WEEK #3, Lecture 1: Sparse Systems, MATLAB Graphics
|
|
- Jody Reed
- 8 years ago
- Views:
Transcription
1 WEEK #3, Lecture 1: Sparse Systems, MATLAB Graphics Visualization of Matrices Good visuals anchor any presentation. MATLAB has a wide variety of ways to display data and calculation results that can be powerful tools in presenting information. Recall our heat distribution example from last week:
2 2 The heat-balance equations: T 1 : 20+T 2 +T 6 3 T 1 = 0 T 2 : T T 3 +T 7 4 T 2 = 0 T 3 : T T 4 +T 8 4 T 3 = 0 T 4 : T T 5 +T 9 4 T 4 = 0 T 5 : T T 10 4 T 5 = 0 T 6 : T 1 +T 7 +T 11 3 T 6 = 0 T 7 :. T 6 +T 2 +T 8 +T 12 4 T 6 = 0
3 Week 3 Sparse Matrices, Inverses, Non-Linear Equations 3 Matrix form T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T 10 T 11. = (20+120)
4 4 Solving and Interpreting Exercise: Download W3 1.m, which creates and solves the plate temperature equations for a 5 5 grid. Explain what each of the final steps does. T = A \ B; Tm = reshape(t, N, N) pcolor(tm) colorbar
5 Week 3 Sparse Matrices, Inverses, Non-Linear Equations 5 Explain why we only see a 4 4 grid on the screen, even though Tm is a 5 5 matrix. Explain what else is wrong with the temperature plot displayed by pcolor, relative to the temperature problem.
6 6 Refining the plot These commands get us close to the image we want. However, we still want to have the x and y axes scaled so that our plate has the correct dimensions, 1 cm 1 cm, and get the vertical ordering of the data correct. To accomplish this, we ll need to introduce a new command, meshgrid.
7 Week 3 Sparse Matrices, Inverses, Non-Linear Equations 7 Adding coordinates with meshgrid Exercise: Use the following commands to plot the grid of temperatures. [X, Y] = meshgrid(linspace(0, 1, 5), linspace(0, 1, 5)) pcolor(x, Y, Tm) colorbar What changes in the pcolor plot?
8 8 How meshgrid works Here are the X and Y matrices created by meshgrid, along with the original Tm temperature matrix. X = Y = Tm =
9 Week 3 Sparse Matrices, Inverses, Non-Linear Equations 9 How does the pcolor(x, Y, Tm) command use all this information to generate the plot?
10 10 With the help of meshgrid, our pcolor plot is now correctly scaled along the x and y axes. Inverting the y ordering Finally, we want to make sure that the top-left corner in our matrix shows up in the top-left corner of our graph. From the meshgrid output, what is the y coordinate assigned to the first row of Y? Exercise: What vector does the command linspace(1, 0, 5) generate? Exercise: Using that new linspace command for the y points in meshgrid, what do the first row and column of Y look like?
11 Week 3 Sparse Matrices, Inverses, Non-Linear Equations 11 Final Display Code [X, Y] = meshgrid(linspace(0, 1, N), linspace(1, 0, N)) pcolor(x, Y, Tm) colorbar Other plotting options: surf(x, Y, Tm) colorbar shading flat shading interp contour(x, Y, Tm, 0:10:100)
12 12 Speed of computation Now that we have successfully graphed a simple 5 5 grid of temperatures, we consider the problem of generating and displaying the equilibrium temperatures using a finer grid. Exercise: Start with the script W3 2.m. Modify it so it generates a grid and run it. Note the approximate run time. Exercise: Incrementally increase N by 20 and re-run the script. When does it start getting slow enough to be annoying?
13 Week 3 Sparse Matrices, Inverses, Non-Linear Equations 13 Matrix Characteristics Unfortunately, the value of N that slows MATLAB to a crawl does not result in a particularly detailed temperature grid. Since the accuracy of our calculation depends on how small each sub-cell in the plate is, we would really like to use a finer grid. Let s investigate further. Exercise: Type clear to clear MATLAB s memory. Then run the script with N = 60. Type spy(a) to see where the non-zero elements are in A. What do you notice? If there are roughly 5 entries per row in A, what fraction of its values are non-zero? How much space have we wasted storing the number 0 over and over again?
