Math 98 - Introduction to MATLAB Programming. Spring Lecture 3

Size: px
Start display at page:

Download "Math 98 - Introduction to MATLAB Programming. Spring Lecture 3"

Transcription

1

2 Reminders Instructor: Chris Policastro Class Website: Assignment Submission: Homework 2 1 Due September 8th by 11:59pm on bcourses. 2 See assignment description for hints. 3 You are encouraged to collaborate. Use the discussion page on bcourses.

3 Exercise Remember the formula for the sum of n consecutive numbers n = We want to compute this number four ways n(n + 1) 2 1 Use the MATLAB function sum. Recall that we studied sum in lecture 2. 2 Set S=0. Use a for loop to redene S at each iteration. Compare to problem about tiling a disk in lecture 2. 3 Set S=0. Use a while loop to redene S at each step. Remember to update the number counting iterations. 4 Write a function called ConSum such that ConSum(n) equals 1 n(n + 1). 2.

4 Lecture 3: Using functions to package code 1 Problems (problem_3_1) Approximating square root (Chapter 5.1 van Loan) (problem_3_2) Transition matrices (Chapter 7.1 van Loan) 2 Concepts functions, local variables matrices, matrix operations

5 (problem_3_1) Approximating square root Let A be a positive number. Think of A as the area of a square with side length A. We can construct a rectangle with side lengths L and W that has area A. This means A = L W If L W, then A L W This gives an approximation to the number A. Let us start with some L and W. Let us bring them nearer in value through averaging.

6 (problem_3_1) Approximating square root We want to write a program to implement the following steps 1 If A = 0, then output 0. 2 Otherwise, start with L = A and W = 1. 3 Replace L by the average (L + W )/2 4 Replace W by W/ 1 2 (L + W ) 5 Repeat until L W is less than a tolerance Tol 6 Output L Here Tol is a small number reecting the accuracy of the approximation. Small tolerance implies good accuracy. We do not know how many times we must repeat the steps. Therefore we should use a while loop. While L-W >= Tol, we can repeat the steps.

7 Functions Remember the following terms 1 code: a statement to be executed by the computer. For example, a conditional statement. The code implements a procedure. 2 program: a collection of codes 3 debug: to remove errors from a program 4 script: a le containing a program. We run a script. For example, run problem_1_1.m. A function packages code. However, it is dierent from a script. function: a le containing a program. We call a function. For example, CalcCents(100). Note the le name is CalcCents.m.

8 Functions We have functions in addition to scripts because functions are 1 reusable A function replaces a repeated block of code in a program 2 simplifying A function organizes groups of code within a program. A function can be written in a separate le with its own variables. 3 changeable If procedures are packaged as a function, then they are easily accessed. Helpful for code that must be modied by user or programmer.

9 Functions The format for functions is 1 %%%Name.m%%% 2 f u n c t i o n [ o utput ] = Name( i n p u t ) 3 4 Code 5 6 end The name of the function should match the name of the M-le. Use a capital letter in le name to avoid conict with built-in MATLAB functions

10 Functions Remember that MATLAB has a built-in function for computing square root. It is sqrt. 1 %%% S q r t.m %%% 2 % Computes s q u a r e r o o t o f a number u s i n g MATLAB f u n c t i o n s q r t 3 % I n p u t : a p o s i t i v e number 4 % Output : t h e s q u a r e r o o t 5 % Example : S q r t ( 4 ) w i l l y i e l d f u n c t i o n [ outputnumber ] = S q r t ( inputnumber ) 8 9 outputnumber=s q r t ( inputnumber ) ; end

11 nargin The command nargin appears within the code for a function. It is the number of inputs specied by the user. 1 %%% S q r t.m %%% 2 % Computes s q u a r e r o o t o f a number u s i n g MATLAB f u n c t i o n s q r t 3 % I n p u t : a p o s i t i v e number 4 % Output : t h e s q u a r e r o o t 5 % Example : S q r t ( 4 ) w i l l y i e l d f u n c t i o n [ outputnumber, x ] = S q r t ( inputnumber ) 8 x=n a r g i n ; 9 outputnumber=s q r t ( inputnumber ) ; 10 end

12 Testing Inputs We cannot assume that the user has correctly entered input. If the user enters incorrect input, then the function cannot be called. This makes it seem that there is an error in the function. You must add comments in the le that explain the proper format for inputs. You should include an example in the comments. Type help filename.m to display comments in command window. Compare to help for built in MATLAB function. Checking that the input has the proper format can avoid an error caused by the user.

13 return The command return halts execution of a program. The user returns to the command window. 1 %%% S q r t.m %%% 2 % Computes s q u a r e r o o t o f a number u s i n g MATLAB f u n c t i o n s q r t 3 % I n p u t : a p o s i t i v e number 4 % Output : t h e s q u a r e r o o t 5 % Example : S q r t ( 4 ) w i l l y i e l d f u n c t i o n [ outputnumber ] = S q r t ( inputnumber ) 8 i f ( n a r g i n ~= 1) 9 f p r i n t f ( ' Read comments! ' ) ; 10 r e t u r n 11 e l s e 12 outputnumber=s q r t ( inputnumber ) ; 13 end 14 end

14 Variables in Functions 1 %%%DoubleSum.m%%% 2 f u n c t i o n [ z, V a r i a b l e s ]=DoubleSum ( x, y ) 3 V a r i a b l e s =[ e x i s t ( ' x ' ) ] ; 4 5 f u n c t i o n t=add ( r, s ) 6 t=r+s ; 7 end 8 9 V a r i a b l e s =[ V a r i a b l e s, e x i s t ( ' r ' ) ] ; 10 V a r i a b l e s =[ V a r i a b l e s, e x i s t ( ' t ' ) ] ; z =2 (Add ( x, y ) ) ; V a r i a b l e s =[ V a r i a b l e s, e x i s t ( ' r ' ) ] ; 15 V a r i a b l e s =[ V a r i a b l e s, e x i s t ( ' t ' ) ] ; 16 end Can you call the subfunction in lines 5-7 called Add?

