From Richard Palais Wed Aug 27 11:15:40 2003 To: Math32 From: Richard Palais Subject: Welcome Dear Math 32a Students, This is a message to welcome the students who have registered for Math 32a, my course on the differential geometry of curves and surfaces. We will have our first meeting in Goldsmith 226 this Friday at 10:40 am. I have created a website for the course at: http://rsp.math.brandeis.edu/Math32/ and I would like to invite you to visit it and become familiar with its structure, since it contains a lot of information about the course, how it will be run, where you can find various useful lecture notes and other resources, and it will also be the place that you can check the details of various homework exercises and projects that I plan to assign during the course. (I have had my own server and have managed my own web-sites for many years, so I decided against using the Web-CT system, with which I am anyway not familiar.) Please keep the above URL private---i.e., do not create any links to it from any other web pages. (There are a number of reasons for this, not the least of which is that I don't want the spam robots harvesting our email addresses.) Please do not hesitate to write if you have questions. I am looking forward to working together these next few months! Dick Palais ---------------------------------------- Richard S. Palais Home: 68 Lexington Street, Weston, MA 02493-2146 VOICE: 781 899 4951 FAX: 781 642 6266 MOBILE: 781 354 3984 Office: Dept. of Math. Brandeis Univ. Waltham, MA 02254-9110 VOICE: 781 736 3050 FAX: 781 736 3085 EMAIL:palais@brandeis.edu WWW Home Page URL: http://rsp.math.brandeis.edu --------------------------------------------------------------------------------- -- From Richard Palais Thu Aug 28 07:52:29 2003 To: Math32 From: Richard Palais Subject: For my records To Math 32a Students: It will be helpful to me in organizing the material for the course and in preparing my lectures if I know a little about your backgrounds, so I hope you won't mind answering a few questions about yourself. Please use return email. Your Name? Your Year? (Freshman, sophomore,...?) Your major? What courses have you taken in the Math department? What Math oriented courses have you taken in other departments? Have you studied any mathematical topics on your own, outside of assigned class works? If so please list them. (Include any topics from reading courses.) Many thanks. See you in class, R. Palais --------------------------------------------------------------------------------- -- From Richard Palais Sun Aug 31 23:59:41 2003 To: Math32 From: Richard Palais Subject: First assignment To Math 32a Students I would like to ask you to download the first assignment from the Math 32 website. You can find a link on the Homework, Projects, and Assignments page at: http://rsp.math.brandeis.edu/math32/ or get it directly at: http://rsp.math.brandeis.edu/Math32/Assignments/Assignment1.pdf Notice that it is in PDF format, and part of what I am interested in is whether you are all able to download this file successfully. Note that the first exercise of this assignment is due on Tuesday and the rest on Friday. R. Palais --------------------------------------------------------------------------------- -- From Richard Palais Thu Sep 04 00:53:08 2003 To: Math32 From: Richard Palais Subject: Various administrative matters. To Math 32a Students. 1) My office hours. I will be available in my office (Goldsmith 206) during the hour just prior to the time we meet---i.e., from 9:30 to 10:30 am Tuesday and Friday. I realize this isn't a good time for several of you, so please feel free to ask me to arrange some other time to meet with you. Also, if you remind me on Friday, perhaps we can work out some other regular time. 2) There will definitely be a TA for the course---one who will be able to help answer your questions about the mathematical and also the programming aspects of the course. However all the details will probably not be worked out before the weekend so I will let you know about this next Tuesday. 3) The computer clusters in the Goldfarb classroom and in the Hughes classroom (Gerstenzang basement) have Matlab installed, and it will be installed in the Farber classroom computers very soon. We will use one of these classrooms next Tuesday for our first full instruction session in the use of Matlab. In the meantime, please start looking at the "Getting Started with Matlab" material on the resources page of the course website (or if you prefer start reading any of the large number of good elementary textbooks on Matlab). You are free to use the computers in any of the above mentioned clusters anytime that there is no class being held there. It is very important to actually sit down at a computer and practice