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### Hw2

Course: MATH 456, Fall 2009
School: Delaware
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Word Count: 166

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456, MATH/CSE Spring 2004: Homework # 2: 1) Let A be a nn matrix that has a linearly independent set of eigenvectors, {u(1) , u(2) , ...u(n) }. Let Au(i) = i u(i) , let P be the matrix whose columns are the vectors u(1) , u(2) , ...u(n) , and let D = diag(1 , 2 , ...n ). Verify that AP = P D. What is P -1 AP ? 2) Prove that if is an eigenvalue of a real matrix with eigenvector x, then is also an eigenvalue with...

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456, MATH/CSE Spring 2004: Homework # 2: 1) Let A be a nn matrix that has a linearly independent set of eigenvectors, {u(1) , u(2) , ...u(n) }. Let Au(i) = i u(i) , let P be the matrix whose columns are the vectors u(1) , u(2) , ...u(n) , and let D = diag(1 , 2 , ...n ). Verify that AP = P D. What is P -1 AP ? 2) Prove that if is an eigenvalue of a real matrix with eigenvector x, then is also an eigenvalue with with eigenvector x. 3) Devise a simple modification of the power method to handle following the case: 1 = -2 > |3 | |4 | ...|n |. 4) Assume that A and B are square matrices and either A or B is nonsingular. Prove that I - AB has the same eigenvalues as I - BA. AB has the same eigenvalues as BA. 5) Use Gershgorin's Theorem to prove that the eigenvalues of the matrix ...

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Delaware - M - 353
University of Delaware Department of Mathematical Sciences Math 353 Engineering Mathematics III 07S C. Bacuta Homework 2: Due 02/21/07 Download, print and read Introduction to MATLAB by exercises- Braun, Chapters 1-7, 10 from the class home-page. Cre
Delaware - M - 353
University of Delaware Department of Mathematical Sciences Math 353 Engineering Mathematics III 07S C. Bacuta Homework 8: Due Wednesday 04/11/071) Section 5.3 Problem 3(ad). 2) Section 5.3 Problem 4-using MATLAB. For each item hand in the Matlab co
Delaware - M - 353
University of Delaware Department of Mathematical Sciences Math 353 Engineering Mathematics III 07S C. Bacuta Homework 6:1) Section 4.3 Problem 2. Use the book function lagran and the MATLAB function polyval to check the result. 2) Section 4.3 Prob
Delaware - M - 353
Review for MIDTERM EXAM I (Wednesday, March 14).CHAPTER 1, Preliminaries: Sections 1.2, 1.3-Binary numbers, Binary fractions, Machine numbers, Computer floating point numbers.-Absolute and relative error, approximation to d significant digits-
Delaware - MATH - 349
Review for the FINAL EXAM I(Monday 12/13/04, 1:00-3:00PM).Review Midterm I and midterm II.CHAPTER 2 SYSTEMS - (Section 1-Section 4)-Consistent (inconsistent) systems. -Augmented matrix.-Row echelon form - Gaussian Elimination-Reduced row e
Delaware - M - 353
Review for MIDTERM EXAM I (Friday, March 17).CHAPTER 1, Preliminaries: Sections 1.2, 1.3-Binary numbers, Binary fractions, Machine numbers, Computer floating point numbers.-Absolute and relative error, approximation to d significant digits- Ro
Delaware - MATH - 451
Review for FINAL.3.1-3.3 Bisection method, Newton's method, Secant method. -Review problem: 3.1 #8; 3.2 #1, 9, 23; 3.3 #1, 2, 4. 4.1 Polynomial Interpolation. -Interpolating Polynomial: Lagrange Form-Interpolating Polynomial: Newton Form an