15 Variables in Functions Remember that the Workspace is a special le called a MAT-le which contains the variables we have dened during our MATLAB session. MATLAB must allocate memory to call a function. Think of a function as having its own Workspace. MATLAB takes the following steps 1 We enter input to execute the function. MATLAB creates a MAT-le. 2 MATLAB goes through the code. Each time a variable is dened, the variable is saved in the MAT-le. 3 MATLAB will not look other places for variable. It will not look in the WorkSpace. Suppose the code species x=7. Suppose we have x= in the Workspace. There will not be an error. 4 After going through the code, MATLAB has the numbers generated by the program.matlab saves the numbers to the output variables specied by the user. These appear in the Workspace. 5 The MAT-le is deleted.

16 Variables in Functions This means that 1 Variables in functions are not global Variables in a function cannot be accessed or modied by other functions. Therefore variables within a function cannot conict with variables of the same name outside the function. 2 Variables in functions do not persist When the code in a function nishes execution, the values assigned to variables are not stored in the Workspace. See lecture 4 for more information on the scope of variables in functions.

17 Review of problem_3_1 The debugger is a tool to help locate errors in a code. Click the dash next to lines 9 of problem_3_1.m. A red dot will appear next to the line. It is called a break point Calling problem_3_1 will prompt the debugger.

18 Review of problem_3_1 We want to see how variables are stored during the execution of the program in problem_3_1. Try x = problem_3_1(8,1e-4). Remember that this is scientic notation for Use the command dbstep to go line by line through the code. 2 Use dbstack to check your spot in the code. This will tell you the function being called. 3 Exit using dbquit. 4 Click on the red dot to make it disappear.

19 (exercise_3_1) Consider the following version of IsMagic.m the program from homework assignment 1. 1 i n p u t M a t r i x=i n p u t ( ' E n t e r a m a t r i x : ' ) ; 2 3 % sum o f t h e columns, rows, and d i a g o n a l s o f t h e i n p u t m a t r i x 4 sumcol = sum ( i n p u t M a t r i x ) ; 5 sumrow = sum ( t r a n s ( i n p u t M a t r i x ) ) ; 6 sumdiag1 = t r a c e ( i n p u t M a t r i x ) 7 sumdiag2 = t r a c e ( f l i p u d ( i n p u t M a t r i x ) ) ; 8 9 % Set r e s u l t = 1 i f a l l t h e sums a r e e q u a l. Otherwise, s e t r e s u l t = r e s u l t = i s e q u a l ( sumcol, sumrow ', sumdiag1 ones ( 1, n ), sumdiag2 ones ( 1, n ) ) ; r e t u r n

20 (exercise_3_1) Try to make the following improvement. 1 Turn into a function 2 Use debugger to locate errors 3 Improve comments 4 Check user input Use [m,n] = size(inputmatrix) to calculate the number of rows (m) and number of columns. Check m == n Check m >= 3 Use all to check that all entries are nonzero. Can you check that all entries are positive? Use Vector=inputMatrix(:)' to create a row vector. Consider the vector Vector01=(Vector > 0) If the user does not specify a matrix, then set the default input to be magic(4).

21 (problem_3_2) Transition matrices Let Let x be a 4 1 column vec P = x 1 x = x 2 x 3 x 4

22 (problem_3_2) Transition matrices Think of the entries of x as corresponding to four possible states of a system. Each state has a probability of occuring. x i is the probability of the system being in the ith state. For example x 1 = 0.4 means there is a 40% chance of the system being in state 1 Think of the (i, j)th entry of P as being the probability of transitioning from state j to state i during one step of a process. We call P a transition matrix because y = P x is the probability of the states after one step of the process.

23 (problem_3_2) Transition matrices We want to write a function called problem_3_2 such that 1 the function has two inputs P an n n matrix x an n 1 vector 2 the function outputs y = P x

24 Review of problem_3_2 The function problem_3_2 helps us study the likelihood of states of the system. In particular, we would like to study the probabilities as the process goes on for a long time. This means that we want to calculate P N x for N a large number. This kind of process is called a Markov process. See Chapter 7.1 of van Loan for more information on matrices and matrix operations.

25 (exercise_3_2) Take P to be P = We want to nd x such that x = P x. The probabilities of the system being in states 1,2,3,4 does not change during the process. Use the command [Vect,Val]=eig(P) to return a 4 4 matrix Vect and a 4 4 matrix Val. Note that the columns of Vect are eigenvectors of P. Note Val is a diagonal matrix consisting of eigenvalues. More precisely is approximately zero. What eigenvector are we looking for? P Vect - Vect Val

Lecture 4: Matrices. Math 98, Spring Math 98, Spring 2018 Lecture 4: Matrices 1 / 20

Lecture 4: Matrices. Math 98, Spring Math 98, Spring 2018 Lecture 4: Matrices 1 / 20 Lecture 4: Matrices Math 98, Spring 2018 Math 98, Spring 2018 Lecture 4: Matrices 1 / 20 Reminders Instructor: Eric Hallman Login:!cmfmath98 Password: c@1analog Class Website: https://math.berkeley.edu/~ehallman/98-fa18/