using the program---you cannot learn to program just be reading notes! Working through one or more of the various tutorials while sitting at the computer is a good way to get going for most people. R. Palais --------------------------------------------------------------------------------- -- From Richard Palais Sat Sep 06 00:36:25 2003 To: Math32 From: Richard Palais Subject: Answers to exercises in First Assignment. Cc: Bcc: X-Attachments: Message-Id: I've posted answers to the exercises in the first assignment at: http://rsp.math.brandeis.edu/Math32/Assignments/Assignment1Answ.pdf R. Palais P.S. Most of you who passed in answers did quite well. --------------------------------------------------------------------------------- -- From Richard Palais Wed Sep 10 14:01:55 2003 To: Math32 From: Richard Palais Subject: TA Office Hours, Assignment 2 etc. Cc: Bcc: X-Attachments: Message-Id: To Math 32a Students: 1) As I mentioned to you yesterday, the TA for the course is: Anna Varvak Goldsmith 111 x6095 Office Hours: Mon, Tues 12pm-1pm 2) As I also mentioned in class, I have posted the lecture notes for the first two lectures: http://rsp.math.brandeis.edu/Math32/LectureNotes/Lecture1.pdf , http://rsp.math.brandeis.edu/Math32/LectureNotes/Lecture2.pdf , and I have also posted my notes for the first computer lab: http://rsp.math.brandeis.edu/Math32/LectureNotes/FirstSteps.m 3) Finally, I have now posted the second assignment: http://rsp.math.brandeis.edu/Math32/Assignments/Assignment2.pdf Although I have put on a due date of next Friday, if you would like to work on it over the week-end and pass it in on Tuesday, that's OK too. --------------------------------------------------------------------------------- -- From Richard Palais Sat Sep 13 10:18:33 2003 To: Math32 From: Richard Palais Subject: An Invitation, New Lecture Notes, and an explanation of yesterdays error :-(. Cc: Bcc: X-Attachments: Message-Id: To Math 32a Students: 1) I would like to invite you all to a cookout at my home. I was originally hoping to make it for tomorrow, but the weather report looks too unpromising, so tentatively lets think about having it next Sunday. I live just a couple of miles from Brandeis, and an easy walk from the Kendall Green Commuter rail station---the next one west of Roberts, but perhaps enough of you have cars to provide rides for everyone. We can discuss it in class this week. Of course our TA Anna Varvak and our Matlab expert Izi Aviyente are also invited. 2) I have now posted the lecture notes for lectures 3 and 4. The URLs are: http://rsp.math.brandeis.edu/Math32/LectureNotes/Lecture3.pdf http://rsp.math.brandeis.edu/Math32/LectureNotes/Lecture3.pdf I have also posted my notes for the first Matlab instruction session, last Tuesday evening in Farber classroom. The URL is: http://rsp.math.brandeis.edu/Math32/LectureNotes/FirstSteps.m There are links to all my lecture notes for the course at: http://rsp.math.brandeis.edu/Math32/Course%20Lecture%20Notes.html and you should get in the habit of checking there every so often. 3) The lecture notes for the fourth lecture have a correction to the mistake I made in class yesterday. That mistake was really worse than it looked like. Please read the notes for lecture 4 carefully, since I have fallen a little behind and would like to review it very rapidly on Tuesday. Here is a discussion of the error and its (easy) fix. Recall that I defined the orthogonal group O(V) of an inner-product space V to be the set of all norm-preserving maps f:V --> V, and I stated a theorem to the effect that norm-preserving maps were linear, but I got hung up during the proof. Well, no wonder!---that "theorem" is WRONG. (The easiest counter-example is for the one-dimensional space V = R, the real numbers. The map f(x) = |x| is clearly norm preserving but just as clearly NOT linear. The correct definition of the orthogonal group O(V) is that it is the set of all DISTANCE PRESERVING maps f:V --> V such that f(0) = 0. (Distance preserving means || f(v) - f(w) || = || v - w ||. While it is clear that a distance preserving map that leaves the origin fixed IS norm preserving, the converse is false as the above counter-example shows.) With this new definition of O(V) it is easy to show that orthogonal transformations really are linear (see the notes). Sorry about that! R. Palais --------------------------------------------------------------------------------- -- From Richard Palais Sat Sep 13 20:49:10 2003 To: Math32 From: Richard Palais Subject: Help with Matlab Cc: Bcc: X-Attachments: Message-Id: To Math 32a Students: Our Matlab Consultant, Izi Aviyente, will be having Office Hours each Friday from 1:40 pm--2:40 pm in the "Masters Room" (Volen 104). If you have any questions about Matlab, or if you are