14 14 Common Matrix Structure Both trusses and heat distribution equation matrices are based on connectivity properties. Trusses: eachbeam(columnofvalues)appearsinatmost2joints,or4equations (horizontal and vertical force equations, for 2 joints). Heat: each row models temperature at a grid point; non-zero coefficients only for neighbouring grid points. In both cases, the number of non-zero elements in the A matrix will be very small relative to the numberof zeros. In these cases, if we stored the matrix in full, the vast majority of the values we store would be largely redundant zero values.
15 Week 3 Sparse Matrices, Inverses, Non-Linear Equations 15 Sparse Matrices Matrices with relatively few non-zero elements are called sparse matrices There are two important implications of using sparse matrices in computer programs. Memory: Much less memory is needed to store sparse matrices. Instead of storing every entry in the matrix, just the non-zero elements are stored, along with their coordinates in the matrix. Run Time: Algorithms like Gaussian elimination can be coded to accept sparse matrices as input. How would knowing a matrix was sparse save time during row operations?
16 16 Notethatseeinglotsofzerosinamatrixmakesitconceptually sparse, butonlyby explicitly creating and storing the matrix in sparse format would it provide any practical advantage. Here is our original, conceptually sparse matrix, in standard form This format requires storing n 2 numbers for an n n matrix. Almost all of these numbers are zeros.
17 Week 3 Sparse Matrices, Inverses, Non-Linear Equations 17 The same matrix can also be stored in an explicit sparse format: A(1,1) = -3; A(1,2) = 1; A(1,5) = 1; A(2,1) = 1; A(2,2) = -4; A(2,3) = 1;. By implication, any element that isn t explicitly listed must be a zero. For each element, sparse storage is a little more expensive, because we store both the location and value, but if 90% of the values in A are zeros, we will still save more space overall by not having to store the zero values.
18 18 Empirical Comparison Exercise: Set N = 80 in W3 2.m. This makes A a matrix, with roughly 40 million elements. Using regular format, solving takes seconds Exercise: Change the command A = zeros(nsq, Nsq) to A = sparse(nsq, Nsq) Using sparse format, solving takes seconds.
19 Week 3 Sparse Matrices, Inverses, Non-Linear Equations 19 We now try an even larger case. Using a plate grid (62, temperature values, or a matrix A with values). Exercise: What happens if we try to create this A matrix using regular format, with A = zeros(nsq, Nsq)? Exercise: If we use the sparse matrix format instead, can the 62,500 62,500 A matrix be built successfully? If so, how long does it take to solve the equations and generate the plot? In fact, using the sparse approach, most computer time in the script is spent creating the A matrix, not solving the final equations!
20 20 Sparse Summary If a problem leads naturally to a sparse matrix, it can be a huge advantage. Storage of sparse matrices is compact, so larger matrices can be stored. Specially designed algorithms can save orders of magnitude of time. Unfortunately, you can t choose sparseness: problem types either lead to sparse matrices, or they don t. Also, even if a system is conceptually sparse, you need to make sure that your software takes advantage of the sparse structure of the equations/matrices. e.g. in MATLAB, creating the matrix with the sparse command, rather than zeros.