More information

Lecture 10: Powers of Matrices, Difference Equations

Lecture 10: Powers of Matrices, Difference Equations Lecture 10: Powers of Matrices, Difference Equations Difference Equations A difference equation, also sometimes called a recurrence equation is an equation that defines a sequence recursively, i.e. each

More information

MAT 1302B Mathematical Methods II

MAT 1302B Mathematical Methods II MAT 1302B Mathematical Methods II Alistair Savage Mathematics and Statistics University of Ottawa Winter 2015 Lecture 19 Alistair Savage (uottawa) MAT 1302B Mathematical Methods II Winter 2015 Lecture

More information

CSCE 155N Fall Homework Assignment 2: Stress-Strain Curve. Assigned: September 11, 2012 Due: October 02, 2012

CSCE 155N Fall Homework Assignment 2: Stress-Strain Curve. Assigned: September 11, 2012 Due: October 02, 2012 CSCE 155N Fall 2012 Homework Assignment 2: Stress-Strain Curve Assigned: September 11, 2012 Due: October 02, 2012 Note: This assignment is to be completed individually - collaboration is strictly prohibited.

More information

MATLAB BASICS. Instructor: Prof. Shahrouk Ahmadi. TA: Kartik Bulusu

MATLAB BASICS. Instructor: Prof. Shahrouk Ahmadi. TA: Kartik Bulusu MATLAB BASICS Instructor: Prof. Shahrouk Ahmadi 1. What are M-files TA: Kartik Bulusu M-files are files that contain a collection of MATLAB commands or are used to define new MATLAB functions. For the

More information

Matrix-Vector Operations

Matrix-Vector Operations Week3 Matrix-Vector Operations 31 Opening Remarks 311 Timmy Two Space View at edx Homework 3111 Click on the below link to open a browser window with the Timmy Two Space exercise This exercise was suggested

More information

Matrix-Vector Operations

Matrix-Vector Operations Week3 Matrix-Vector Operations 31 Opening Remarks 311 Timmy Two Space View at edx Homework 3111 Click on the below link to open a browser window with the Timmy Two Space exercise This exercise was suggested

More information

22A-2 SUMMER 2014 LECTURE Agenda

22A-2 SUMMER 2014 LECTURE Agenda 22A-2 SUMMER 204 LECTURE 2 NATHANIEL GALLUP The Dot Product Continued Matrices Group Work Vectors and Linear Equations Agenda 2 Dot Product Continued Angles between vectors Given two 2-dimensional vectors

More information

Example Example Example Example

Example Example Example Example 7. More Practice with Iteration and Conditionals Through Graphics We will Draw Pictures Using Three User-Defined* Graphics Functions For-Loop Problems Introduce While-Loops DrawRect DrawDisk DrawStar Rectangles

More information

1 Overview of Simulink. 2 State-space equations

1 Overview of Simulink. 2 State-space equations Modelling and simulation of engineering systems Simulink Exercise 1 - translational mechanical systems Dr. M. Turner (mct6@sun.engg.le.ac.uk 1 Overview of Simulink Simulink is a package which runs in the

More information

Math Assignment 3 - Linear Algebra

Math Assignment 3 - Linear Algebra Math 216 - Assignment 3 - Linear Algebra Due: Tuesday, March 27. Nothing accepted after Thursday, March 29. This is worth 15 points. 10% points off for being late. You may work by yourself or in pairs.

More information

COMS 6100 Class Notes

COMS 6100 Class Notes COMS 6100 Class Notes Daniel Solus September 20, 2016 1 General Remarks The Lecture notes submitted by the class have been very good. Integer division seemed to be a common oversight when working the Fortran

More information

WILEY. Differential Equations with MATLAB (Third Edition) Brian R. Hunt Ronald L. Lipsman John E. Osborn Jonathan M. Rosenberg

WILEY. Differential Equations with MATLAB (Third Edition) Brian R. Hunt Ronald L. Lipsman John E. Osborn Jonathan M. Rosenberg Differential Equations with MATLAB (Third Edition) Updated for MATLAB 2011b (7.13), Simulink 7.8, and Symbolic Math Toolbox 5.7 Brian R. Hunt Ronald L. Lipsman John E. Osborn Jonathan M. Rosenberg All

More information

CS Homework 3. October 15, 2009

CS Homework 3. October 15, 2009 CS 294 - Homework 3 October 15, 2009 If you have questions, contact Alexandre Bouchard (bouchard@cs.berkeley.edu) for part 1 and Alex Simma (asimma@eecs.berkeley.edu) for part 2. Also check the class website

More information

A First Course on Kinetics and Reaction Engineering Example S3.1

A First Course on Kinetics and Reaction Engineering Example S3.1 Example S3.1 Problem Purpose This example shows how to use the MATLAB script file, FitLinSR.m, to fit a linear model to experimental data. Problem Statement Assume that in the course of solving a kinetics

More information

Note: The command name is upper case in the description given by help, but must be lower case in actual use. And the backslash Anb is dierent when A i

Note: The command name is upper case in the description given by help, but must be lower case in actual use. And the backslash Anb is dierent when A i MATLAB Tutorial You need a small number 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 to

More information

Introduction to Matlab

Introduction to Matlab History of Matlab Starting Matlab Matrix operation Introduction to Matlab Useful commands in linear algebra Scripts-M file Use Matlab to explore the notion of span and the geometry of eigenvalues and eigenvectors.