Delaware - M - 351
Review for MIDTERM EXAM II( Friday- November 11).CHAPTER 9 Vector space. Sections 9.6-9.9 -Generalized Vector Spaces, inner product, norm -Span and Subspace -Bases, Expansions, Dimension Review problems: 9.6 # 10, 12 a, d 9.7 # 1
Delaware - MATH - 456
The Final Project :Directions:1) Choose a topic of interest to you but related with the topics of the book. 2) Learn about the topic: Use references beyond the text book. 3) The project should include programming and numerical experiments
Delaware - M - 351
Review for MIDTERM EXAM I (Friday- September 30).CHAPTER 2 Differential Equations. Sections 2.2, 2.3, 2.4-First Order Linear Equations -Bernoulli's equation (Section 2.2 Exercise #9) -Applications: Radioactivity decay; Mixing problem. -Separat
Delaware - MATH - 342
Review for the FINAL EXAM (Thursday-December 16, 10:30-12:30).I) Review the the Midterm I and Midterm II. II) From Leon's book: CHAPTER 3- VECTOR SPACES (Section 1-Section 4)-Definitions, Properties, Vector spaces axioms -Subspaces, Span and
Delaware - M - 353
Review for MIDTERM EXAM II (Wednesday, April 25).CHAPTER 4, Interpolation Section 4.3 -Interpolation, extrapolation, Lagrange coefficient polynomials (Cardinal functions) , Lagrange Interpolation, Theorem 4.3, page 211. -Review problems:4.3
Delaware - M - 353
Review for MIDTERM EXAM II (Friday, April 28).CHAPTER 4, Interpolation Section 4.3 -Interpolation, extrapolation, Lagrange coefficient polynomials (Cardinal functions) , Lagrange Interpolation, Theorem 4.3, page 211. -Review problems:4.3 #1,
Delaware - MATH - 455
Review for the FINAL EXAM (Thursday, December 18, 12:20PM in 107 Wartik).3.1-3.3 Bisection method, Newton's method, Secant method. -Review problems: 3.1 #2, 4. 3.2 #6, 7, 9, 10, 23. 3.3 #2, 4. 3.4-CMT, fixed points.-Example 3, page 103,
Delaware - M - 349
Review for MIDTERM EXAM II(Tuesday -November 22).CHAPTER 3, Real Vector Space. Section: 3.8 -Rank of a matrix-Theorem 3.20 (page 210)- Rank Theorem (page 211)-Review problems: 3.8 # 13a, 17a, 26, 28, 29, 39. CHAPTER 4 INNER PRODUCT SPACE
Delaware - M - 426
UNIVERSITY OF DELAWARE Mathematical Sciences Department Math 426/Cisc 410 Introduction to Numerical Analysis and Algorithmic Computation, Fall 2008 Instructor: Dr. Richard J. Braun, Ewing 509, (302) 831-1869, braun@math.udel.edu, http:/www.math.udel.
Delaware - M - 810
UNIVERSITY OF DELAWARE Mathematical Sciences Department Math 810 Asymptotic and Perturbation Methods Spring 2009 Instructor: Dr. Richard J. Braun, Ewing 509, (302) 831-1869, braun@math.udel.edu Lecture times: MWF 2:30-3:20 in Willard 135. Office Hour
Delaware - M - 426
Math 426/CISC 410 08F, All sections R.J. Braun Homework 3 Solutions, Hints and AnswersProblem 1.1.1(c). The answer is P6 (x) = 1 + x2 x4 x6 + + . 2 24 720Problem 1.1.2(a). Note that log is denoting the natural log here. Then d[log(1 + x)]/dx = (1
Delaware - M - 426
Math 426/CISC 410 08F, All sections R.J. Braun Homework 2 SolutionProblem 1. The script and what it does in the command window are shown below. The plot is A shown in Figure 1; its not part of the L TEX le that I used to make this document. Note tha
Delaware - M - 426
Math 426/CISC 410 08F, All Sections R.J. Braun Homework 7 Solutions, Hints and AnswersProblem 3.2.1. Here we use the normal equation approach. AT A = 8 -6 -6 6 , AT b = -10 6 .Solving AT Ax = AT b gives x = [-2 - 1]T . 3.2.2. We can compute all th
