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### Fall2011-HW7

Course: MATH 371, Fall 2011
School: Michigan
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Word Count: 372

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7 Math Homework 371, Fall 2011 Assigned: Saturday, October 28, 2011 Due: Thursday, November 3, 2011 Clearly label all plots using title, xlabel, ylabel, legend Use the subplot command to compare multiple plots Include printouts of all Matlab code, labeled with your name, date, and section. Add 12 for page numbers in the international edition. (1) (Condition numbers) For two n n matrices A and B , prove...

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7 Math Homework 371, Fall 2011 Assigned: Saturday, October 28, 2011 Due: Thursday, November 3, 2011 Clearly label all plots using title, xlabel, ylabel, legend Use the subplot command to compare multiple plots Include printouts of all Matlab code, labeled with your name, date, and section. Add 12 for page numbers in the international edition. (1) (Condition numbers) For two n n matrices A and B , prove that (AB )1 = B 1 A1 , 1 and (A)1 = A1 . Now do P. 187 #1, 2. (2) (Residual errors vs. forward errors) As in the previous homework, let A be the n n matrix of the form 2 1 1 2 1 .. . 1 2 . .. .. . 1 . 1 2 Write a matlab function named [rerr, ferr, condnum] = ResidualFwdErrors(n)which takes n as an argument, generates the matrix A as above, and then a vector x, and then a vector b = Ax, and returns the relative residual error, relative forward error, and the condition number. Make a table with four columns corresponding to n and the three numbers returned by the function, taking n to be 50, 100, 500, 1000, 2000, 4000. Connect the table to the theorem on page 182 of the textbook. Which part of the theorem does the table this illustrate? For problem, generate A using the diag() command as shown in class, but do not use sparse(). (3) (Cholesky Factorization) Consider the following matrix factorization: a11 a21 a21 a22 = l11 l21 l22 l11 l21 l22 . Write this down as three equations for a11, a21 , a22 . Explain in what order and in what way the equation can be solved for the entries of the L matrix. (4) (Cholesky Factorization) Consider a matrix factorization as in the previous problem, but now with 3 3 matrices. Write down 6 equations for the aij s. Explain how these equations can be ordered and used to solve for the lij s. (5) (Triangular Factorization) Consider the following matrix factorization A = LU : l11 1 u12 a11 a12 a21 a22 a23 l21 l22 1 u23 = . a32 a33 a34 l32 l33 1 u34 1 a43 a44 l43 l44 Write this down as a set of 10 equations and explain how to solve for the lij s and uij s given the aij s. Verify the operation count of 3n 3 given on p. 219. 1 2 Suppose you want to solve b1 x1 x2 b2 A x3 = b3 b4 x4 for x. Explain how to do that using the LU factorization. Verify the operation count of 5n 4 on p. 219.
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Michigan - MATH - 371
Homework 8Math 371, Fall 2011Assigned: Thursday, November 3, 2011Due: Thursday, November 10, 2011Clearly label all plots using title, xlabel, ylabel, legendUse the subplot command to compare multiple plotsInclude printouts of all Matlab code, labele
Michigan - MATH - 371
Homework 9Math 371, Fall 2011Assigned: Thursday, November 10, 2011Due: Thursday, November 17, 2011Clearly label all plots using title, xlabel, ylabel, legendUse the subplot command to compare multiple plotsInclude printouts of all Matlab code, label
Michigan - MATH - 371
Homework 10Math 371, Fall 2011Assigned: Thursday, November 17, 2011Due: Thursday, December 1, 2011Clearly label all plots using title, xlabel, ylabel, legendUse the subplot command to compare multiple plotsInclude printouts of all Matlab code, label
Michigan - MATH - 371