More information

Matlab Exercise 0 Due 1/25/06

Matlab Exercise 0 Due 1/25/06 Matlab Exercise 0 Due 1/25/06 Geop 523 Theoretical Seismology January 18, 2006 Much of our work in this class will be done using Matlab. The goal of this exercise is to get you familiar with using Matlab

More information

Math Camp Notes: Linear Algebra II

Math Camp Notes: Linear Algebra II Math Camp Notes: Linear Algebra II Eigenvalues Let A be a square matrix. An eigenvalue is a number λ which when subtracted from the diagonal elements of the matrix A creates a singular matrix. In other

More information

Finite Math - J-term Section Systems of Linear Equations in Two Variables Example 1. Solve the system

Finite Math - J-term Section Systems of Linear Equations in Two Variables Example 1. Solve the system Finite Math - J-term 07 Lecture Notes - //07 Homework Section 4. - 9, 0, 5, 6, 9, 0,, 4, 6, 0, 50, 5, 54, 55, 56, 6, 65 Section 4. - Systems of Linear Equations in Two Variables Example. Solve the system

More information

Introduction to Computational Neuroscience

Introduction to Computational Neuroscience CSE2330 Introduction to Computational Neuroscience Basic computational tools and concepts Tutorial 1 Duration: two weeks 1.1 About this tutorial The objective of this tutorial is to introduce you to: the

More information

(t) ), ( ) where t is time, T is the reaction time, u n is the position of the nth car, and the "sensitivity coefficient" λ may depend on un 1(

(t) ), ( ) where t is time, T is the reaction time, u n is the position of the nth car, and the sensitivity coefficient λ may depend on un 1( Page 1 A Model of Traffic Flow Everyone has had the experience of sitting in a traffic jam, or of seeing cars bunch up on a road for no apparent good reason MATLAB and SIMULINK are good tools for studying

More information

This appendix provides a very basic introduction to linear algebra concepts.

This appendix provides a very basic introduction to linear algebra concepts. APPENDIX Basic Linear Algebra Concepts This appendix provides a very basic introduction to linear algebra concepts. Some of these concepts are intentionally presented here in a somewhat simplified (not

More information

MOL410/510 Problem Set 1 - Linear Algebra - Due Friday Sept. 30

MOL410/510 Problem Set 1 - Linear Algebra - Due Friday Sept. 30 MOL40/50 Problem Set - Linear Algebra - Due Friday Sept. 30 Use lab notes to help solve these problems. Problems marked MUST DO are required for full credit. For the remainder of the problems, do as many

More information

Section 4.1 The Power Method

Section 4.1 The Power Method Section 4.1 The Power Method Key terms Dominant eigenvalue Eigenpair Infinity norm and 2-norm Power Method Scaled Power Method Power Method for symmetric matrices The notation used varies a bit from that

More information

1 R.V k V k 1 / I.k/ here; we ll stimulate the action potential another way.) Note that this further simplifies to. m 3 k h k.

1 R.V k V k 1 / I.k/ here; we ll stimulate the action potential another way.) Note that this further simplifies to. m 3 k h k. 1. The goal of this problem is to simulate a propagating action potential for the Hodgkin-Huxley model and to determine the propagation speed. From the class notes, the discrete version (i.e., after breaking

More information

Lab 2 Worksheet. Problems. Problem 1: Geometry and Linear Equations

Lab 2 Worksheet. Problems. Problem 1: Geometry and Linear Equations Lab 2 Worksheet Problems Problem : Geometry and Linear Equations Linear algebra is, first and foremost, the study of systems of linear equations. You are going to encounter linear systems frequently in

More information

Math/Phys/Engr 428, Math 529/Phys 528 Numerical Methods - Summer Homework 3 Due: Tuesday, July 3, 2018

Math/Phys/Engr 428, Math 529/Phys 528 Numerical Methods - Summer Homework 3 Due: Tuesday, July 3, 2018 Math/Phys/Engr 428, Math 529/Phys 528 Numerical Methods - Summer 28. (Vector and Matrix Norms) Homework 3 Due: Tuesday, July 3, 28 Show that the l vector norm satisfies the three properties (a) x for x

More information

PageRank: The Math-y Version (Or, What To Do When You Can t Tear Up Little Pieces of Paper)

PageRank: The Math-y Version (Or, What To Do When You Can t Tear Up Little Pieces of Paper) PageRank: The Math-y Version (Or, What To Do When You Can t Tear Up Little Pieces of Paper) In class, we saw this graph, with each node representing people who are following each other on Twitter: Our

More information

Lab 2: Static Response, Cantilevered Beam

Lab 2: Static Response, Cantilevered Beam Contents 1 Lab 2: Static Response, Cantilevered Beam 3 1.1 Objectives.......................................... 3 1.2 Scalars, Vectors and Matrices (Allen Downey)...................... 3 1.2.1 Attribution.....................................

More information

Properties of Linear Transformations from R n to R m

Properties of Linear Transformations from R n to R m Properties of Linear Transformations from R n to R m MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Topic Overview Relationship between the properties of a matrix transformation

More information

22A-2 SUMMER 2014 LECTURE 5

22A-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 information

Optimization and Calculus

Optimization and Calculus Optimization and Calculus To begin, there is a close relationship between finding the roots to a function and optimizing a function. In the former case, we solve for x. In the latter, we solve: g(x) =

More information

AMSC/CMSC 466 Problem set 3

AMSC/CMSC 466 Problem set 3 AMSC/CMSC 466 Problem set 3 1. Problem 1 of KC, p180, parts (a), (b) and (c). Do part (a) by hand, with and without pivoting. Use MATLAB to check your answer. Use the command A\b to get the solution, and

More information

Math 515 Fall, 2008 Homework 2, due Friday, September 26.