Question 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 informationBeginner s Matlab Tutorial
Christopher Lum lum@u.washington.edu Introduction Beginner s Matlab Tutorial This document is designed to act as a tutorial for an individual who has had no prior experience with Matlab. For any questions
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 informationDepartment of Chemical Engineering ChE-101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VI
Department of Chemical Engineering ChE-101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VI Solving a System of Linear Algebraic Equations (last updated 5/19/05 by GGB) Objectives:
More informationMatLab - Systems of Differential Equations
Fall 2015 Math 337 MatLab - Systems of Differential Equations This section examines systems of differential equations. It goes through the key steps of solving systems of differential equations through
More information2+2 Just type and press enter and the answer comes up ans = 4
Demonstration Red text = commands entered in the command window Black text = Matlab responses Blue text = comments 2+2 Just type and press enter and the answer comes up 4 sin(4)^2.5728 The elementary functions
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 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 informationProgramming Exercise 3: Multi-class Classification and Neural Networks
Programming Exercise 3: Multi-class Classification and Neural Networks Machine Learning November 4, 2011 Introduction In this exercise, you will implement one-vs-all logistic regression and neural networks
More informationPolynomial Neural Network Discovery Client User Guide
Polynomial Neural Network Discovery Client User Guide Version 1.3 Table of contents Table of contents...2 1. Introduction...3 1.1 Overview...3 1.2 PNN algorithm principles...3 1.3 Additional criteria...3
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 information14:440:127 Introduction to Computers for Engineers. Notes for Lecture 06
14:440:127 Introduction to Computers for Engineers Notes for Lecture 06 Rutgers University, Spring 2010 Instructor- Blase E. Ur 1 Loop Examples 1.1 Example- Sum Primes Let s say we wanted to sum all 1,
More informationCOS702; Assignment 6. Point Cloud Data Surface Interpolation University of Southern Missisippi Tyler Reese December 3, 2012
COS702; Assignment 6 Point Cloud Data Surface Interpolation University of Southern Missisippi Tyler Reese December 3, 2012 The Problem COS 702, Assignment 6: Given appropriate sets of Point Cloud data,
More informationThe Matrix Stiffness Method for 2D Trusses
The Matrix Stiffness Method for D Trusses Method CEE 4L. Matrix Structural Analysis Department of Civil and Environmental Engineering Duke University Henri P. Gavin Fall, 04. Number all of the nodes and
More informationManifold Learning Examples PCA, LLE and ISOMAP
Manifold Learning Examples PCA, LLE and ISOMAP Dan Ventura October 14, 28 Abstract We try to give a helpful concrete example that demonstrates how to use PCA, LLE and Isomap, attempts to provide some intuition
More informationAMATH 352 Lecture 3 MATLAB Tutorial Starting MATLAB Entering Variables
AMATH 352 Lecture 3 MATLAB Tutorial MATLAB (short for MATrix LABoratory) is a very useful piece of software for numerical analysis. It provides an environment for computation and the visualization. Learning
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 informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationUsing MATLAB to Measure the Diameter of an Object within an Image
Using MATLAB to Measure the Diameter of an Object within an Image Keywords: MATLAB, Diameter, Image, Measure, Image Processing Toolbox Author: Matthew Wesolowski Date: November 14 th 2014 Executive Summary
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 information(!' ) "' # "*# "!(!' +,
MATLAB is a numeric computation software for engineering and scientific calculations. The name MATLAB stands for MATRIX LABORATORY. MATLAB is primarily a tool for matrix computations. It was developed
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 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 informationx y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are
Solving Sstems of Linear Equations in Matri Form with rref Learning Goals Determine the solution of a sstem of equations from the augmented matri Determine the reduced row echelon form of the augmented
More informationReduced echelon form: Add the following conditions to conditions 1, 2, and 3 above:
Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (i.e. left most nonzero entry) of a row is in
More information1 Finite difference example: 1D implicit heat equation
1 Finite difference example: 1D implicit heat equation 1.1 Boundary conditions Neumann and Dirichlet We solve the transient heat equation ρc p t = ( k ) (1) on the domain L/2 x L/2 subject to the following
More informationThe Image Deblurring Problem
page 1 Chapter 1 The Image Deblurring Problem You cannot depend on your eyes when your imagination is out of focus. Mark Twain When we use a camera, we want the recorded image to be a faithful representation
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 information1 Topic. 2 Scilab. 2.1 What is Scilab?
