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18.085 Fall 2010 Class stuff: MWF 11-12 2-190Lecturer: Gilbert StrangOffice: 2-240e-mail: gs@math.mit.edu TAs: To be announced Tomorrow (Wednesday's) exam will cover Fourier series, a discreteconvolution computation, circulant matrices, and a Fourier integral.2010: We will follow the 2008 outline or indeedthe video lectures fairly closely perhaps with a slightmore emphasis on computing. For 2010 exams will be in class, open book and open notes, notevenings, no computers. Homeworks will be due on Fridays in spring 2010. The exam dates will take place on the classes numbered 13, 26, and 39.The dates are Wednesday March 3, Friday April 9, and Wednesday May 12.Strang's Videos (terrific) Quizzes and Solutions for Spring 2010 Quiz 1 (pdf) and Solutions (pdf) .Quiz 2(pdf) and Solutions (pdf) Quiz 3(pdf) and Solutions (pdf) HOMEWORK for Spring 2010 #11 (Final HW) Due Friday April 30 (Solutions to HW 11) 4.3: 6,7,9,204.4: 1,2,4,5,6,74.5: 3,4,5,7,11#10 Due Friday April 23 (Solutions to HW 10) Note: we may not cover this material in class, but it's very useful andsimilar in many ways to material we saw in class.4.2: 1,2,3,7,8,10,20,21Hint for problem 8: Write down T4(x) and combine until you get x^4.By popular request (matlab to watch Fourier Series for square wave):x=pi*[-1:.01:1];for n=3:5:51, k=(1:2:n)'; plot(x, sum(diag(1./k)*sin(k*x)));title(['n=' num2str(n)]); pause(0.1); end#9 (Solutions to HW 9) Section 4.1: 2,3,4,5,7,8,9Hints: 4: assume f(x) is general, not any one particular functionHints: 9: Note the taylor series for 2*log(1+z)is 2(z-z^2/2+...). It's not that imaginary part of the taylor seriesis 2 log(1+z). Guess this could have been written clearer, butif you knew the series, you'd be ok. The real problem is a bit ofa trig identity.#8 (Solutions to HW 8) Due April 12 1pmSection 3.4: 1,2,3,12,13Section 3.5: 8,9,10#7 (Solutions to #7) Due April 2 1pmSection 3.2: 6. 8, 9. 10 Section 3.3: 1,2,3,4,5,6, 9Hint on 3.2,6: I think we must remember that w=u\" to think clearly here. Perhaps also it is good to realizethat we will need all the functions to cancel out to get higher derivatives 0.Hint on 3.3,9c: I think s(x,y)=arctan(y/x) might be a good stream function#6 (Solutions to #6) Due March 19 1pm3.1:3,4,7,10,12,13,15#5 (Solutions to #5) Due March 12 (Friday) 1pm2.7:4,5,8,9,10,13 (Note 12 is a copy of 8)#4 (Solutions to #4) Due March 8 (monday), 1pm 12. (My survey of students was that extra time would be nice given theexam, but the homeworks are useful for mastering the material).2.2: 7 2.3: 3,4,10Problem 2.2:7 requires understanding that lambda=exp(i*theta).If you expand lambda as a power series of h, it looks likethe power series of exp(i*h) for a few terms. The problem asksyou to recognize where they differ. Problem 2.3:3 is long but I hope very instructive. The U matrix in theHouseholder code has the information to multiply the Householder's in reverseorder.I got the exact answer using matlab's symbolic toolbox and then roundedonly at the end. I got 1.000000000705105e+000 -2.964328419073236e-007 -7.683186917434977e+000 -2.206297290004058e-004 9.840487549806722e+000 -9.324611932899178e-003 -5.011975877477831e+000 -4.685413770053830e-002 1.417778585705657e+000 2.287193842956169e-002 -3.135724049839490e-001 7.196408559132141e-002If you have the tollbox you can typeAAA=sym(0:49).'/49;AAA=(AAA*ones(1,12) ).^ (ones(50,1)*(0:11));bbb= 0:sym(8/100):4-sym(8/100);bbb=cos(bbb).';digits(100);xxx=double(vpa(AAA.'*AAA)\\vpa(AAA.'