Math 515 Fall, 2008 Homework 2, due Friday, September 26. Math 515 Fall, 2008 Homework 2, due Friday, September 26 In this assignment you will write efficient MATLAB codes to solve least squares problems involving block structured matrices known as Kronecker

More information

Designing Information Devices and Systems I Fall 2018 Homework 3

Designing Information Devices and Systems I Fall 2018 Homework 3 Last Updated: 28-9-5 :8 EECS 6A Designing Information Devices and Systems I Fall 28 Homework 3 This homework is due September 4, 28, at 23:59. Self-grades are due September 8, 28, at 23:59. Submission

More information

Numerical Linear Algebra

Numerical Linear Algebra Math 45 David Arnold David-Arnold@Eureka.redwoods.cc.ca.us Abstract In this activity you will learn how to solve systems of linear equations using LU decomposition, with both forward and substitution.

More information

4. Linear transformations as a vector space 17

4. Linear transformations as a vector space 17 4 Linear transformations as a vector space 17 d) 1 2 0 0 1 2 0 0 1 0 0 0 1 2 3 4 32 Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation

More information

MATH 341 MIDTERM 2. (a) [5 pts] Demonstrate that A and B are row equivalent by providing a sequence of row operations leading from A to B.

MATH 341 MIDTERM 2. (a) [5 pts] Demonstrate that A and B are row equivalent by providing a sequence of row operations leading from A to B. 11/01/2011 Bormashenko MATH 341 MIDTERM 2 Show your work for all the problems. Good luck! (1) Let A and B be defined as follows: 1 1 2 A =, B = 1 2 3 0 2 ] 2 1 3 4 Name: (a) 5 pts] Demonstrate that A and

More information

Numerical Linear Algebra

Numerical Linear Algebra Introduction Numerical Linear Algebra Math 45 Linear Algebra David Arnold David-Arnold@Eureka.redwoods.cc.ca.us Abstract In this activity you will learn how to solve systems of linear equations using LU

More information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors MAT 67L, Laboratory III Contents Instructions (1) Read this document. (2) The questions labeled Experiments are not graded, and should not be turned in. They are designed for

More information

MATH3200, Lecture 31: Applications of Eigenvectors. Markov Chains and Chemical Reaction Systems

MATH3200, Lecture 31: Applications of Eigenvectors. Markov Chains and Chemical Reaction Systems Lecture 31: Some Applications of Eigenvectors: Markov Chains and Chemical Reaction Systems Winfried Just Department of Mathematics, Ohio University April 9 11, 2018 Review: Eigenvectors and left eigenvectors

More information

= main diagonal, in the order in which their corresponding eigenvectors appear as columns of E.

= main diagonal, in the order in which their corresponding eigenvectors appear as columns of E. 3.3 Diagonalization Let A = 4. Then and are eigenvectors of A, with corresponding eigenvalues 2 and 6 respectively (check). This means 4 = 2, 4 = 6. 2 2 2 2 Thus 4 = 2 2 6 2 = 2 6 4 2 We have 4 = 2 0 0

More information

Homework 6: Image Completion using Mixture of Bernoullis

Homework 6: Image Completion using Mixture of Bernoullis Homework 6: Image Completion using Mixture of Bernoullis Deadline: Wednesday, Nov. 21, at 11:59pm Submission: You must submit two files through MarkUs 1 : 1. a PDF file containing your writeup, titled

More information

Eigenvalue and Eigenvector Homework

Eigenvalue and Eigenvector Homework Eigenvalue and Eigenvector Homework Olena Bormashenko November 4, 2 For each of the matrices A below, do the following:. Find the characteristic polynomial of A, and use it to find all the eigenvalues

More information

Lecture 3: Special Matrices

Lecture 3: Special Matrices Lecture 3: Special Matrices Feedback of assignment1 Random matrices The magic matrix commend magic() doesn t give us random matrix. Random matrix means we will get different matrices each time when we

More information

CSCI 2033 Spring 2016 ELEMENTARY COMPUTATIONAL LINEAR ALGEBRA

CSCI 2033 Spring 2016 ELEMENTARY COMPUTATIONAL LINEAR ALGEBRA CSCI 2033 Spring 2016 ELEMENTARY COMPUTATIONAL LINEAR ALGEBRA Class time : MW 4:00-5:15pm Room : Vincent Hall 16 Instructor : Yousef Saad URL : www-users.cselabs.umn.edu/classes/spring-2016 /csci2033 afternoon

More information

Image Registration Lecture 2: Vectors and Matrices

Image Registration Lecture 2: Vectors and Matrices Image Registration Lecture 2: Vectors and Matrices Prof. Charlene Tsai Lecture Overview Vectors Matrices Basics Orthogonal matrices Singular Value Decomposition (SVD) 2 1 Preliminary Comments Some of this

More information

COLUMN VECTORS a=[1;2;3] b. a(1) = first.entry of a

COLUMN VECTORS a=[1;2;3] b. a(1) = first.entry of a Lecture 2 vectors and matrices ROW VECTORS Enter the following in SciLab: [1,2,3] scilab notation for row vectors [8]==8 a=[2 3 4] separate entries with spaces or commas b=[10,10,10] a+b, b-a add, subtract

More information

Exercise 1.3 Work out the probabilities that it will be cloudy/rainy the day after tomorrow.

