2019 Spring - MAT 2680 - Reitz

"...how it differs from the rocks"

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OpenLab #2: Numerical Methods PROJECT – Part 2

March 19, 2019 / / 39 Comments

This post contains additional information related to the Numerical Methods Project earlier in the week, due April 4th.

Submission guidelines. Your completed project will consist of three elements:

  1. A working numerical methods calculator using your choice of technology (spreadsheet, programming, mathematical software), as described in the previous post.
    HOW TO SUBMIT:

    • If your project is a spreadsheet, either share it with me (if it is in Google Docs or a similar cloud-based platform), or email the file to me as an attachment.
    • If your project is code, please submit it using an online coding site like ideone.com – once your code is working on the site, you can simply submit a link.  If you are using a programming language not supported by ideone.com, you can email the source code to me.
    • If your project uses mathematical software, either share it with me (if it is in MatLab Online or a similar cloud-based platform), or email the file to me as an attachment.
  2. Solutions to the following example using each of the three methods studied in class (Euler’s Method, Improved Euler’s Method, and Runge-Kutta), generated by your numerical methods calculator.  Submit using the same method as in part 1.

Project Example.  Given the differential equation dy/dx = \frac{x^2}{y-1} and initial condition y(-1)=-0.5, approximate the value of y(2) using Euler’s Method, Improved Euler’s Method, and Runge-Kutta.

For each method, choose a step size that gives four correct digits following the decimal point.  How many steps are required to obtain this level of precision?

NOTE: The actual solution is y(2)=-1.87228…

  1. Writing assignment.  Write one or two paragraphs (minimum 300 words) responding to the following.  Leave your response as a comment on this post.
    1. Describe your project and how it works.
    2. Describe the process of building your numerical methods calculator.  What kind of technology did you decide to use, and why? Did you encounter any unexpected challenges in completing this project?
    3. Why do we need numerical methods in addition to the other methods studied in the class?
    4. Why is this assignment included in the class (instead of just computing the various methods using a calculator)?
    5. Please include a link to your project (if it is online), or clearly state that you will be sending me the files by email (and don’t forget to do it!).

 

Update regarding WeBWorK: 6-BasicSecondOrder

March 17, 2019 / / 0 Comments

Hi everyone,

Looking through this assignment, I see that the first 3 problems ask about the “characteristic polynomial,” which we have not yet discussed in class.  We will discuss this concept on Tuesday (it is not difficult), but in the meantime you should be able to solve problems 4-8 on this assignment.

Regards,
Prof. Reitz

Exam 1 SPECIAL OFFER Scoring Guide

March 16, 2019 / / 0 Comments

For your reference, here is the scoring guide I will be using for the EXAM 1 Special Offer

-Prof. Reitz

 

Name: ____________________

 

____ Includes Name, Date, Problem #s, original scores (up to 6 points deduction)

____ Presentation is neat, well-organized, readable (up to 4 points deduction)

____ Includes Original Exam

____ Max bonus (30 points for <50%, 20 points 50%-59%, 15 points 60%-69%, 10 points 70%-79%, 5 points 80%-89%)

 

First problem #:  ____

____ Original Score (out of 25)

____ Revised Score

____ (up to 5 points deduction if incomplete) Written explanation,  2 sentences,  what you did wrong OR how to solve the problem.

____ Bonus points earned for problem 1

 

Second problem #: ____

____ Original Score (out of 25)

____ Revised Score

____ (up to 5 points deduction if incomplete) Written explanation,  2 sentences,  what you did wrong OR how to solve the problem.

____ Bonus points earned for problem 2

 

EXAM 1 SPECIAL OFFER BONUS POINTS:   _______

(Bonus points for problems 1 and 2, with maximum bonus based on original exam score, minus any deductions, ).

EXAM 1 REVISED SCORE: ______

Exam #2 Date Change

March 16, 2019 / / 0 Comments

Hi everyone,

Because we are running slightly behind the class calendar, I am pushing back the second exam by 1 week.  The new date for the second exam is Tuesday, April 2nd.  A review sheet will be provided at least one week in advance.

Regards,
Prof. Reitz

Preliminary Midsemester Grades are posted

March 14, 2019 / / 0 Comments

Hi everyone,

Your Midsemester Grades are posted under Dashboard/GradeBook).