having problems with your Matlab projects, that will be a good opportunity for you to get some help. If Izi's office hour gets to busy---as is likely to happen later in the semester when I assign some more difficult projects---then he has expressed a willingness to have a second such hour on Tuesdays. Izi will also be helping me out in the Matlab instructional computer labs on Tuesday evenings in the Farber classroom, and you will have a chance to meet him this Tuesday. R. Palais --------------------------------------------------------------------------------- -- From Richard Palais Thu Sep 18 14:28:28 2003 To: Math32 From: Richard Palais To Math 32a Students: http://rsp.math.brandeis.edu/Math32/Assignments/MatlabProject1.pdf --------------------------------------------------------------------------------- -- From Richard Palais Thu Sep 18 15:37:57 2003 To: Math32 From: Richard Palais Subject: Cookout Sunday afternoon Cc: Bcc: X-Attachments: :DeskStar:424188:LocalMap.JPG: :DeskStar:424188:Map1.jpg: :DeskStar:424188:Map2.jpg: :DeskStar:424188:RailSched.pdf: Message-Id: Math 32a Students: It looks like the weather will be perfect Sunday afternoon, so I am confirming the cookout invitation for 3:30 PM on 9/21. The attached train schedule shows a train leaving Roberts at 3:23 pm and arriving at Kendall Green at 3:28 pm. (It is about a ten minute walk from Kendall Green Station to our home.) Remind me tomorrow that we should discuss possible car pooling. Our home at 68 Lexington Street in Weston is on the west side of the street, just opposite a large old red house (dating from the 1700s!). I have attached a few maps. The star marks our home and the crosses mark Roberts and Kendall Green stations. It's about a ten minute walk from the Kendall Green station to our home. See you tomorrow, R. Palais --------------------------------------------------------------------------------- ---- From Richard Palais Fri Sep 19 17:55:57 2003 To: Math32 From: Richard Palais To Math 32a Students: I have posted the third exercise set (due in a week) at: http://rsp.math.brandeis.edu/Math32/Assignments/Assignment3.pdf The notes for todays lecture are at: http://rsp.math.brandeis.edu/Math32/LectureNotes/Lecture5 I am also posting my Matlab notes called "Second Steps" at: http://rsp.math.brandeis.edu/Math32/LectureNotes/SecondSteps.m These are an edited version of the notes from last Tuesdays lab. I have still not got them in a form that satisfies me, but several of you asked for them so I am posting them in preliminary form. (In particular, the final topic, on function M-Files is missing.) I'll let you know when the final version is available. Finally, I have posted the First Matlab Project (on Gram-Schmidt). This is the original assignment plus some addenda. Several of you have already passed in this project, and you might want to look this over and consider if you want to revise your first version in the light of the addenda there and the remarks I made in class today. http://rsp.math.brandeis.edu/Math32/Assignments/MatlabProject1.pdf If you have not passed in this assignment you might want to also read through "Second Steps" before finishing writing your version of the GramSchmidt function M-file. But please, try to have it done by Monday evening, so Izi and I can look them over in time for the Tuesday evening meeting. Remember that the plan is to discuss the project and your various answers at the Tuesday evening Lab session and I hope as many of you as can will come. See you all Sunday afternoon, R. Palais --------------------------------------------------------------------------------- -- From Richard Palais Fri Sep 19 23:22:47 2003 To: Math32 From: Richard Palais Subject: Assignment 2 Answers To Math 32a Students: I have posted the answers to the second assignment exercises at: http://rsp.math.brandeis.edu/Math32/Assignments/Assignment2Answ.pdf R. Palais --------------------------------------------------------------------------------- -- From Richard Palais Mon Sep 22 10:32:04 2003 To: Math32 From: Richard Palais Subject: revised Matlab notes Cc: Bcc: X-Attachments: Message-Id: To Math 32a Students: I have revised my first and second sets of Matlab notes and posted the revisions at: http://rsp.math.brandeis.edu/Math32/LectureNotes/FirstSteps.m and http://rsp.math.brandeis.edu/Math32/LectureNotes/SecondSteps.m These notes should now be better for self-help. The idea is to go through the notes, reading the comments and entering the commands in a Matlab Command Window, and be sure you understand what the responses from Matlab mean and why you are getting them before going on to the next topic. While the notes are "improved" they are not really in final form, and I hope eventually to publish them. So please, if you spot any errors or