Exercise 1.3 Work out the probabilities that it will be cloudy/rainy the day after tomorrow. 54 Chapter Introduction 9 s Exercise 3 Work out the probabilities that it will be cloudy/rainy the day after tomorrow Exercise 5 Follow the instructions for this problem given on the class wiki For the

More information

Eigenvalues and eigenvectors

Eigenvalues and eigenvectors Roberto s Notes on Linear Algebra Chapter 0: Eigenvalues and diagonalization Section Eigenvalues and eigenvectors What you need to know already: Basic properties of linear transformations. Linear systems

More information

Matlab Instruction Primer; Chem 691, Spring 2016

Matlab Instruction Primer; Chem 691, Spring 2016 1 Matlab Instruction Primer; Chem 691, Spring 2016 This version dated February 10, 2017 CONTENTS I. Help: To obtain information about any instruction in Matlab 1 II. Scripting 1 III. Loops, determine an

More information

Lecture 5b: Starting Matlab

Lecture 5b: Starting Matlab Lecture 5b: Starting Matlab James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University August 7, 2013 Outline 1 Resources 2 Starting Matlab 3 Homework

More information

3.3 Real Zeros of Polynomial Functions

3.3 Real Zeros of Polynomial Functions 71_00.qxp 12/27/06 1:25 PM Page 276 276 Chapter Polynomial and Rational Functions. Real Zeros of Polynomial Functions Long Division of Polynomials Consider the graph of f x 6x 19x 2 16x 4. Notice in Figure.2

More information

[ Here 21 is the dot product of (3, 1, 2, 5) with (2, 3, 1, 2), and 31 is the dot product of

[ Here 21 is the dot product of (3, 1, 2, 5) with (2, 3, 1, 2), and 31 is the dot product of . Matrices A matrix is any rectangular array of numbers. For example 3 5 6 4 8 3 3 is 3 4 matrix, i.e. a rectangular array of numbers with three rows four columns. We usually use capital letters for matrices,

More information

Homework Assignment 4 Root Finding Algorithms The Maximum Velocity of a Car Due: Friday, February 19, 2010 at 12noon

Homework Assignment 4 Root Finding Algorithms The Maximum Velocity of a Car Due: Friday, February 19, 2010 at 12noon ME 2016 A Spring Semester 2010 Computing Techniques 3-0-3 Homework Assignment 4 Root Finding Algorithms The Maximum Velocity of a Car Due: Friday, February 19, 2010 at 12noon Description and Outcomes In

More information

Designing Information Devices and Systems I Spring 2018 Homework 5

Designing Information Devices and Systems I Spring 2018 Homework 5 EECS 16A Designing Information Devices and Systems I Spring 2018 Homework All problems on this homework are practice, however all concepts covered here are fair game for the exam. 1. Sports Rank Every

More information

MATLAB and Mathematical Introduction Will be discussed in class on 1/24/11

MATLAB and Mathematical Introduction Will be discussed in class on 1/24/11 MATLAB and Mathematical Introduction Will be discussed in class on 1/24/11 GEOP 523; Theoretical Seismology January 17, 2011 Much of our work in this class will be done using MATLAB. The goal of this exercise

More information

Convex Optimization / Homework 1, due September 19

Convex Optimization / Homework 1, due September 19 Convex Optimization 1-725/36-725 Homework 1, due September 19 Instructions: You must complete Problems 1 3 and either Problem 4 or Problem 5 (your choice between the two). When you submit the homework,

More information

Designing Information Devices and Systems I Fall 2017 Homework 3

Designing Information Devices and Systems I Fall 2017 Homework 3 EECS 6A Designing Information Devices and Systems I Fall 07 Homework 3 This homework is due September 8, 07, at 3:9. Self-grades are due September, 07, at 3:9. Submission Format Your homework submission

More information

Math 552 Scientific Computing II Spring SOLUTIONS: Homework Set 1

Math 552 Scientific Computing II Spring SOLUTIONS: Homework Set 1 Math 552 Scientific Computing II Spring 21 SOLUTIONS: Homework Set 1 ( ) a b 1 Let A be the 2 2 matrix A = By hand, use Gaussian elimination with back c d substitution to obtain A 1 by solving the two

More information

Homework 1 Solutions

Homework 1 Solutions 18-9 Signals and Systems Profs. Byron Yu and Pulkit Grover Fall 18 Homework 1 Solutions Part One 1. (8 points) Consider the DT signal given by the algorithm: x[] = 1 x[1] = x[n] = x[n 1] x[n ] (a) Plot

More information

2. The Power Method for Eigenvectors

2. The Power Method for Eigenvectors 2. Power Method We now describe the power method for computing the dominant eigenpair. Its extension to the inverse power method is practical for finding any eigenvalue provided that a good initial approximation

More information

Model-building and parameter estimation

Model-building and parameter estimation Luleå University of Technology Johan Carlson Last revision: July 27, 2009 Measurement Technology and Uncertainty Analysis - E7021E MATLAB homework assignment Model-building and parameter estimation Introduction

More information

Designing Information Devices and Systems I Fall 2018 Homework 5

Designing Information Devices and Systems I Fall 2018 Homework 5 Last Updated: 08-09-9 0:6 EECS 6A Designing Information Devices and Systems I Fall 08 Homework 5 This homework is due September 8, 08, at 3:59. Self-grades are due October, 08, at 3:59. Submission Format

More information

Econ Slides from Lecture 8

Econ Slides from Lecture 8 Econ 205 Sobel Econ 205 - Slides from Lecture 8 Joel Sobel September 1, 2010 Computational Facts 1. det AB = det BA = det A det B 2. If D is a diagonal matrix, then det D is equal to the product of its

More information

Using Microsoft Excel

Using Microsoft Excel Using Microsoft Excel Objective: Students will gain familiarity with using Excel to record data, display data properly, use built-in formulae to do calculations, and plot and fit data with linear functions.