Delaware - M - 426
Math 426/CISC 410 08F, All Sections R.J. Braun Homework 5 Solutions, Hints and AnswersProblem 2.4.1(d). Take limits. 1 n+ n = lim 1 + = 1; lim n n n n this implies that n + n n. Likewise, n+2 n 2 lim = lim 1 + = 1; n n n n this implies that n
Delaware - M - 426
Math 426/CISC 410 08F, All sections R.J. Braun Homework 4 Solutions, Hints and AnswersProblem 1.6.2. Note that log is denoting the natural log here. Here is one way to solve this problem. See the web page for a published version of this problem. % S
Delaware - M - 428
UNIVERSITY OF DELAWARE Mathematical Sciences Department Math 428/Cisc 411 Algorithmic &amp; Numerical Solution of Differential Equations Spring 2008 Honors Sections Instructor: Dr. Richard J. Braun, Ewing 509, (302) 831-1869, braun@math.udel.edu. Text:
Delaware - M - 426
Beginning Matlab ExercisesR. J. Braun Department of Mathematical Sciences University of Delaware Version of September 5, 20081IntroductionThis collection of exercises is intended to help you start learning Matlab. Matlab is a huge package with
Delaware - M - 426
University of Delaware Department of Mathematical Sciences Math 426/CISC 410 Intro to Numerical Analysis and Algorithmic Computing 08F R. J. Braun Your name: Project 3: SVD Image Compression In class, we used the SVD to compress an image. We treated
Delaware - M - 428
University of Delaware Department of Mathematical Sciences Math 428 Algorithmic and Numerical Solution of Differential Equations 08S R. J. Braun Project 2: FitzHugh-Nagumo Equations Due date: 5/1/08, 3:30 pm. YOUR NAME: In this project you will use M
Delaware - M - 426
Delaware - M - 428
Math 428 Numerical and Algorithmic Solution of Dierential Equations 08S R. J. Braun Department of Mathematical Sciences University of Delaware Honors Project 1. Due 4/17/08, 3:30pm. In this project you will reproduce some of the eorts to compute 10 d
Delaware - M - 428
UNIVERSITY OF DELAWARE Mathematical Sciences Department Math 428/Cisc 411 Algorithmic &amp; Numerical Solution of Differential Equations Spring 2008 Instructor: Dr. Richard J. Braun, Ewing 509, (302) 831-1869, braun@math.udel.edu. Text: &quot;Numerical Analys
Delaware - M - 426
Math 426/CISC 410 08F, All Sections R.J. Braun Homework 9 Solutions, Hints and AnswersProblem 4.3.1. The ratios below lead one to the hypothesis that = |1 /2 | where 1 is the smallest eigenvalue and 2 is the next largest. I used the absolute value
Delaware - M - 426
Math 426/CISC 410 08F, All Sections R.J. Braun Homework 6 Solutions, Hints and AnswersProblem 2.7.1. We assume that the relative change in b is eps; that is, |b|/|b| =eps. This script will do the job. % Script: Prob2_7_1.m % Problem 2.7.1: verify re
Delaware - M - 426
Math 426/CISC 410 08F, All Sections R.J. Braun Homework 10 Solutions, Hints and AnswersProblem 4.6.2. (a) Assume that A is n-by-n and nonsingular. The SVD is A = U SV ; here S is a diagonal matrix with the singular values on the diagonal, and U and
Delaware - M - 353
University of Delaware Department of Mathematical Sciences Math 353 Engineering Mathematics III 05F R. J. Braun Homework 12: Due Wednesday, 11/16/05, 2:30pm1. Read 9.1,9.2,9.3,9.5. 2. 9.2, Problems 9.2.2(ab). Do h = 0.2 by hand; you may (and are en
Delaware - M - 428