if you find some parts difficult to understand, or if you have some suggestions for improvements, please write to me with your comments. Tomorrow in the evening session in Farber Classroom we will discuss the various solutions to Matlab Project #1. Please try to finish up (and test) your GramSchmidt function M-File, and send it as an attachment to both me and Izi Aviente before this evening. R. Palais --------------------------------------------------------------------------------- -- From Richard Palais Wed Sep 24 14:14:32 2003 To: Math32 From: Richard Palais Subject: various Cc: Bcc: X-Attachments: Message-Id: To Math 32a Students: 1) At the computer lab last night, several of you requested that I postpone the due date for Assignment 3 until next Tuesday, and I agreed to that. We also agreed that future assignments would be due on Tuesdays too, so asto give you time to work on them over a week-end. 2) I edited my notes from last night in which I discussed one way of going about solving the first Matlab project (on implementing GramSchmidt as a Matlab function M-File). I have put these revised notes on our web-site at: http://rsp.math.brandeis.edu/Math32/Assignments/Project1Discussion.m In addition to my own suggested solution, I also posted the code from several other solutions submitted by class members that had some interesting twist on how to go about things. R. Palais --------------------------------------------------------------------------------- -- From Richard Palais Wed Sep 24 15:43:58 2003 To: Math32 From: Richard Palais Subject: Assignment Cc: Bcc: X-Attachments: Message-Id: Math 32a Students: I have posted Assignment Number 4. The URL is: http://rsp.math.brandeis.edu/Math32/Assignments/Assignment4.pdf It isn't due until October 6 (a week from next Tuesday), but just in case you would like to get a head start... R. Palais --------------------------------------------------------------------------------- -- From Richard Palais Fri Sep 26 22:24:39 2003 To: Math32 From: Richard Palais Subject: Cc: Bcc: X-Attachments: Message-Id: To Math 32a Students: I have posted the notes for "Lecture 6". The URL is: http://rsp.math.brandeis.edu/Math32/LectureNotes/Lecture6.pdf (This covers material from both lectures of this week.) R. Palais --------------------------------------------------------------------------------- -- To: Math32 From: Richard Palais To Math 32a Students; I have put my notes from last weeks (i.e., the third) Computer Lab online. (This is a discussion of different ways to go about solving the first Matlab project.) You can find it at: http://rsp.math.brandeis.edu/Math32/LectureNotes/Project1Discussion.m --------------------------------------------------------------------------------- --- To: Math32 From: Richard Palais Subject: To Math 32 Students: Anna made up an answer sheet for the third assignment (the one that was due today), and I have posted it on the web-site. The URL is: http://rsp.math.brandeis.edu/Math32/Assignments/Assignment3Answ.pdf R. Palais --------------------------------------------------------------------------------- --- To: Math32 From: Richard Palais Subject: Lecture Notes and 2nd Matlab project To Math 32a Students: Preparing the notes for my recent lectures took substantially longer than I had expected. Sorry. The notes for the last couple of lectures are now available on the course web-site. The URL is: http://rsp.math.brandeis.edu/Math32/LectureNotes/Lecture7.pdf Actually, these notes go beyond where I left off on last Friday and cover material I will lecture on next Friday as well. Moreover, the final three pages of these notes are an exposition of numerical integration (the Trapezoidal Rule and Simpson's Rule) which you perhaps (I hope) already saw in your basic calculus course. The last of these three pages explains the Second Matlab Project, which involves implementing the Trapezoidal Rule and Simpson's Rule as Matlab code and doing some experiments with it. You can also find the same material from those three pages at: http://rsp.math.brandeis.edu/Math32/Assignments/MatlabProject2.pdf Please make an effort to understand this material before the Computer Lab Tuesday evening, since I would like to begin discussing it then. (But don't worry if you have difficulty in understanding the material. It will all be discussed in class and you will have plenty of opportunity to ask questions about it.) BTW, we will only BEGIN to discuss the second project this Tuesday. It will not be due until a week later. R. Palais --------------------------------------------------------------------------------- --- To: Math32 From: Richard Palais Subject: Notes for Matlab lecture To Math 32a Students: I have posted the notes for the fourth Matlab Computer Lab on our web-site. The URL