*bbb)); 2.4: 2,5,6,7,9#3 (Solutions to #3) for Friday February 261.6: 15, 16, 17, 21, 232.1: 4,5,62.2: 5Recommended but do not hand in: Do the MATLAB homework in Problem set 3 by scrollingall the way down to \"First MATLAB Homework in 18.085\" from the previoussemester. #2 (Solutions to #2) for Friday February 19 1.3: 5,6,7,131.4: 2,6,12,151.5: 2,10,11,121.6: 12#1 (Solutions to #1) for Friday February 12 (Please submit in class, or by 1pm near 2-343)(Please write your name extra extra neatly)1.1: 1,7,10,12Hint on 10: The idea is to check that UU' truly is H and then invertusing the rule inv(A*B)=inv(B)*inv(A).1.2: 3,4,7,18Last semester's stuff: [announcements] [homework assignments] [quizzes and solutions] [links] [resources and old exams] Course Topics Applied Linear Algebra Applied Differential Equations Fourier Methods Algorithms Course outline 2008 Goals for the Course: See applications of calculus, ODE, linear algebra, and discrete methods without going into too much proof. Textbook: Computational Science and Engineering (Wellesley-Cambridge, 2007). Grades: Homework 40%, 3 evening quizzes 60%, no final. Homework: Due Wednesdays. Please use MATLAB notation to describe algorithms. Use of MATLAB for tedious calculations is encouraged, however you need to know how to do the basic algorithms taught in the course by hand (at least for small matrices) for the quizzes. Announcements Exam 3 and 18.085 grades are all done and the Exam 3 average was 80. The graded exams are available from Debbie Bower in 2-108 (desk at right)I hope you enjoyed the course and will remember those A'CA ideas forever.It was a pleasure for me to work with you. Very best wishes! Gilbert Strang Exam 3 - Solutions Professor Strang's favorite -1,2,-1 matrix in cupcakes Bigger version Exam 2 - Solutions [top] HOMEWORK Please include your 18.085 number (if it was circled on hwk 1)at the top of your first homework page to help us order the hwksalphabetically THANK YOU #10 for Fri Dec 4 (Quiz TUES 8th IS IN WALKER !!!!) - Solutions Section 4.4: 1, 2, 3, 7, 8, 9, 10 Section 4.5: 1, 2, 8, 10, 11, 12, 14, 22, 24, 25 #9 for Mon Nov 23 - Solutions Section 4.1: 1, 6, 10, 13, 14, 18 Section 4.2: 4 Section 4.3: 2, 6, 8, 10, 15, 22 Problem 4.3.22 was added in the latest printing of the book. Here are the two important parts of the problem: a) Write out F2D = kron(F,F) for N=2. It will be 4 by 4 d) Why does F2D times a vector u2D need only O(N^2 log N)operations MATLAB HomeworkThe homework is about the important \"Gibbs phenomenon\".Create a figure like Fig. 4.3 showing the partial sums of theFourier series for the SQUARE WAVE in Exercise 4.1.2.It jumps from -1 to +1 at x = 0.Can you determine the amount of the overshoot In what way doesthis same overshoot number appear in Fig. 4.2 for the delta functionWhat is the size of the overshoot if the step fcn jumps from 0 to A Find a relation between the WIDTH of the overshoot (the first lobeabove +1) and the NUMBER of terms in the partial sum for the square wave.New question (to me): Beyond the overshoot in Fig. 4.3it looks to me that there is an UNDERSHOOT below +1. Is this truein your graph of partial sums What depth of undershootI hope you can blow up the figure near the jump.Extra: Also graph and blow up the Fourier series for the rampfunction f(x) = max(0,x) near the corner (not a jump!) at x = 0. #8 for Fri Nov 6 - Solutions Section 3.3: 7, 8, 11, 16, 24 (harder), 27 Section 3.4: 4, 17, 18 Section 3.5: 1, 2 (the book explains the kron command) #7 for Fri October 30 - Solutions