Math 428 Numerical and Algorithmic Solution of Dierential Equations 08S Project 4. Due Friday, 5/16/05, 4:30pm. R. J. Braun Your name: In this project you will examine the numerical solution of the boundary value problem u + eu = 0, u(0) = u(1) = 0,
Delaware - M - 353
University of Delaware Department of Mathematical Sciences Math 353 Engineering Mathematics III 05F R. J. Braun Your name: Due date: 11/30/05, 5pm. Project 2: Solving a Nonlinear ODE IVP In this project you will study the solution of a nonlinear seco
Delaware - M - 353
University of Delaware Department of Mathematical Sciences Math 353 Engineering Mathematics III 05F R. J. Braun Homework 8: Due Tuesday, 10/11/05, 4:00pm1. Read 3.3-5. Reading 3.1,2 is optional; do so if you're not sure about all this matrix-vector
Delaware - M - 353
University of Delaware Department of Mathematical Sciences Math 353 Engineering Mathematics III 05F R. J. Braun Homework 7: Due Monday, 10/3/05, 5:00pm 1. Read 2.2, 2.3, 2.4. 2. 2.2, Problems 2.2.3(a,d), 2.2.11. 3. Apply the method of bisection 3 tim
Delaware - M - 426
qriter_testThe approximation to the smallest eigenvalue:ans = -0.059675578871905 1.564191203698703 1.987691412628296 2.349297531449158 3.409078620563635 -1.263166795003825 0.386479838574264 0.760280531365414 0.8920698056836
Delaware - M - 426
type Hw1_07f.m% Script for Hw 1, 07f. % Problem 3; display the output for second part of 11.2 of the Beginning% Matlab Problemsg = [1 2 3 4; 5 6 7 8; 9 10 11 12]h = [3 3 4 4; 5 5 6 6; 7 7 8 8]h &gt;= gg = hbigger = (g &gt;= h)g(bigger)g([0 0 0
Delaware - M - 426
type TestESeq% Script TestESeq.m% try approximation for exp(1) approx = zeros(1,15);format short e;for k = 1:15 approx(k) = (1+10^(-k)^(10^k);endabserror = abs(exp(1)-approx)%% make a nice table, like that from the basic matlab problem
Delaware - MATH - 611
A=[2 1 1 0; 4 3 3 1; 8 7 9 5; 6 7 9 8 ][L,U] = lufact(A);LUL*U - Ab = [1;2;3;4]y = L\bx = U\yA\bA = [ 1e-12 1; 1 1A = [ 1e-12 1; 1 1 ]norm(A)cond(A)
Delaware - MATH - 611
t = linspace(0,2*pi,200);A=[1 3;4 2]/4;for i=1:200, v=A*[cos(t(i);sin(t(i)]; plot(v(1),v(2),'o'), hold on, endginput(1)norm(ans)norm(A)
Delaware - MATH - 611
A = magic(8)A=A+A';eig(A)lam = poweriter(A);plot(lam,'-o')plot(520-lam,'-o')shgsemilogy(abs(520-lam),'-o')shg[x,y]=ginput(2)diff(log(y) / diff(x)exp(ans)eigeig(A)(ans(end-1)/ans(end)^2eig(A)lam=inviter(A,160);plot(lam,'-o')shglam=i
Delaware - MATH - 611
edit lufactA=[2 1 1 0; 4 3 3 1; 8 7 9 5; 6 7 9 8 ]A([1 3],:) = A([3 1],:)L1=eye(4); L1(2:4,1) = -A(2:4,1)/A(1,1)format ratL1A = L1*AP1=eye(4); P1([1 3],:)=P1([3 1],:);P2=eye(4); P2([2 4],:)=P1([2 4],:);P2*AP2=eye(4); P2([2 4],:)=P1([4 2],:)
Delaware - MATH - 611
m=5;A = toeplitz([1;-ones(m-1,1)],[1 zeros(1,m-1)]); A(:,m)=1[L,U]=lu(A)m=50;A=[2 1 1 0; 4 3 3 1; 8 7 9 5; 6 7 9 8 ];A = toeplitz([1;-ones(m-1,1)],[1 zeros(1,m-1)]); A(:,m)=1;x = rand(50,1);b = A*x;[L,U]=lu(A);x1 = U\(L\b);norm(x-x1)/norm(x
Delaware - MATH - 611
A = magic(5)A = A(:,1:3)[Q,R] = gs(A)Q'*Qans-eye(3)Q*R - A[Q,R]=qr(A,0)[Q,R]=qr(A)N = 40;N = 400;x = linspace(-1,1,N+1)';A = [ x.^0 x.^1 x.^2 x.^3 x.^4 ];[Q,R]=qr(A,0);plot(x,Q)
Delaware - MATH - 611
% MATLAB introduction2+23^4exp(1)log(1)sin(pi/2)exp(1i*pi)sqrt(-4)xpiformat longpiepseps=1e-100pix = 2sin(x)x = x+1v = [ 1, 2, 3, 4 ]v = [ 1; 2; 3; 4 ]v'A = [ 1, 2, 3; 10, 20, 30 ]vB = [ v, v, v ]-3*v-v - 2*v4*BA = [ 1, 2,
Delaware - MATH - 611
Delaware - CIS - 361