is: http://rsp.math.brandeis.edu/Math32/LectureNotes/MoreAboutM-Files.m R. Palais --------------------------------------------------------------------------------- --- To: Math32 From: Richard Palais Subject: To Math 32a Students: Several students have asked me to delay posting the answers to Assignment 4, to give them more time to work on it, and your TA, Anna Varvak also feels that is a good idea. I am agreeable, and we can discuss the timing further tomorrow. BTW, Exercise 4, the proof of the Chain Rule, is admittedly fairly difficult. Do your best with it, but don't feel bad if you cannot get it. R. Palais --------------------------------------------------------------------------------- --- To: Math32 From: Richard Palais Subject: Midterm take-home To Math 32a Students: I hope the subject line did not panic anybody. All that is happening is that I am combining what would have been Assignments 4 and 5 and I am calling it a "take-home midterm exam". You will have until Friday October 24 to complete the exam, and I would like everyone to please hand it in at class-time on the twenty-fourth. You will find the exam at: http://rsp.math.brandeis.edu/Math32/Assignments/MidTerm.pdf Please do NOT put off working on it until the last moment! There are fifteen problems, all of them asking you to prove something---except for Problem 8, which asks you to write a Matlab function that implements the Banach Contraction Principle (aka, Successive Approximations). Each problem builds on the previous problems, so take them in order. The first few are quite easy and they gradually get a little harder (but there are lots of hints.) I think the best way to proceed for most of you will be to first read over all of the problems carefully, and then try to do a few problems each day. Perhaps try to do a problem first without looking at the hints. If you do not see how to do a the problem after thinking about it, read any hints over and think about it some more and then try again. I have tried quite hard to make it an interesting exam that will will not only be challenging and enjoyable, but will also teach you some interesting mathematics. If you do not understand something about what a question means, don't hesitate to ask me or Anna either in person or by email. Although I do not mind you conferring with each other on the other assignments, I would like you all to work out the problems on the midterm on your own. Please look over the exam before the class on Tuesday and we can discuss any questions you may have then. --------------------------------------------------------------------------------- --- To: Math32 From: Richard Palais Subject: Re: Midterm take-home To Math 32a Students: Sally LeGore pointed out that I made an error in the message I just sent you---I should have said "Assignments 5 and 6", not "Assignments 4 and 5". Of course Assignment 4 is the one that is supposed to be handed in either already or very shortly. Sorry, R. Palais --------------------------------------------------------------------------------- --- To: Math32 From: Richard Palais Subject: ODE Lecture Notes To Math 32a Students: I have posted the notes for this morning's lecture on ODE. They are at: http://rsp.math.brandeis.edu/Math32/LectureNotes/Lecture8.pdf Please note that this is a preliminary version. I will let you know when I have posted the final version. R. Palais --------------------------------------------------------------------------------- --- To: Math32 From: Richard Palais Subject: Answers to Fourth Assignment problems To Math 32a Students: The answers to the Assignment 4 problems have been placed on our web-site. The URL is: http://rsp.math.brandeis.edu/Math32/Assignments/Assignment4Answ.pdf R. Palais --------------------------------------------------------------------------------- --- To: Math32 From: Richard Palais Subject: Matlab notes To Math 32a Students: Rather than post new notes for last Tuesday's Computer lab, I have revised and updated the notes from the preceding week, correcting the error I mentioned to you on Tuesday. The revised notes are at: http://rsp.math.brandeis.edu/Math32/LectureNotes/MoreAboutM-Files.m R. Palais --------------------------------------------------------------------------------- --- To: Math32 From: Richard Palais Subject: ODE Lecture Notes Cc: Bcc: X-Attachments: To Math 32a Students: The final version of my lecture notes on ODE is now posted on the course web-site at: http://rsp.math.brandeis.edu/Math32/LectureNotes/Lecture8.pdf This includes all the material through todays lecture, including the material on numerical integration (Euler's Method and Runge-Kutta). R. Palais --------------------------------------------------------------------------------- --- To: Math32 From: Richard Palais Subject: Answers to Fourth Assignment problems Cc: Bcc: X-Attachments: To Math 32a Students: The answers to the Assignment 4 problems have been placed on our web-site. The URL is: http://rsp.math.brandeis.edu/Math32/Assignments/Assignment4Answ.pdf R. Palais --------------------------------------------------------------------------------- --- To: Math32 From: Richard Palais Subject: Matlab notes Cc: Bcc: X-Attachments: To Math 32a Students: Rather than post new notes for last Tuesday's Computer lab, I have revised and updated the notes from the preceding week, correcting the error I mentioned to you on Tuesday. The revised notes are at: http://rsp.math.brandeis.edu/Math32/LectureNotes/MoreAboutM-Files.m R. Palais --------------------------------------------------------------------------------- --- To: Math32 From: Richard Palais Subject: ODE Lecture Notes Cc: Bcc: X-Attachments: To Math 32a Students: The final version of my lecture notes on ODE is now posted on the course web-site at: http://rsp.math.brandeis.edu/Math32/LectureNotes/Lecture8.pdf This includes all the material through todays lecture, including the material on numerical integration (Euler's Method and Runge-Kutta). R. Palais --------------------------------------------------------------------------------- --- To: Math32 From: Richard Palais Subject: A couple of typos in the midtrm exam Cc: Bcc: X-Attachments: To Math 32a sudents: Chandni has pointed out that there are a couple of typos in the midterm exam. Namely, in problems 13 and 15 there is a small f denoting a function that should be an upper case F. R. Palais --------------------------------------------------------------------------------- --- To: Math32 From: Richard Palais Subject: Amazing speedup of Trapezoidal! Cc: Bcc: X-Attachments: To Math 32a Students I did some more experimenting with Nick Dufresne's vectorized versions of Trapezoidal Rule and Simpson's rule. The results are pretty amazing, particularly as concerns Trapezoidal. The ELAPSED time is about 1/5000 the elapsed time for the unvectorized version, and it is even in the same ballpark as Simpson's when used to compute pi to 10 decimal places. I think that the speedup of the vectorized Trapezoidal with respect to the unvectorized version is real and shows how expensive looping is. On the other hand, I think that the comparison of Trapezoidal and Simpson's is fake. I am sure that Trapezoidal for n = 100000 takes longer by a very large factor than Simpson's with n = 14, however there is a fixed "startup" time for running any M-File, and this is big enough to swamp the much smaller compute times of both Trapezodal and Simpson's. That is, if F is the fixed startup time and T and S are the computing times for Trapezoidal and Simpson's, then If F is much larger than T, And S is only a small fraction of T, F + T = F(1 + T/F) and F+ S = F(1 + S/F) differ only by the factor (1+ S/F)/(1 + T/F) which will be close to 1 if T/F is small, even if S/F is very, very small. Izi, does that seem right to you, or do you have a better explanation? Anyone else have a competing theory? Anyway, here are Nick's M-Files, and below them the "shootout". Congratulations Nick! R. Palais %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function thesum = TrapezoidalRule (f,a,b,n) delta = (b-a)/n; arguments = [a:delta:b]; %calculate all the values of the function %in summing all n parts of the trapezoidal method we count internal points %twice so we will multiply by 2 values = 2*feval(f,arguments); %since we have calculated the value of the endpoints twice we need to %subtract the values and we need to multiply by 0.5*delta = 1/2*(b-a)/n thesum = 0.5*delta*(sum(values)-feval(f,a)-feval(f,b)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function thesum = SimpsonsRule (f,a,b,n) delta = (b-a)/n; arguments = [a:delta:b]; midpointArgs = [a+delta/2:delta:b-delta/2]; %calculate all the values of the function. %in summing all n parts of the trapezoidal method, %we need to count internal pointstwice so we multiply by 2. values = 2*feval(f,arguments); %in the formula all midpoint values are multiplied by 4 midpointValues = 4*feval(f,midpointArgs); %since we have calculated the value of the endpoints twice we need to %subtract the values and we need to multiply by (1/6)*delta = 1/6*(b-a)/n thesum = (1/6)*delta*(sum(midpointValues)+sum(values)-feval(f,a)-feval(f,b)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Shootout f = inline('4./(1+ x.