What is this for?

These grades are not entered into the transcript, but serve as important guidance to students and an opportunity to review their progress in the course.  – Provost August

The grades are based on work completed so far:

  • WeBWorK performance (weighted 35%)
  • the first OpenLab assignment (weighted 15%)
  • Exam 1 (weighted 50%)

The grading scheme is as follows:

  • P=Passing work
  • BL= Borderline
  • U=Unsatisfactory
  • SA=Stopped Attending

NOTE: For those of you that decide to take advantage of the EXAM 1 SPECIAL OFFER, your score on Exam 1 may change.  This, in turn, may affect your standing in the course.    Feel free to ask me for an updated Midsemester Grade  after the SPECIAL OFFER is returned.

Heads up: The last day to drop the class with a “W” grade is April 1st.  If you decide to drop, it is *much* better to do so before that date (but you should be aware of any financial aid impact that dropping a class may have).  Please let me know if you wish to discuss it.

-Prof. Reitz

OpenLab #2: Numerical methods PROJECT – Example Data

March 14, 2019 / / 0 Comments

Below you will find complete solutions for the Example given the previous post – you can use this to test your project.

Example.  Given the differential equation dy/dx=x^2-0.5xy and initial condition y(1.5)=2.2, approximate the value of y(2.5) using step size h=0.05

ACTUAL ANSWER: y(2.5) = 3.49201…

Euler’s Method
i h x_i y_i k = f(x_i,y_i) y_(i+1)
0 0.05 1.5 2.2 0.6 2.23
1 0.05 1.55 2.23 0.67425 2.2637125
2 0.05 1.6 2.2637125 0.74903 2.301164
3 0.05 1.65 2.301164 0.8240397 2.342365985
4 0.05 1.7 2.342365985 0.8989889128 2.387315431
5 0.05 1.75 2.387315431 0.9735989982 2.435995381
6 0.05 1.8 2.435995381 1.047604158 2.488375588
7 0.05 1.85 2.488375588 1.120752581 2.544413217
8 0.05 1.9 2.544413217 1.192807443 2.60405359
9 0.05 1.95 2.60405359 1.26354775 2.667230977
10 0.05 2 2.667230977 1.332769023 2.733869428
11 0.05 2.05 2.733869428 1.400283836 2.80388362
12 0.05 2.1 2.80388362 1.465922199 2.87717973
13 0.05 2.15 2.87717973 1.52953179 2.95365632
14 0.05 2.2 2.95365632 1.590978049 3.033205222
15 0.05 2.25 3.033205222 1.650144125 3.115712428
16 0.05 2.3 3.115712428 1.706930708 3.201058964
17 0.05 2.35 3.201058964 1.761255718 3.289121749
18 0.05 2.4 3.289121749 1.813053901 3.379774445
19 0.05 2.45 3.379774445 1.862276305 3.47288826
20 0.05 2.5 y(2.5) = 3.47288826

 

 

 

 

 

Improved Euler’s Method
i h x_i y_i k1 z_(i+1) k2 y_(i+1)
0 0.05 1.5 2.2 0.6 2.23 0.67425 2.23185625
1 0.05 1.55 2.23185625 0.6728114063 2.26549682 0.7476025438 2.267366599
2 0.05 1.6 2.267366599 0.746106721 2.304671935 0.8211456538 2.306547908
3 0.05 1.65 2.306547908 0.8195979758 2.347527807 0.8946013641 2.349402892
4 0.05 1.7 2.349402892 0.8930075421 2.394053269 0.9677033899 2.395920665
5 0.05 1.75 2.395920665 0.9660694182 2.444224136 1.040198278 2.446077357
6 0.05 1.8 2.446077357 1.038530378 2.498003876 1.111846414 2.499836777
7 0.05 1.85 2.499836777 1.110150981 2.555344326 1.18242289 2.557151124
8 0.05 1.9 2.557151124 1.180706432 2.616186446 1.251718216 2.61796174
9 0.05 1.95 2.61796174 1.249987303 2.680461105 1.319538895 2.682199895
10 0.05 2 2.682199895 1.317800105 2.7480899 1.385707852 2.749787594
11 0.05 2.05 2.749787594 1.383967716 2.81898598 1.450064721 2.820638405
12 0.05 2.1 2.820638405 1.448329675 2.893054889 1.512465995 2.894658297
13 0.05 2.15 2.894658297 1.510742331 2.970195413 1.572785045 2.971746481
14 0.05 2.2 2.971746481 1.571078871 3.050300425 1.630912022 3.051796253
15 0.05 2.25 3.051796253 1.629229215 3.133257714 1.686753629 3.134695825
16 0.05 2.3 3.134695825 1.685099802 3.218950815 1.740232793 3.220329139
17 0.05 2.35 3.220329139 1.738613261 3.307259802 1.791288237 3.308576677
18 0.05 2.4 3.308576677 1.789707988 3.398062076 1.839873957 3.399316225
19 0.05 2.45 3.399316225 1.838337624 3.491233107 1.885958617 3.492423631
20 0.05 2.5 y(2) = 3.492423631