The need for File SystemsNeed to store data and programs in files Must be able to store lots of data Must be nonvolatile and survive crashes and power outages Must allow multiple processes concurrent access Store on disks OS manages files in a file
Delaware - CIS - 361
Solaris Management Facility (SMF) - WorkshopGanesh Hiregoudar Renaud Manus OP/N1 RPE ApproachabilitySun MicrosystemsThanks! We would like to thanks the following engineers who participated in writing and delivering this SMF workshop. &gt; Jarod Nas
Delaware - CIS - 361
Memory ManagementOne of the most important OS jobs.Review memory hierarchy sizes, speed, costGrowing memories -&gt; growing programs need for swapping and paging The simplest way to use memory is to have one program in memory sharing it with the OS
Delaware - CIS - 361
ZFSNew filesystem and volume manager from Sun Open source after 5 years development First appeared in OpenSolaris, later in Solaris 10 6/06 (u2). Has been ported to FreeBSD. Also being including on Macs (Leopard*) Linux FUSE port Jeff Bonwick is a U
Delaware - CIS - 361
Process SynchronizationConsider two threads using and modifying a shared global variable. What problems could occur? Example: a bank account balance (a shared global variable) Balance \$200 A: Deposit \$10 B: Deposit \$10,000Process SynchronizationA
Delaware - CIS - 361
ThreadsTraditional processes have one thread of control sequential program. Multiple threads of control are possible to get parallelism. Processes - own resources (resource grouping) - are scheduled for running (have states (Running, Ready, Blocked
Delaware - CIS - 361
CPU SchedulingScheduling processes (or kernel-level threads) onto the cpu is one of the most important OS functions. The cpu is an expensive resource so utilize it well. May have more than 1. With multiprogramming will have many processes (threads)
Delaware - CIS - 361
Solaris Zones: Operating System Support for Consolidating Commercial WorkloadsDaniel Price and Andrew Tucker Sun Microsystems, Inc. ABSTRACTServer consolidation, which allows multiple workloads to run on the same system, has become increasingly im
Delaware - CIS - 361
DeadlocksLots of resources can only be used by one process at a time. Exclusive access is needed. Suppose process A needs R1 + R2 and B needs R1 + R2. A gets R1, B gets R2 A waits for R2, B waits for R1 A and B can't continue without second resource
Delaware - CIS - 361
Zones - ContainersServer Consolidation Run multiple workloads on system Improve utilization of resources Reduce costs Run workloads in isolation Cannot observe others Security Isolation Running apps as different user not enough - privilege escalatio
Delaware - CIS - 361
Dining PhilosophersFive philosophers sit around a table with five forks and spaghetti to eat. Philosophers think for a while and they want to eat, only spaghetti, for a while. To eat a philosopher requires two forks, one from the left and one from r
Delaware - CIS - 361
The ShellWhat does a shell do? - execute commands, programs - but how? For built in commands run some code to do the command For other commands find program executable and . run it. Other features: wildcards, pipes, redirection.ProcessesThe shell
Delaware - CIS - 361
Architecture BackgroundVon Neumann architecture - cpu = ALU + control unit - memory - devices Above connected with buses ALU carry out instruction, may have more than one execution unit Program counter memory address of next instruction to fetch.