Homework 11Math 371, Fall 2012Assigned: Friday, December 2, 2011Due: Tuesday, December 13, 2011Clearly label all plots using title, xlabel, ylabel, legendUse the subplot command to compare multiple plotsInclude printouts of all Matlab code, labeled
Michigan - MATH - 371
Homework 1 SolutionsMath 371, Fall 20111. (Finite precision numbers) The oating point representation of a realnumber takes the form x = (0.a1 a2 . . . an ) e , where a1 = 0,M e M . Suppose that = 2, n = 4, and M = 5.(a) Find the smallest and largest
Michigan - MATH - 371
Homework 2 SolutionsMath 371, Fall 2011Page numbers in the International Edition are +12.Here is an example of two MATLAB scripts which, together, allow you to pass a pass a function into yourcode. This allows you to reuse your code for other function
Michigan - MATH - 371
Homework 3, Solution SketchesMath 371, Fall 20111. (Rootnding and Optimization)(a)(b) Approximate this local maximum using six iterations of the bisectionmethod with starting interval [4, 6].(see table)(c) Approximate this local maximum using six i
Michigan - MATH - 371
Homework 4 Solution SketchesMath 371, Fall 2011Assigned: Thursday, September 29, 2011Due: Thursday, October 6, 2011(1) (Matrix Operations) P. 148 #1, 2, 3, 4, 5, 6, 7, 8, 91a.6186 1 61b. DN E8 4 22 1 34 3b.3a.6 2 142a.2b.5b. DN E9 14 118
Michigan - MATH - 371
Homework 5 Solution SketchesMath 371, Fall 2011Assigned: Sunday, October 9, 2011Due: Thursday, October 20, 2011(1) (Pivoting)(a) Prove that the matrix0111does not have an LU decomposition.Use direct decomposition. Suppose that01=1112101u1
Michigan - MATH - 371
Homework 6 Solution SketchesMath 371, Fall 2011Assigned: Thursday, October 20, 2011Due: Thursday, October 26, 2011 Clearly label all plots using title, xlabel, ylabel, legend Use the subplot command to compare multiple plots Include printouts of all
Michigan - MATH - 371
Homework 7 solution sketchesMath 371, Fall 2011Assigned: Saturday, October 28, 2011Due: Thursday, November 3, 2011Clearly label all plots using title, xlabel, ylabel, legendUse the subplot command to compare multiple plotsInclude printouts of all Ma
Michigan - MATH - 371
Homework 8 SolutionMath 371, Fall 2011Assigned: Thursday, November 3, 2011Due: Thursday, November 10, 2011Clearly label all plots using title, xlabel, ylabel, legendUse the subplot command to compare multiple plotsInclude printouts of all Matlab cod
Michigan - MATH - 371
Homework 9 SolutionsMath 371, Fall 20111. (Interpolating Polynomials) The function f (x) = 1/(x + 1) is given at the four points x0 = 1,x1 = 2, x2 = 3, x3 = 4.(a) Write the interpolating polynomial in Lagrange form.(x 2)(x 3)(x 4) 1 (x 1)(x 3)(x 4) 1
Michigan - MATH - 371
Homework 10 SolutionsMath 371, Fall 20111. (Chebyshev polynomials) Do problem #3 on page 385 of Bradie.Let Tn (x) = cos(n cos1 x). We must compute the integral11Tn (x) Tm (x)dx.1 x2First, substitute x = cos to obtain /22cos(n) cos(m) d./2App
Michigan - MATH - 371
Homework 11 SolutionsMath 371, Fall 2011(1) (Richardson Extrapolation) P. 454 #11(a) Adding the Taylor series for f (x0 + h) and f (x0 h) givesf (x0 + h) + f (x0 h) =h4h8 (8)h82f (x0 ) + h2 f (x0 ) + f (4) (x0 ) +f (x0 ) +[f (8) ( ) f (8) (x0 )]
Michigan - MATH - 371
Math 371 (Fall 2011): Midterm Exam13 October 2011Name:Q120 pointsQ220 pointsQ320 pointsQ420 pointsQ520 pointsTotal100 points11. (20 points) Answer each of the following questions briey.(a) For scientic measurements, which of relative erro
Michigan - MATH - 371
Math 371-001/002Time:Place:Instructor:Oce:Phone:Email:O.hours:Textbook:Numerical MethodsFall 2011TuTh 9:00-10:30 a.m. (Section 002)TuTh:12:00-1:30 p.m. (Section 001)3427 EECS (Section 002)2150 DOW (Section 001)Divakar Viswanath3856 East Ha
Michigan - ME - 320
The University of MichiganDepartment of Mechanical EngineeringME 320 - Sections 1 &amp; 2Homework #9 - Due November 23, 2011Problem 1: A simplied form of the stepped thrust bearing consists of a stationary plateheld parallel to a moving surface as shown.