1 Topic Data Mining with Scilab. I know the name "Scilab" for a long time (http://www.scilab.org/en). For me, it is a tool for numerical analysis. It seemed not interesting in the context of the statistical
More informationMulti scale random field simulation program
Multi scale random field simulation program 1.15. 2010 (Updated 12.22.2010) Andrew Seifried, Stanford University Introduction This is a supporting document for the series of Matlab scripts used to perform
More informationMATLAB and Big Data: Illustrative Example
MATLAB and Big Data: Illustrative Example Rick Mansfield Cornell University August 19, 2014 Goals Use a concrete example from my research to: Demonstrate the value of vectorization Introduce key commands/functions
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 informationCS 5614: (Big) Data Management Systems. B. Aditya Prakash Lecture #18: Dimensionality Reduc7on
CS 5614: (Big) Data Management Systems B. Aditya Prakash Lecture #18: Dimensionality Reduc7on Dimensionality Reduc=on Assump=on: Data lies on or near a low d- dimensional subspace Axes of this subspace
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 informationGeoGebra. 10 lessons. Gerrit Stols
GeoGebra in 10 lessons Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It was developed by Markus Hohenwarter
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 informationMatrix Multiplication
Matrix Multiplication CPS343 Parallel and High Performance Computing Spring 2016 CPS343 (Parallel and HPC) Matrix Multiplication Spring 2016 1 / 32 Outline 1 Matrix operations Importance Dense and sparse
More informationis in plane V. However, it may be more convenient to introduce a plane coordinate system in V.
.4 COORDINATES EXAMPLE Let V be the plane in R with equation x +2x 2 +x 0, a two-dimensional subspace of R. We can describe a vector in this plane by its spatial (D)coordinates; for example, vector x 5
More informationEQUATIONS and INEQUALITIES
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
More informationIntroduction to Matlab
Introduction to Matlab Social Science Research Lab American University, Washington, D.C. Web. www.american.edu/provost/ctrl/pclabs.cfm Tel. x3862 Email. SSRL@American.edu Course Objective This course provides
More informationLinear Programming I
Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins
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 informationGetting to know your TI-83
Calculator Activity Intro Getting to know your TI-83 Press ON to begin using calculator.to stop, press 2 nd ON. To darken the screen, press 2 nd alternately. To lighten the screen, press nd 2 alternately.
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 informationArithmetic and Algebra of Matrices
Arithmetic and Algebra of Matrices Math 572: Algebra for Middle School Teachers The University of Montana 1 The Real Numbers 2 Classroom Connection: Systems of Linear Equations 3 Rational Numbers 4 Irrational
More informationA Direct Numerical Method for Observability Analysis
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL 15, NO 2, MAY 2000 625 A Direct Numerical Method for Observability Analysis Bei Gou and Ali Abur, Senior Member, IEEE Abstract This paper presents an algebraic method
More informationLecture Notes 2: Matrices as Systems of Linear Equations
2: Matrices as Systems of Linear Equations 33A Linear Algebra, Puck Rombach Last updated: April 13, 2016 Systems of Linear Equations Systems of linear equations can represent many things You have probably
More informationPOISSON AND LAPLACE EQUATIONS. Charles R. O Neill. School of Mechanical and Aerospace Engineering. Oklahoma State University. Stillwater, OK 74078
21 ELLIPTICAL PARTIAL DIFFERENTIAL EQUATIONS: POISSON AND LAPLACE EQUATIONS Charles R. O Neill School of Mechanical and Aerospace Engineering Oklahoma State University Stillwater, OK 74078 2nd Computer
More information@ MAXSTOR=1000000. 1 Note that it s tempting to use the auxiliary variables representing loga, logb or logc since, as we noted last time,
Bifurcation Analysis with AUTO Marc R. Roussel May 20, 2004 In this lecture, we will learn how to do a bifurcation analysis with the computer program AUTO. AUTO is built into xppaut, which is where the
More informationBreak-even analysis. On page 256 of It s the Business textbook, the authors refer to an alternative approach to drawing a break-even chart.
