MS213: Numerical Methods Computing Assignments with Matlab

Transcription

1 MS213: Numerical Methods Computing Assignments with Matlab SUBMISSION GUIDELINES & ASSIGNMENT ADVICE 1 Assignment Questions & Supplied Codes 1.1 The MS213 Numerical Methods assignments require students to solve 6 numerical problems using Matlab. The problems are related to 6 areas of the syllabus, namely (1) roots of a single nonlinear equation; (2) direct methods for linear systems; (3) iterative methods for linear systems; (4) numerical differentiation; (5) interpolation; and (6) numerical integration. 1.2 All problems require a small amount of Matlab programming but the typical requirement is for the student to piece together existing Matlab codes for problem-solving and comparison purposes. Emphasis should be placed on program design, and all numerical experiments should be carefully structured to allow for subsequent results analysis and meaningful comparisons. Marks will be awarded especially on the content and quality of presentation of a final report, detailing the major quantitative and qualitative results. 1.3 Matlab codes have been specially written for each of the 6 syllabus areas mentioned above. There are 5 methods available for solving nonlinear equations, 3 methods each for direct and iterative linear systems, a general purpose code for numerical differentiation (for derivatives of orders 1, 2 and 3), one method (two codes) for Newton polynomial interpolation, and 2 methods for numerical integration. All available codes are accompanied by sample scripts (with associated function files, where necesary) and will generate sample output when executed. The codes are available on Moodle (see the Matlab Assignments folder). 2 Submission Date Students are requested to submit the assignments by 1pm on Thursday of week 10 of Semester 1. Each of the 6 assignments is structured in the form of a tutorial question (requiring computational analysis) and it is intended that one question at a minimum should be tackled each week. 3 Submission Format 3.1 It is intended that the output for each question should require no more than a single A4 page so please consult with the Tutor or Lecturer if you are having difficulties in this regard. For the MS213 Computing Assignments overall, you are to submit at most 25 pages, 4 pages per question (three pages at most for the main code and any newly-created function and a single page for the output) together with a completed Assignment Cover Page which is available on Moodle (see the Matlab Assignments folder). 1

2 MS213: Numerical Mathematics Submission Guidelines You are requested to store the full code (which was used to generate the output presented) for each question on a CD using file names to make clear which question is being answered (see example below). MS213: NUMERICAL METHODS COMPUTING ASSIGNMENTS WITH MATLAB Assignment Question Numbers To be Completed by the Lecturer MS213 Student Name Student Name: In Block Capitals ID Number: Student ID Number DECLARATION: I, the undersigned, declare that the assignment material, which I hereby submit, is my own work. Any assistance received by way of borrowing from others has been cited and duly acknowledged within this submission. I make this declaration in the knowledge that a breach of the rules pertaining to project submission may carry serious consequences. SIGNATURE: DATE: 3.3 Please submit the duly-completed cover sheet, a correctly formatted print-out of the main code and output, and a CD containing all the codes used for the 8 assignment questions in a standard lightweight A4 display book having 20 polypropylene pockets: MS213 Student Name Student ID Number MS213: NUMERICALMETHODS COMPUTINGASSIGNMENTSWITHMATLAB AssignmentQuestionNumbers To be CompletedbytheLecturer Student Name: In Block Capitals ID Number: DECLARATION: I, the undersigned, declare that the assignment material, which I hereby submit, is my own work. Any assistance received by way of borrowing from others has been cited and duly acknowledged within this submission. I make this declaration in the knowledge that a breach of the rules pertaining to project submissionmaycarryseri- ous consequences. SIGNATURE: DATE: 3.4 The first comment statements of all files must contain your name, ID number, etc., as well as the question number, filename and date created. Output should be formatted in a similar manner. Submissions which do not conform to these guidelines will not be accepted. 3.5 Comment statements should also state the purpose of the program and guide the reader through its various steps. Every program that accepts input from the user and/or prints results must prompt the user to enter input and/or label the output.

3 MS213: Numerical Mathematics Submission Guidelines 3 % MS213: Matlab Computing Project % =============================== % % Student: Jack O Shea % ID Number: % Class Group: ACM2 % Module: MS213 % Question: Q.N % Date: 1/10/ Test your code carefully prior to submission. Always use the exact data supplied (if any) or the kind of data suggested (if any) to test the version of the code that you intend to submit. Send an to the Lecturer if you are unable to locate any required data files. 4 Grading & Academic Honesty 4.1 Failure to submit any of the assignment questions will result in a fail grade being awarded for that question and, as a consequence, this will undermine any decision to record an overall pass in the continuous assessment element of the module. To pass the overall module will require an award of pass in both the continuous assessment and examination elements of the module. 4.2 You are expected to consult with your Tutors for advice in solving the assigned problems. However, each student must submit his or her own individual solutions. A code of academic honesty is expected in this module. Students who are found to have submitted material which is in breach of this code will suffer the following two consequences: (a) a fail grade will be given for the assignment in question; and (b) a report of the incident will be filed with the Programme Board. Breaches of the code include the provision of assignment code segments to someone else, the receipt of assignment code segments from someone else, and plagiarism. 5 Advice on Assignment Questions Q.1 By way of illustration, suppose that we wish to find a root of f(x) = x 1 8 e2x 1 (which we will refer to as the Boltzmann problem) which is close to x = 2 (there is another root close to x = 0). A key requirement for the fixed-point method is to choose g(x) correctly and this depends on the location of the root. There are two obvious choices for g(x), namely Simple analysis shows that g 1 (x) = 1 8 e2x 1 and g 2 (x) = 1 (1 + log 8x) 2 g 1(x) = 1 4 e2x 1 < 1 for x < 1 (1 + log 4) and g 2(x) = 1 2x < 1 for x > 1 2