More information

New Mexico Tech Hyd 510

New Mexico Tech Hyd 510 Vectors vector - has magnitude and direction (e.g. velocity, specific discharge, hydraulic gradient) scalar - has magnitude only (e.g. porosity, specific yield, storage coefficient) unit vector - a unit

More information

PowerPoints organized by Dr. Michael R. Gustafson II, Duke University

PowerPoints 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 information

TOPIC 2 Computer application for manipulating matrix using MATLAB

TOPIC 2 Computer application for manipulating matrix using MATLAB YOGYAKARTA STATE UNIVERSITY MATHEMATICS AND NATURAL SCIENCES FACULTY MATHEMATICS EDUCATION STUDY PROGRAM TOPIC 2 Computer application for manipulating matrix using MATLAB Definition of Matrices in MATLAB

More information

Notes on singular value decomposition for Math 54. Recall that if A is a symmetric n n matrix, then A has real eigenvalues A = P DP 1 A = P DP T.

Notes on singular value decomposition for Math 54. Recall that if A is a symmetric n n matrix, then A has real eigenvalues A = P DP 1 A = P DP T. Notes on singular value decomposition for Math 54 Recall that if A is a symmetric n n matrix, then A has real eigenvalues λ 1,, λ n (possibly repeated), and R n has an orthonormal basis v 1,, v n, where

More information

Math 304 Handout: Linear algebra, graphs, and networks.

Math 304 Handout: Linear algebra, graphs, and networks. Math 30 Handout: Linear algebra, graphs, and networks. December, 006. GRAPHS AND ADJACENCY MATRICES. Definition. A graph is a collection of vertices connected by edges. A directed graph is a graph all

More information

Lab 1: Dynamic Simulation Using Simulink and Matlab

Lab 1: Dynamic Simulation Using Simulink and Matlab Lab 1: Dynamic Simulation Using Simulink and Matlab Objectives In this lab you will learn how to use a program called Simulink to simulate dynamic systems. Simulink runs under Matlab and uses block diagrams

More information

Math 504 (Fall 2011) 1. (*) Consider the matrices

Math 504 (Fall 2011) 1. (*) Consider the matrices Math 504 (Fall 2011) Instructor: Emre Mengi Study Guide for Weeks 11-14 This homework concerns the following topics. Basic definitions and facts about eigenvalues and eigenvectors (Trefethen&Bau, Lecture

More information

1 Linearity and Linear Systems

1 Linearity and Linear Systems Mathematical Tools for Neuroscience (NEU 34) Princeton University, Spring 26 Jonathan Pillow Lecture 7-8 notes: Linear systems & SVD Linearity and Linear Systems Linear system is a kind of mapping f( x)

More information

chapter 5 INTRODUCTION TO MATRIX ALGEBRA GOALS 5.1 Basic Definitions

chapter 5 INTRODUCTION TO MATRIX ALGEBRA GOALS 5.1 Basic Definitions chapter 5 INTRODUCTION TO MATRIX ALGEBRA GOALS The purpose of this chapter is to introduce you to matrix algebra, which has many applications. You are already familiar with several algebras: elementary

More information

14. Properties of the Determinant

14. Properties of the Determinant 14. Properties of the Determinant Last time we showed that the erminant of a matrix is non-zero if and only if that matrix is invertible. We also showed that the erminant is a multiplicative function,

More information

MATH 315 Linear Algebra Homework #1 Assigned: August 20, 2018

MATH 315 Linear Algebra Homework #1 Assigned: August 20, 2018 Homework #1 Assigned: August 20, 2018 Review the following subjects involving systems of equations and matrices from Calculus II. Linear systems of equations Converting systems to matrix form Pivot entry

More information

CSC 1700 Analysis of Algorithms: Warshall s and Floyd s algorithms

CSC 1700 Analysis of Algorithms: Warshall s and Floyd s algorithms CSC 1700 Analysis of Algorithms: Warshall s and Floyd s algorithms Professor Henry Carter Fall 2016 Recap Space-time tradeoffs allow for faster algorithms at the cost of space complexity overhead Dynamic

More information

LAB 1: MATLAB - Introduction to Programming. Objective:

LAB 1: MATLAB - Introduction to Programming. Objective: LAB 1: MATLAB - Introduction to Programming Objective: The objective of this laboratory is to review how to use MATLAB as a programming tool and to review a classic analytical solution to a steady-state

More information

Image Compression Using Singular Value Decomposition

Image Compression Using Singular Value Decomposition Image Compression Using Singular Value Decomposition Ian Cooper and Craig Lorenc December 15, 2006 Abstract Singular value decomposition (SVD) is an effective tool for minimizing data storage and data

More information

Section 29: What s an Inverse?

Section 29: What s an Inverse? Section 29: What s an Inverse? Our investigations in the last section showed that all of the matrix operations had an identity element. The identity element for addition is, for obvious reasons, called

More information

Gaussian Elimination and Back Substitution

Gaussian 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 information

Announcements. Problem Set 6 due next Monday, February 25, at 12:50PM. Midterm graded, will be returned at end of lecture.