Break-even analysis On page 256 of It s the Business textbook, the authors refer to an alternative approach to drawing a break-even chart. In order to survive businesses must at least break even, which
More informationCHAPTER 3. INTRODUCTION TO MATRIX METHODS FOR STRUCTURAL ANALYSIS
1 CHAPTER 3. INTRODUCTION TO MATRIX METHODS FOR STRUCTURAL ANALYSIS Written by: Sophia Hassiotis, January, 2003 Last revision: February, 2015 Modern methods of structural analysis overcome some of the
More informationhttp://school-maths.com Gerrit Stols
For more info and downloads go to: http://school-maths.com Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It
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 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 informationHow To Understand And Solve A Linear Programming Problem
At the end of the lesson, you should be able to: Chapter 2: Systems of Linear Equations and Matrices: 2.1: Solutions of Linear Systems by the Echelon Method Define linear systems, unique solution, inconsistent,
More informationUnit 1 Equations, Inequalities, Functions
Unit 1 Equations, Inequalities, Functions Algebra 2, Pages 1-100 Overview: This unit models real-world situations by using one- and two-variable linear equations. This unit will further expand upon pervious
More informationDSP First Laboratory Exercise #9 Sampling and Zooming of Images In this lab we study the application of FIR ltering to the image zooming problem, where lowpass lters are used to do the interpolation needed
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 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 and TI 89
Linear Algebra and TI 89 Abdul Hassen and Jay Schiffman This short manual is a quick guide to the use of TI89 for Linear Algebra. We do this in two sections. In the first section, we will go over the editing
More informationMATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3
MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................
More informationMachine Learning and Data Mining. Regression Problem. (adapted from) Prof. Alexander Ihler
Machine Learning and Data Mining Regression Problem (adapted from) Prof. Alexander Ihler Overview Regression Problem Definition and define parameters ϴ. Prediction using ϴ as parameters Measure the error
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
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 informationMATH 551 - APPLIED MATRIX THEORY
MATH 55 - APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
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 informationIntroduction to Numerical Math and Matlab Programming
Introduction to Numerical Math and Matlab Programming Todd Young and Martin Mohlenkamp Department of Mathematics Ohio University Athens, OH 45701 young@math.ohiou.edu c 2009 - Todd Young and Martin Mohlenkamp.
More informationComponent Ordering in Independent Component Analysis Based on Data Power
Component Ordering in Independent Component Analysis Based on Data Power Anne Hendrikse Raymond Veldhuis University of Twente University of Twente Fac. EEMCS, Signals and Systems Group Fac. EEMCS, Signals
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 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 informationHow long is the vector? >> length(x) >> d=size(x) % What are the entries in the matrix d?
MATLAB : A TUTORIAL 1. Creating vectors..................................... 2 2. Evaluating functions y = f(x), manipulating vectors. 4 3. Plotting............................................ 5 4. Miscellaneous
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 informationLinear Algebra Notes
Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note
More information6 Scalar, Stochastic, Discrete Dynamic Systems
47 6 Scalar, Stochastic, Discrete Dynamic Systems Consider modeling a population of sand-hill cranes in year n by the first-order, deterministic recurrence equation y(n + 1) = Ry(n) where R = 1 + r = 1
More informationCHAPTER 11: BASIC LINEAR PROGRAMMING CONCEPTS
Linear programming is a mathematical technique for finding optimal solutions to problems that can be expressed using linear equations and inequalities. If a real-world problem can be represented accurately
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 informationMatrix Algebra in R A Minimal Introduction
A Minimal Introduction James H. Steiger Department of Psychology and Human Development Vanderbilt University Regression Modeling, 2009 1 Defining a Matrix in R Entering by Columns Entering by Rows Entering
More informationTutorial 2: Using Excel in Data Analysis