4 MS213: Numerical Mathematics Submission Guidelines 4 Therefore, we choose g(x) = g 2 (x) for the root near x = 2. As starting values, we use [a, b] = [1.5, 2.5] for the Bisection method and x 0 = 3 for all the other methods except the Secant method where we use x 0 = a and x 1 = b. In addition to the standard headings, the data output required has the following format: MS213 Project Sample Problem Q1 :: The Boltzmann Problem =============================== ===================== Method TOL ITER NFE Appr Root Error Bisection 1.0e e-005 Fixed Pt 1.0e e-005 Newton 1.0e e-014 Secant 1.0e e-008 Steffensen 1.0e e-012 Bisection 1.0e e-007 Fixed Pt 1.0e e-007 Newton 1.0e e-014 Secant 1.0e e-013 Steffensen 1.0e e-016 Bisection 1.0e e-009 Fixed Pt 1.0e e-009 Newton 1.0e e+000 Secant 1.0e e-016 Steffensen 1.0e e-016 As we do not know the true value of the root, we can substitute the result obtained from Matlab s fzero function using a very small tolerance, e.g : accurate_value = fzero(fx, (a+b)/2, optimset( Tolx, 1.0e-14)); In the actual assignment question, it is advisable to declare β as a global variable in order to loop through the 3 values in the same code run. Q.2 A key requirement in Question 2 is the need to compute the l norm of the error in the computed solution: x ˆx = max 1 i n x i ˆx i where x and ˆx denote the true and approximate solutions respectively. Suppose that x pp denotes the solution provided by the partial pivoting algorithm. Then, Matlab s norm function will compute the l norm of the error as follows: pp_error = norm(x_pp - xtrue, inf);

5 MS213: Numerical Mathematics Submission Guidelines 5 As an example of the output format, we ran the code when A was the Hilbert matrix of order n which is the n n matrix defined by a ij = (i + j 1) 1, 1 i, j n. It is often used for test purposes because of its ill-conditioned nature. If we define n b i = a ij, j=1 then the solution of the system of equations Ax = b is x = [1, 1,, 1] T. The corresponding output (for a different range of n) is then: Hilbert Matrix: L_inf Norm of the Error ================================================ n No Pivoting Partial Pivoting Scaled Pivoting e e e e e e e e e e e e e e e Q.3 As with question 2, you will also require the infinity error norm function. To illustrate the required output format, we used the code to solve an almost identical (but numerically easier) problem, defined by the equations: 33x 1 16x 2 = 17, 16x i x i 16x i+1 = 1, 2 i n 1, 16x n x n = 17. Under the same conditions as in the actual assignment question, namely 6 values of n (8, 16, 32, 64, 128 and 256), an accuracy requirement of 10 6, an upper limit of 2,000 on the number of iterations allowed and a zero initial guess, the following results were obtained: Iterative methods applied to the Sample Tridiagonal System Jacobi Gauss-Seidel SOR Method =========================================================== n Iter Error Iter Error Iter Error e e e e e e e e e e e e e e e e e e

6 MS213: Numerical Mathematics Submission Guidelines 6 For this problem, we used slightly different values of ω for the SOR method: n ω Q.4 The main issue for question 4 is the required output format. Using the sample function f(x) = sin(x) + cos(x), we ran the code for a slightly different range of h to produce the following: Centered Differencing Methods: f(x) = sin(x) + cos(x) ============================= ====================== x h 2nd Deriv. 3rd Deriv. f^(3)+f*f^(2)/ e e e e e e e e e e e e e e e The absolute errors in the respective approximations appear on the second line for each value of h. For this example, the optimum value of h occurs close to h = which 10 corresponds well with the results presented above. Q.5 The task in this question is to apply interpolation in two directions, namely (a) to find temperature values for the required new x-value at all the existing y-values; and (b) to use these new temperature values at the given y-values to find the temperature at the new y-value. As an illustration, we applied the code using a different set of temperature values: y x Suppose that we require the temperature at the three coordinate points: Then, the output might look as follows: (0.35, 1.45), (1.25, 2.25) and (1.75, 0.55).

7 MS213: Numerical Mathematics Submission Guidelines 7 Interpolated Values at the given x-value Temperature u(0.35, 1.45) u(1.25, 2.25) u(1.75, 0.55) The 6 data values alongside x = 0.35 represent the temperature values at (0.35, y) for y running from 0.00 to Taking these 6 y-values and the newly-found temperature values, we are then able to interpolate to find the temperature at the new value y = 1.45 to obtain u(0.35, 1.45) You should set up the code so that you can loop through the three cases in one run. Q.6 In order for the value of k to be visible (within the declaration of f(x)) to the numerical integrators, it is necessary to make this parameter global by declaring it as a global variable in both the main script and the function containing the definition of f(x). As mentioned in the assignment question statement, it will suffice to use Matlab s quad function as a replacement for the exact value of the integral. exact_value = quad(fcn, a, b, 1.0e-14); Also, a safe starting value to use for the backward recurrence is close to k = 30 but you should experiment with different starting values to observe accuracy. The suggested output format is as follows: -- Exact Trapezoidal Simpson Forward Backward k Integral Rule Error Rule Error Recurrence Recurrence e e : : JCarroll MS213: Numerical Methods Computing Assignments: Submission Guidelines JC 15/8/2013