Announcements. Problem Set 6 due next Monday, February 25, at 12:50PM. Midterm graded, will be returned at end of lecture. Turing Machines Hello Hello Condensed Slide Slide Readers! Readers! This This lecture lecture is is almost almost entirely entirely animations that that show show how how each each Turing Turing machine

More information

Solution Set 7, Fall '12

Solution Set 7, Fall '12 Solution Set 7, 18.06 Fall '12 1. Do Problem 26 from 5.1. (It might take a while but when you see it, it's easy) Solution. Let n 3, and let A be an n n matrix whose i, j entry is i + j. To show that det

More information

VECTORS AND MATRICES

VECTORS AND MATRICES VECTORS AND MATRICES COMPUTER SESSION C1 BACKGROUND PREPARATIONS The session is divided into two parts. The first part involves experimenting in the Mathematics Laboratory and the second part involves

More information

Laboratory 11 Control Systems Laboratory ECE3557. State Feedback Controller for Position Control of a Flexible Joint

Laboratory 11 Control Systems Laboratory ECE3557. State Feedback Controller for Position Control of a Flexible Joint Laboratory 11 State Feedback Controller for Position Control of a Flexible Joint 11.1 Objective The objective of this laboratory is to design a full state feedback controller for endpoint position control

More information

Programming Assignment 4: Image Completion using Mixture of Bernoullis

Programming Assignment 4: Image Completion using Mixture of Bernoullis Programming Assignment 4: Image Completion using Mixture of Bernoullis Deadline: Tuesday, April 4, at 11:59pm TA: Renie Liao (csc321ta@cs.toronto.edu) Submission: You must submit two files through MarkUs

More information

The Principal Component Analysis

The Principal Component Analysis The Principal Component Analysis Philippe B. Laval KSU Fall 2017 Philippe B. Laval (KSU) PCA Fall 2017 1 / 27 Introduction Every 80 minutes, the two Landsat satellites go around the world, recording images

More information

MATRICES. a m,1 a m,n A =

MATRICES. a m,1 a m,n A = MATRICES Matrices are rectangular arrays of real or complex numbers With them, we define arithmetic operations that are generalizations of those for real and complex numbers The general form a matrix of

More information

Multivariable Control Laboratory experiment 2 The Quadruple Tank 1

Multivariable Control Laboratory experiment 2 The Quadruple Tank 1 Multivariable Control Laboratory experiment 2 The Quadruple Tank 1 Department of Automatic Control Lund Institute of Technology 1. Introduction The aim of this laboratory exercise is to study some different

More information

Class President: A Network Approach to Popularity. Due July 18, 2014

Class President: A Network Approach to Popularity. Due July 18, 2014 Class President: A Network Approach to Popularity Due July 8, 24 Instructions. Due Fri, July 8 at :59 PM 2. Work in groups of up to 3 3. Type up the report, and submit as a pdf on D2L 4. Attach the code

More information

Introduction to Hartree-Fock calculations in Spartan

Introduction to Hartree-Fock calculations in Spartan EE5 in 2008 Hannes Jónsson Introduction to Hartree-Fock calculations in Spartan In this exercise, you will get to use state of the art software for carrying out calculations of wavefunctions for molecues,

More information

Math 103, Summer 2006 Determinants July 25, 2006 DETERMINANTS. 1. Some Motivation

Math 103, Summer 2006 Determinants July 25, 2006 DETERMINANTS. 1. Some Motivation DETERMINANTS 1. Some Motivation Today we re going to be talking about erminants. We ll see the definition in a minute, but before we get into ails I just want to give you an idea of why we care about erminants.

More information

Chapter 1 Linear Equations. 1.1 Systems of Linear Equations

Chapter 1 Linear Equations. 1.1 Systems of Linear Equations Chapter Linear Equations. Systems of Linear Equations A linear equation in the n variables x, x 2,..., x n is one that can be expressed in the form a x + a 2 x 2 + + a n x n = b where a, a 2,..., a n and

More information

Review of Basic Concepts in Linear Algebra

Review of Basic Concepts in Linear Algebra Review of Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University September 7, 2017 Math 565 Linear Algebra Review September 7, 2017 1 / 40 Numerical Linear Algebra

More information

Designing Information Devices and Systems I Fall 2018 Lecture Notes Note Introduction to Linear Algebra the EECS Way

Designing Information Devices and Systems I Fall 2018 Lecture Notes Note Introduction to Linear Algebra the EECS Way EECS 16A Designing Information Devices and Systems I Fall 018 Lecture Notes Note 1 1.1 Introduction to Linear Algebra the EECS Way In this note, we will teach the basics of linear algebra and relate it

More information

Problem set 1. (c) Is the Ford-Fulkerson algorithm guaranteed to produce an acyclic maximum flow?

Problem set 1. (c) Is the Ford-Fulkerson algorithm guaranteed to produce an acyclic maximum flow? CS261, Winter 2017. Instructor: Ashish Goel. Problem set 1 Electronic submission to Gradescope due 11:59pm Thursday 2/2. Form a group of 2-3 students that is, submit one homework with all of your names.

More information

Homework and Computer Problems for Math*2130 (W17).

Homework and Computer Problems for Math*2130 (W17). Homework and Computer Problems for Math*2130 (W17). MARCUS R. GARVIE 1 December 21, 2016 1 Department of Mathematics & Statistics, University of Guelph NOTES: These questions are a bare minimum. You should

More information

Eigenvalues, Eigenvectors, and Diagonalization

Eigenvalues, Eigenvectors, and Diagonalization Week12 Eigenvalues, Eigenvectors, and Diagonalization 12.1 Opening Remarks 12.1.1 Predicting the Weather, Again Let us revisit the example from Week 4, in which we had a simple model for predicting the

More information