^2)','x'); % must be vectorized for third versions! a = 0; b = 1; n=100000; tic, A = TrapezoidalRule3(f,a,b,n); toc % elapsed_time = 0.07617 Error = abs(A-pi) % 1.664046678229170e-11 n=14; tic, A = SimpsonsRule3(f,a,b,n); toc % elapsed_time = 0.002523 Error = abs(A-pi) % 8.234657400407741e-11 n = 100000; tic, A = TrapezoidalRule2(f,a,b,n); toc % elapsed_time = 49.330224 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --------------------------------------------------------------------------------- --- To: Math32 From: Richard Palais Subject: Interesting experiment! Cc: Bcc: X-Attachments: To Math 32a Students: Well, I have done some more experiments, and they seems to confirm our suspicion that while the Trapezoidal rule is indeed acting like a second order method for our function 1/(1 + x^2), on the other hand Simpson's Rule is NOT behaving like a fourth order method, but rather like a sixth order method! I have to admit that this has me baffled. R. Palais P.S. I have posted my edited notes of the Computer Lab session last night in which we discussed the second computer project. It is at: http://rsp.math.brandeis.edu/Math32/Assignments/Project2Discussion.m and contains the "Shootout" from my previous message and also the results below from my experiments on how errors depend on the number of subdivisions. R. Palais %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %%% Let's check how the errors vary as a function %%% of the number n of subdivisions. %%% First Trapezoidal: % n = 5; A = TrapezoidalRule1(f,a,b,n); Error5 = abs(A-pi) % 0.00666653977880 n=10; A = TrapezoidalRule1(f,a,b,n); Error10 = abs(A-pi) % 0.00166666468263 % % Since Trapezoidal is a quadratic method, the ratio % of Error5 to Error 10 should be about (10/5)^2 = 4 % ratio = Error5/Error10 % 3.99992862887449 % %%%% Now Simpson's % n = 5; A = SimpsonsRule1(f,a,b,n); Error5 = abs(A-pi) % 3.965057793209326e-08 n=10; A = SimpsonsRule1(f,a,b,n); Error10 = abs(A-pi) % 6.200080449048073e-10 % % Since SimpsonsRule1 is a fourth order method, the ratio % of Error5 to Error 10 should be about (10/5)^4 = 16 % BUT,IN FACT: ratio = Error5/Error10 % 63.95171523650308 ~ (10/5)^6 % n = 20; A = SimpsonsRule1(f,a,b,n); Error20 = abs(A-pi) % 9.687362023669266e-12 % ratio = Error10/Error20 % 64.00174200055011 ~ (20/10)^6 % So, the conclusion seems nearly inescapable that for % the present function, Simpson's is behaving as a % sixth order method !!!! --------------------------------------------------------------------------------- --- To: Math32 From: Richard Palais Subject: Third and Fourth Matlab projects Cc: Bcc: X-Attachments: To Math 32a Students Recall that the third Matlab project (on successive approximation) was assigned as part of the Midterm exam, and the fourth project is an exercise at the end of the notes for "Lecture 8"---the section of the notes on the IVP for first order ODEs. I have now posted each of these as separate pdf files, with just the exposition that is relevant to each project. They are located at: http://rsp.math.brandeis.edu/Math32/Assignments/MatlabProject3.pdf and http://rsp.math.brandeis.edu/Math32/Assignments/MatlabProject4.pdf Of course you will need to complete Project 3 before Friday when the Midterm is due. Please try to work on Project 4 over the weekend, since I would like to discuss both of these projects during the Computer Lab next Tuesday evening. R. Palais --------------------------------------------------------------------------------- --- To: Math32 From: Richard Palais Subject: Programming groups To Math 32a Students Recall that we discussed dividing up into groups to carry out the upcoming more complex Matlab projects, and we decided that the size of a group should not be greater than three, to avoid logistical problems. Since Lacra and Chandni live in the same suite they asked to be together in one group. That left twelve other members of the class---enough for four more groups, and on the advice of Izi I would like to propose the following groupings. Group 1: Chandni Valiathan and Lacra Bintu Group 2: * Nicholas DuFresne, Gregory Berlinrut, Dina Shapiro Group 3: * Steve Gindi, Steve Dupree, Qaiser Saify Group 4: * David Diamondstone, Matthew Roberts, Ilya Bronshtein Group 5: * Simon Slutsky, Nicholas Lee, Margaret Jones The first named person in each of the latter four groups is our recommendation for the group leader, who will be in charge of setting up meetings, and seeing to it that the group submits its projects on time. There is of course considerable arbitrariness in these choices. If two people in different groups both agree that they would like to switch to the other's group let me know and I will make the change. Also, if when a group first gets together and discusses things, if the group leader would prefer not to have that responsibility and can convince someone else in the group to accept the job, that is OK, but let me know about it. R. Palais --------------------------------------------------------------------------------- --- --------------------------------------------------------------------------------- ---