Michigan - ME - 320
The University of MichiganDepartment of Mechanical EngineeringME 320 - Sections 1 &amp; 2Homework #10 - Due November 30, 2011Problem 1: Problems 7.10, 7.21 and 7.23 of the text.Problem 2: Problems 7.30 and 7.33 of the text.Problem 3: Problems 7.50, 7.56
Michigan - ME - 320
The University of MichiganDepartment of Mechanical EngineeringME 320 - Sections 1 &amp; 2Homework #11 - Due December 7, 2010Problem 1: Problems 8.61, 8.66 and 8.72 of the text.Problem 2: Problems 8.77 and 8.78 of the text.Problem 3: Problems 8.79, 8.90
Michigan - ME - 320
The University of MichiganDepartment of Mechanical EngineeringME 320 - Sections 1 &amp; 2Homework #12 - Due by midnight December 13, 2011Problem 1: Problems 9.19 and 9.20 of the text.Problem 2: Problems 9.45, 9.52 and 9.70 of the text.Problem 3: Problem
Michigan - ME - 320
The University of MichiganDepartment of Mechanical EngineeringME 320 - Section 1 &amp; 2Homework #2 - Due September 28, 2011Problem 1: Problems 2.7 and 2.9 of the text.Problem 2: Problems 2.14 and 2.15 of the text.Problem 3: Problems 2.36 and 2.37 of th
Michigan - ME - 320
The University of MichiganDepartment of Mechanical EngineeringME 320 - Sections 1 &amp; 2Homework #7 - Due November 9, 2011Problem 1: Problems 6.15, 6.21 and 6.22 of the text.Problem 2: Problems 6.24 and 6.26 of the text.Problem 3: Problem 6.20 of the t
Michigan - ME - 320
The University of MichiganDepartment of Mechanical EngineeringME 320 - Sections 1 &amp; 2Homework #9 - Due November 23, 2011Problem 1: A simplied form of the stepped thrust bearing consists of a stationary plateheld parallel to a moving surface as shown.
Michigan - ME - 320
The University of MichiganDepartment of Mechanical EngineeringME 320 - Sections 1 &amp; 2Homework #10 - Due November 30, 2011Problem 1: Problems 7.10, 7.21 and 7.23 of the text.Problem 2: Problems 7.30 and 7.33 of the text.Problem 3: Problems 7.50, 7.56
Michigan - ME - 320
The University of MichiganDepartment of Mechanical EngineeringME 320 - Sections 1 &amp; 2Homework #1 - Due September 21, 2011Problem 1: Problems 1.11, 1.12 and 1.13 of Munson, Young, Okiishi, Huebsch, 6th edition(hereafter referred to as the text)Proble
Michigan - ME - 320
The University of MichiganDepartment of Mechanical EngineeringME 320 - Sections 1 &amp; 2Homework #3 - Due October 5, 2011Problem 1: Problems 2.56, 2.60 and 2.72 of the text.Problem 2: Problems 2.83, 2.87 and 2.94 of the text.Problem 3: Problems 2.89, 2
Michigan - ME - 320
The University of MichiganDepartment of Mechanical EngineeringME 320 - Sections 1 &amp; 2Homework #4 - October 12, 2011Problem 1: problems 4.13, 4.16 and 4.19 of the text.Problem 2: Problems 4.32, 4.35 and 4.49 of the text.Problem 3: Problems 3.35 and 3
Michigan - ME - 320
The University of MichiganDepartment of Mechanical EngineeringME 320 - Sections 1 &amp; 2Homework #5 - Due October 19, 2010Problem 1: Problems 3.46, 3.88 and 3.90 of the text. Assume quasi-steady ow in allproblems.Problem 2: Consider the reservoir syste
Michigan - ME - 320