Section 3.1: 1, 2, 5, 9, 11, 14, 17 Section 3.2: 5, 7, 17, 19 #6 for Wed October 21 - Solutions Section 2.7: 1, 2, 3, 6, 7, 11 MATLAB1 : Create the matrix A for Problem 5 (Truss E, with 30-30-120triangle at the top)Check det (A'*A) Find 2 solutions to Au = 0. MATLAB2: Bar 2 of a 4-node truss connects joints 1 and 3 at a 45 degreeangle.That row of A is row2 = [-1 -1 0 0 1 1 0 0]/sqrt(2)] With stiffness c2 =6, print thestiffness matrix E2 = c2*row2' *row2. The true element matrix e2 is 4 by4,without all the zeros in E2. Print e2 and execute a Matlab command to placee2 into E2. #5 for Wed October 14 - Solutions Section 2.4: 7, 8, 9, 11, 14, 15, 17, 18 #4 due the evening of Thursday October 8 (when you hand in the exam) Section 2.2: 5 + this question:Solve u'' = -Tu for T = [1 -1; -1 2]. Looking for the solution with 4constants A B as on page 116 (which was fixed-fixed) Section 2.3: 1, 7, 8, 9, 12 Section 2.4: 1, 3Please keep this homework for reference during the exam / plan to collectlater #3 for Wed September 30 - SolutionsSection 1.6: 20, 22, 24, 27Section 2.1: 1, 3, 7, 8, plus this questionFirst MATLAB Homework in 18.085 Find the displacements x(1),...,x(100) of 100 masses connected by springs all with c = 1. You may take each force f(i) = .01 and consider two boundary conditions at the bottom: (a) Spring 101 connects the last mass to a support as in Figure 1.7 (b) Mass 100 hangs free at the end of the line of springs. Submit GRAPHS of the displacements in these two cases. Here are a few MATLAB hints, mostly correct: d = ones(100,1) is a column vector of 100 ones and diag(d) is a diagonal matrix (in fact I) with d on the diagonal. diag(d,1) puts d on an off-diagonal, maybe this matrix has order 101. After computing the vector x try plot(x,'+') xlabel('mass number') ylabel('displacement') print Suggested problems not to turn in Section 1.5: 9 Section 1.6: 15, 16NOTE: The mysterious numbers on the graded Hwk 1 are in alphabeticalorder -- if you write your number at the top of future homeworks, thathelps to put them quickly in the right order. (Some don't yet have anumber, still to do) #2 for Wed September 23 - Solutions Section 1.3: 1, 5, 9, 11 Section 1.4: 1, 4, 7, 15 Section 1.5: 1, 7, 9, 13 Section 1.6: 4, 6 #1 for WED September 16: Section 1.1: 2, 5, 17, 20 Section 1.2: 1, 4, 7, 16 from the CSE textbook NOTESApologies that question 1.2.1 involves delta functions !! The slope of u(x)jumps from A to B.So the next derivative is a delta function at x = 0 of magnitude ... (seepage 38)No boundary conditions in this question, it runs from minus infinity toinfinity.And the last line of 1.2.14 needs a multiplication by -1. One more erroris hidden in the book. PLEASE PRINT YOUR NAME CLEARLY The class list will be created from the homeworks -- and Hwk 1 will not be graded in full detail, it is a start for this course. [top] Quizzes and Solutions [top] Previous Course Content This section contains information from the previous courses, videos, practiceexams, and extra homework and solutions. (Please report broken linksso I can fix them). 18.085 from Fall 2008 is now completely onlineat OCW-18.085Videos: The special event for Fall 2008 is that the lectures will berecorded for OpenCourseWare You will already find a partial earlier set on the website but the course has since evolved. The videos of 18.06 Linear Algebrahave been successful on OCW(please use them for help!!). The Lord Foundation gave a newgrant for 18.085. 153554b96e

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