 

 

 

 

Runge-Kutta
i h x_i y_i k1 = f(x_i,y_i) k2 = f(x_i+.5h,y_i+.5hk1) k3 = f(x_i+.5h, y_i+.5hk2) k4 = f(x+h,y+hk3) Runge-Kutta y_(i+1) = y_i + h*(k1+2k2+2k3+k4)/6
0 0.05 1.5 2.2 0.6 0.6366875 0.6359881445 0.6728554594 2.23181839
1 0.05 1.55 2.23181839 0.6728407481 0.709821466 0.7090934081 0.746181552 2.267292157
2 0.05 1.6 2.267292157 0.7461662747 0.7832936203 0.7825394711 0.8197042176 2.306438296
3 0.05 1.65 2.306438296 0.8196884061 0.8568207014 0.856043244 0.8931456109 2.349259645
4 0.05 1.7 2.349259645 0.8931293019 0.9301304558 0.9293326184 0.9662395087 2.395745436
5 0.05 1.75 2.395745436 0.9662227434 1.002962858 1.002147687 1.038732462 2.445871905
6 0.05 1.8 2.445871905 1.038715285 1.075071194 1.074241825 1.110384803 2.499602956
7 0.05 1.85 2.499602956 1.110367265 1.146222996 1.145382627 1.180971517 2.556890873
8 0.05 1.9 2.556890873 1.18095367 1.216200837 1.215352702 1.250282954 2.617677071
9 0.05 1.95 2.617677071 1.250264856 1.284802979 1.283950319 1.318125413 2.681892878
10 0.05 2 2.681892878 1.318107122 1.351843875 1.350989913 1.384321567 2.749460347
11 0.05 2.05 2.749460347 1.384303145 1.417154527 1.416302445 1.448710757 2.820293079
12 0.05 2.1 2.820293079 1.448692267 1.480582715 1.479735625 1.51114915 2.894297063
13 0.05 2.15 2.894297063 1.511130657 1.541993079 1.541154007 1.57150976 2.971371518
14 0.05 2.2 2.971371518 1.57149133 1.601267084 1.600438946 1.629682352 3.051409732
15 0.05 2.25 3.051409732 1.629664051 1.658302858 1.657488442 1.685573222 3.134299898
16 0.05 2.3 3.134299898 1.685555117 1.713014923 1.712216872 1.739104879 3.219925928
17 0.05 2.35 3.219925928 1.739087035 1.765333814 1.764554613 1.79021561 3.308168257
18 0.05 2.4 3.308168257 1.790198091 1.815205609 1.814447568 1.838858971 3.398904619
19 0.05 2.45 3.398904619 1.838841842 1.862591365 1.861856614 1.885003188 3.492010794
20 0.05 2.5 y(2.5) = 3.492010794

OpenLab #2: Numerical methods PROJECT – Part 1

March 12, 2019 / / 2 Comments

Numerical methods provide a way to compute (approximate) values of solutions to differential equations, even when we cannot solve the equations exactly.  The drawback is the large number of numerical calculations required to obtain a desired value and level of precision.  In this project, you will use technology to implement the various numerical methods and use your technological solution to solve differential equations problems.