The University of MichiganDepartment of Mechanical EngineeringME 320 - Sections 1 &amp; 2Homework #6 - Due Friday October 28, 2010 by 5:00pmProblem 1: Problems 5.48, 5.66 and 5.70 of the text.Problem 2:(a) Problem 5.60 of the text.(b) The results of a
Michigan - ME - 320
The University of MichiganDepartment of Mechanical EngineeringME 320 - Sections 1 &amp; 2Homework #8 - Due November 16, 2011Problem 1: Problems 6.90 and 6.93 of the text.Problem 2: Problems 6.99 and 6.102 of the text.Problem 3: Problems 6.106 and 6.107
Michigan - ME - 320
The University of MichiganDepartment of Mechanical EngineeringME 320 - Sections 1 &amp; 2Homework #11 - Due December 7, 2010Problem 1: Problems 8.61, 8.66 and 8.72 of the text.Problem 2: Problems 8.77 and 8.78 of the text.Problem 3: Problems 8.79, 8.90
Michigan - ME - 320
The University of MichiganDepartment of Mechanical EngineeringME 320 - Sections 1 &amp; 2Homework #12 - Due by midnight December 13, 2011Problem 1: Problems 9.19 and 9.20 of the text.Problem 2: Problems 9.45, 9.52 and 9.70 of the text.Problem 3: Problem
Michigan - ME - 320
- .-(Ne J 2o _FqlJ 2 ou- z.-h=u~lotf-)cfw_2)1f( J()D);.2.~h= ~ ( J - 9. S'xM 1 ) = To . o 571f.oS.xlOLWJF;,.r &quot;' ~ 4-to 1 ) (u!cD) = Z1; 3 'r 1 N(cl5'~' -/'/']--&quot;-cv\~( j)~';,.X\- it-cLe.[b ,.-.-y,~(c_)- - - !_ _Arr1C.BY\S
Michigan - ME - 320
The University of MichiganDepartment of Mechanical EngineeringME 320 - Sections 1 &amp; 2Final Exam - December 15, 2011Problem 1 (23 points)Two thin rectangular plates, of length L and width w into the page, are connected at one endby a closed-end exibl
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Me :sz.o - f41J-z.o11-l?e!&gt;b) ~ ( z3 rnit i&gt;/ ( o,)COV1 ~ v&quot;a.i(b)D.r1n ool5 (wtl14-f.J .,. ,. , ().1).0Is~), 7 'cfw_) sU J /J.cfw_)Il n uf-lt:.- Wee. .-to cfw_/ow &amp;.tlUl.t? r lle.,l [La.tQ l i,t)&quot;~o.(-*) =PerjdF&gt;.:p&lt;x)fa- fl&quot;
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ME 320, Fluid Mechanics - Fall 2011Sections 1 &amp; 2Section 1 MW 8:30 - 10:00 (Rm. 1311 EECS)Section 2 MW 3:30 - 5:00 (Rm. 1670 CSE)Instructors:Prof. Rayhaneh AkhavanRm. 2018 G. G. Browne-mail: raa@umich.eduOce Hours: M 10:00 - 11:00amProf. Massoud
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NAME:_LABSECTIONNUMBER:_ME350 Fall 2011Homework #4 (35 points total)Due Thursday October 6, 2011 by 1:30pmHand in at the beginning of Lecture1. (25 pts) A bolted joint is subjected to a fluctuating tensile load whose maximum value is 8 kips(kilopou
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ME 350 Design and Manufacturing II: Motor HomeworkDue Date: October 20, 2011To get partial credit you need to show all your work.Problem 1:Rank the following actuation technologies on the design metrics from 1 (best, highest) to 4(worst, lowest). (th
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Michigan - ME - 350
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USC - ECON - 303
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USC - ECON - 303
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USC - ECON - 303
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USC - ECON - 303
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