Tutorial 2: Using Excel in Data Analysis This tutorial guide addresses several issues particularly relevant in the context of the level 1 Physics lab sessions at Durham: organising your work sheet neatly,
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent
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 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 informationFACTORING SPARSE POLYNOMIALS
FACTORING SPARSE POLYNOMIALS Theorem 1 (Schinzel): Let r be a positive integer, and fix non-zero integers a 0,..., a r. Let F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0. Then there exist finite sets S
More informationChapter 6: Break-Even & CVP Analysis
HOSP 1107 (Business Math) Learning Centre Chapter 6: Break-Even & CVP Analysis One of the main concerns in running a business is achieving a desired level of profitability. Cost-volume profit analysis
More informationChapter 2. Software: (preview draft) Getting Started with Stella and Vensim
Chapter. Software: (preview draft) Getting Started with Stella and Vensim Stella and Vensim are icon-based programs to support the construction and testing of system dynamics models. I use these programs
More informationCS171 Visualization. The Visualization Alphabet: Marks and Channels. Alexander Lex alex@seas.harvard.edu. [xkcd]
CS171 Visualization Alexander Lex alex@seas.harvard.edu The Visualization Alphabet: Marks and Channels [xkcd] This Week Thursday: Task Abstraction, Validation Homework 1 due on Friday! Any more problems
More informationNext Generation Intrusion Detection: Autonomous Reinforcement Learning of Network Attacks
Next Generation Intrusion Detection: Autonomous Reinforcement Learning of Network Attacks James Cannady Georgia Tech Information Security Center Georgia Institute of Technology Atlanta, GA 30332-0832 james.cannady@gtri.gatech.edu
More informationFinancial Econometrics MFE MATLAB Introduction. Kevin Sheppard University of Oxford
Financial Econometrics MFE MATLAB Introduction Kevin Sheppard University of Oxford October 21, 2013 2007-2013 Kevin Sheppard 2 Contents Introduction i 1 Getting Started 1 2 Basic Input and Operators 5
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 information521493S Computer Graphics. Exercise 2 & course schedule change
521493S Computer Graphics Exercise 2 & course schedule change Course Schedule Change Lecture from Wednesday 31th of March is moved to Tuesday 30th of March at 16-18 in TS128 Question 2.1 Given two nonparallel,
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More informationBayesian Machine Learning (ML): Modeling And Inference in Big Data. Zhuhua Cai Google, Rice University caizhua@gmail.com
Bayesian Machine Learning (ML): Modeling And Inference in Big Data Zhuhua Cai Google Rice University caizhua@gmail.com 1 Syllabus Bayesian ML Concepts (Today) Bayesian ML on MapReduce (Next morning) Bayesian
More informationBedford, Fowler: Statics. Chapter 4: System of Forces and Moments, Examples via TK Solver
System of Forces and Moments Introduction The moment vector of a force vector,, with respect to a point has a magnitude equal to the product of the force magnitude, F, and the perpendicular distance from
More informationName: Section Registered In:
Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are
More informationFigure 1. An embedded chart on a worksheet.
8. Excel Charts and Analysis ToolPak Charts, also known as graphs, have been an integral part of spreadsheets since the early days of Lotus 1-2-3. Charting features have improved significantly over the
More informationScatter Plot, Correlation, and Regression on the TI-83/84
Scatter Plot, Correlation, and Regression on the TI-83/84 Summary: When you have a set of (x,y) data points and want to find the best equation to describe them, you are performing a regression. This page
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 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 informationAPPM4720/5720: Fast algorithms for big data. Gunnar Martinsson The University of Colorado at Boulder
APPM4720/5720: Fast algorithms for big data Gunnar Martinsson The University of Colorado at Boulder Course objectives: The purpose of this course is to teach efficient algorithms for processing very large
More informationu = [ 2 4 5] has one row with three components (a 3 v = [2 4 5] has three rows separated by semicolons (a 3 w = 2:5 generates the row vector w = [ 2 3
MATLAB Tutorial You need a small numb e r of basic commands to start using MATLAB. This short tutorial describes those fundamental commands. You need to create vectors and matrices, to change them, and
More information