to Calculate Volume
The Determinant: a Means to Calculate Volume Bo Peng August 20, 2007 Abstract This paper gives a definition of the determinant and lists many of its well-known properties Volumes of parallelepipeds are
More informationLecture 2 Matrix Operations
Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrix-vector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or
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 informationUsing row reduction to calculate the inverse and the determinant of a square matrix
Using row reduction to calculate the inverse and the determinant of a square matrix Notes for MATH 0290 Honors by Prof. Anna Vainchtein 1 Inverse of a square matrix An n n square matrix A is called invertible
More informationTypical Linear Equation Set and Corresponding Matrices
EWE: Engineering With Excel Larsen Page 1 4. Matrix Operations in Excel. Matrix Manipulations: Vectors, Matrices, and Arrays. How Excel Handles Matrix Math. Basic Matrix Operations. Solving Systems of
More informationPractical Guide to the Simplex Method of Linear Programming
Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear
More information5.5. Solving linear systems by the elimination method
55 Solving linear systems by the elimination method Equivalent systems The major technique of solving systems of equations is changing the original problem into another one which is of an easier to solve
More informationLecture 4: Partitioned Matrices and Determinants
Lecture 4: Partitioned Matrices and Determinants 1 Elementary row operations Recall the elementary operations on the rows of a matrix, equivalent to premultiplying by an elementary matrix E: (1) multiplying
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationUniversity of Lille I PC first year list of exercises n 7. Review
University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients
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 informationMethods for Finding Bases
Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,
More informationSolving Linear Programs
Solving Linear Programs 2 In this chapter, we present a systematic procedure for solving linear programs. This procedure, called the simplex method, proceeds by moving from one feasible solution to another,
More information9 MATRICES AND TRANSFORMATIONS
9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the
More informationK80TTQ1EP-??,VO.L,XU0H5BY,_71ZVPKOE678_X,N2Y-8HI4VS,,6Z28DDW5N7ADY013
Hill Cipher Project K80TTQ1EP-??,VO.L,XU0H5BY,_71ZVPKOE678_X,N2Y-8HI4VS,,6Z28DDW5N7ADY013 Directions: Answer all numbered questions completely. Show non-trivial work in the space provided. Non-computational
More informationNumerical Methods I Eigenvalue Problems
Numerical Methods I Eigenvalue Problems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course G63.2010.001 / G22.2420-001, Fall 2010 September 30th, 2010 A. Donev (Courant Institute)
More informationA numerically adaptive implementation of the simplex method
A numerically adaptive implementation of the simplex method József Smidla, Péter Tar, István Maros Department of Computer Science and Systems Technology University of Pannonia 17th of December 2014. 1
More informationCHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES. From Exploratory Factor Analysis Ledyard R Tucker and Robert C.
CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES From Exploratory Factor Analysis Ledyard R Tucker and Robert C MacCallum 1997 180 CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES In
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More informationSolutions to Math 51 First Exam January 29, 2015
Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not
More informationOPRE 6201 : 2. Simplex Method
OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2
More informationSensitivity Analysis 3.1 AN EXAMPLE FOR ANALYSIS
Sensitivity Analysis 3 We have already been introduced to sensitivity analysis in Chapter via the geometry of a simple example. We saw that the values of the decision variables and those of the slack and
More informationDETERMINANTS TERRY A. LORING
DETERMINANTS TERRY A. LORING 1. Determinants: a Row Operation By-Product The determinant is best understood in terms of row operations, in my opinion. Most books start by defining the determinant via formulas
More informationSection V.3: Dot Product
Section V.3: Dot Product Introduction So far we have looked at operations on a single vector. There are a number of ways to combine two vectors. Vector addition and subtraction will not be covered here,
More informationMatrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.
Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that
More informationNumerical Methods I Solving Linear Systems: Sparse Matrices, Iterative Methods and Non-Square Systems
Numerical Methods I Solving Linear Systems: Sparse Matrices, Iterative Methods and Non-Square Systems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course G63.2010.001 / G22.2420-001,
More informationMath 215 HW #6 Solutions
Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T
More informationSection 8.2 Solving a System of Equations Using Matrices (Guassian Elimination)
Section 8. Solving a System of Equations Using Matrices (Guassian Elimination) x + y + z = x y + 4z = x 4y + z = System of Equations x 4 y = 4 z A System in matrix form x A x = b b 4 4 Augmented Matrix
More information26. Determinants I. 1. Prehistory
26. Determinants I 26.1 Prehistory 26.2 Definitions 26.3 Uniqueness and other properties 26.4 Existence Both as a careful review of a more pedestrian viewpoint, and as a transition to a coordinate-independent
More informationSystems of Linear Equations
Chapter 1 Systems of Linear Equations 1.1 Intro. to systems of linear equations Homework: [Textbook, Ex. 13, 15, 41, 47, 49, 51, 65, 73; page 11-]. Main points in this section: 1. Definition of Linear
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
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 informationMath 312 Homework 1 Solutions
Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please
More information5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1
5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition
More information3 Orthogonal Vectors and Matrices
3 Orthogonal Vectors and Matrices The linear algebra portion of this course focuses on three matrix factorizations: QR factorization, singular valued decomposition (SVD), and LU factorization The first
More information1 Short Introduction to Time Series
ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The
More informationDerivative Free Optimization
Department of Mathematics Derivative Free Optimization M.J.D. Powell LiTH-MAT-R--2014/02--SE Department of Mathematics Linköping University S-581 83 Linköping, Sweden. Three lectures 1 on Derivative Free
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More informationOrthogonal Bases and the QR Algorithm
Orthogonal Bases and the QR Algorithm Orthogonal Bases by Peter J Olver University of Minnesota Throughout, we work in the Euclidean vector space V = R n, the space of column vectors with n real entries
More informationLecture notes on linear algebra
Lecture notes on linear algebra David Lerner Department of Mathematics University of Kansas These are notes of a course given in Fall, 2007 and 2008 to the Honors sections of our elementary linear algebra
More informationLogistic Regression. Jia Li. Department of Statistics The Pennsylvania State University. Logistic Regression
Logistic Regression Department of Statistics The Pennsylvania State University Email: jiali@stat.psu.edu Logistic Regression Preserve linear classification boundaries. By the Bayes rule: Ĝ(x) = arg max
More informationLinear Algebraic Equations, SVD, and the Pseudo-Inverse
Linear Algebraic Equations, SVD, and the Pseudo-Inverse Philip N. Sabes October, 21 1 A Little Background 1.1 Singular values and matrix inversion For non-smmetric matrices, the eigenvalues and singular
More information7.4. The Inverse of a Matrix. Introduction. Prerequisites. Learning Style. Learning Outcomes
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 =. Similarly a square matrix A may have an inverse B = A where AB =
More informationA linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form
Section 1.3 Matrix Products A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form (scalar #1)(quantity #1) + (scalar #2)(quantity #2) +...
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More information