Assignment (Due Tuesday, April 4th).  Create a numerical methods calculator.  You can choose your technology tool for this job – use any one of the following:

  1. a spreadsheet (Excel, Google Sheets, or other spreadsheet)
    • if you choose to create a spreadsheet, you should have columns forx, y, f(x,y) , and so on, and each stage should appear in its own row
  2. a programming language (Java, Perl, or other programming language)
    • if you choose to write code, your program should output the values of x, y, f(x,y) and so on at each stage
  3. mathematical software (MatLab, Maple, Mathematica, or other mathematical software)
    • if you choose to use mathematical software, your program should output the values of x, y, f(x,y) and so on at each stage

How to submit. Part 2 of this project will talk about how to submit your project – you will be asked to upload your solution (spreadsheet, or code, or mathematical software document) and to write  about what you did.  For now, focus on getting your solution working.

Requirements:

  1. Your solution should allow you to solve problems of this type:

Example.  Given the differential equation dy/dx=x^2-0.5xy and initial condition y(1.5)=2.2, approximate the value of y(2.5) using step size h=0.05

  1. Your solution must be able to carry out Euler’s Method, Improved Euler’s Method, and Runge-Kutta (you may implement these as three separate spreadsheets or programs if you wish).
  2. Your solution should display all the points (x,y) found along the way, not just the final point.
  3. Your solution should also display other values found while carrying out each method:
    1. Euler’s Method: display the slope f(x,y) at each stage
    2. Improved Euler’s: display the values of k1, k2 at each stage
    3. Runge-Kutta: display the values of k1, k2, k3, k4 at each stage
    4. You can display other values as well, if you wish (for example, the intermediate y-value in the Improved Euler method that we refer to as z).
  4. Your solution may NOT use any built-in version of these methods (for example, most mathematical software contains a built-in command for Euler’s Method – you can use this to check your work, but you need to create your own solution).
  5. You should be able to relatively easily change the initial condition, step size, and target value.
  6. You should be able to relatively easily change the differential equation (it is ok if the equation is hard-coded into your program).

Test your project.  Solution data for the above example using Euler, Improved Euler, and Runge-Kutta will be posted this weekend so you can test your project.  You can also use examples from class to test your work, since you know what the solutions are.

 

Exam 1 Grades are posted, and SPECIAL OFFER

March 4, 2019 / / 0 Comments

Hi everyone,

The grades for Exam 1 are posted (under Dashboard/OpenLab GradeBook).

With some exceptions, you will notice that the scores are not as high as you might have liked!  This exam covered a lot of material, and relied on a great deal of prior knowledge and skills (especially Calculus and Algebra).  With that in mind, I am giving you the option to improve your score through the ONE-TIME SPECIAL OFFER below (note: this offer will almost certainly *not* be repeated on future exams), due in two weeks on Tuesday 3/19.

Let me know if you have any questions,
Prof. Reitz

Exam 1 Special Offer – earn bonus points.  You can improve your grade on the exam, by doing the following:

  1. Choose ONE OR TWO problems in which you did NOT earn full points.  You are working to earn back (some of ) the points you missed on this problem.  For each problem you chose:
  2. Do the problem over, neatly and completely, start to finish, on a separate sheet of paper.
  3. Include the following information on the sheet:
    1. Your name
    2. The date
    3. The problem number
    4. Your original score on the problem (out of 25)
  4. On the same sheet, write a short paragraph (at least two complete sentences) explaining why you lost points.
    1. If you lost points due to a mistake or mistakes, explain what you did wrong (this is to let me know that you understand your error).
    2. If you lost points due to not completing the problem, then explain how to solve it (this is to let me know that you have learned how to solve it).
  5. Hand in your original exam and your corrected problem(s) and explanation(s), stapled together, in class on Tuesday, 3/19.
  6. Bonus points will be added to your Exam score based on the number of points you missed on the chosen problem(s),  the accuracy of your corrections and explanation, and your overall grade on the exam.  Bonus points are limited as follows:
    1. If you received less than 50% on the exam, you can earn a maximum of 30 bonus points.
    2. If you received between 50% – 59% on the exam, you can earn a maximum of 20 bonus points.
    3. If you received between 60% – 69% on the exam, you can earn a maximum of 15 bonus points.
    4. If you received between 70% – 79% on the exam, you can earn a maximum of 10 bonus points.
    5. If you received between 80% – 89% on the exam, you can earn a maximum of 5 bonus points.
    6. If you received 90% or more on the exam, you may not earn any bonus points.

 

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