Numerical Analysis II - Math 4510/6510
Instructor: Caner Kazanci. Best way to reach me is e-mail: caner@uga.eduOffices: 440 Boyd Graduate Studies and 410 Driftmier Engineering Center, (706) 542 0863.
Office Hours: 10:45am-11:30am Tuesdays and Thursdays, and anytime by appointment.
Course Information: You can download the syllabus here.
Announcements
- HW 12 is the last HW, no new HWs will be assigned. Good luck on the final tomorrow!
- Final exam date is Thursday, May 3, 2007; between 8:00 - 11:00 am; in the same room.
- HW 11 is now due Wednesday, 4/18/07. HW 12 will be assigned on 4/20/07.
- The midterm will be on Thursday, March 29.
- Here's a code you can use for Hw 9 to plot your spline.
- To print the output of your code in Octave or Matlab, you can use the "diary" command. To activate it, type "diary on", and deactivate it using "diary off". Anything that you write on the Octave prompt, including code outputs, will be stored in a text file named "diary". Type "help diary" for more information.
- Octave or Matlab help: You can download Octave for Windows here. You can find a nice and short tutorial here. A longer tutorial is here. And Octave official documentation can be found here.
- Homework Policy:
- Unless announced in class, or on this website; homework assignments will be posted here Fridays; and they will be due Fridays by noon in the mailbox of Jianbao Wu. His mailbox is located at 4th floor of Boyd Graduate Studies building. No late HW's will be accepted.
- For computer problems in HW assignments, please attach all your codes to a single e-mail and send it to Jianbao (jwu@math.uga.edu) before the due date. Also do not forget to include printed hard copies of your codes along with the solution of the regular HW problems; please hand them in together.
- If a HW assignment contains problems that require you to write matlab/octave codes, half of that HW assignment grade will be determined by the computer problems.
- Some HW assignments may contain a starred problem. These problems are optional for students registered in MATH 4510; but a must for those who are registered in MATH 6510. It is a rule of the graduate school, that we (all instructors) must have different (greater) expectations for the graduate students in split-level classes.
Homework Assignments
- Homework Set 1 (due Monday, 1/22/07)
Section 7.1 : 1., 3., 5., 7(a)., 7(c).
(Computer) : Write a Matlab/Octave routine for the Naive Gaussian elimination algorithm, which takes a matrix and vector as input, and outputs the solution. Run your code for all five examples in 7.1.7(a-e), print the output and attach it to your solutions. -
Homework Set 2 (due Friday, 1/26/07)
1. Prove that the condition number of an invertible matrix must be at least one.
2. Compute the condition number of the matrix [ 0 1 ; -2 0 ] using L-1, L-2 and L-infinity norms.
4. Repeat the previous problem for the matrix [ c+1 c ; c c-1 ].
5. Prove that the condition number has the following property: K(cA)=K(A).
Section 7.2 : 2., 3., 9. - Homework Set 3 (due Friday, 2/02/07)
(Computer) : Write a Matlab/Octave routine for the Gaussian elimination with scaled partial Pivoting, which takes a matrix and vector as input, and outputs the solution. Feel free to modify naive_gauss.m. Run your code for all five examples in 7.2.13(a-e), print the output and attach it to your solutions. - Homework Set 4 (due Monday, 2/12/07)
Section 7.3 : 4., 5.
(Computer) : 7.3.4. - Homework Set 5 (due Friday, 2/16/07)
Section 8.1 : 2., 4., 6., 11., 19., 20.
(Computer) : 8.2.9. Write a matlab code based on the procedure given in class (page 326), that takes a in a matrix of any size, and returns the Cholesky factorization matrix L. - Homework Set 6 (due Friday, 2/23/07)
1. Use Theorem 1 (p348) to determine the convergence of all four linear systems given in 8.2.1 (computer, p355) for Jacobi and Gauss-Seidel methods. (You can use Matlab to find the eigen values, but this is not considered a computer problem)
Section 8.2 : 3., 4., 5., 6., 7., 8., 9. You need not give detailed explanations for these problems.
(Computer) : Write an extremely simple procedure (function) that takes in a matrix A and a vector b as inputs, and applies Gauss-Seidel iteration 20 times, starting from a vector of zeros, and prints each iteration result. Show the output of your code for all systems in 8.2.1 (computer, p355). - Homework Set 7 (due Friday, 3/2/07)
Please explain all answers in detail, and give proofs when necessary.
Section 8.3 : 2., 4., 8., 10., 11.
Section 9.1 : 1., 10., 11., 12. - Homework Set 8 (due Friday, 3/9/07)
Section 9.2 : 2., 3., 4., 7., 9., 14., 23
Hint: On 9.2.23, only the boundary equations change. - Homework Set 9 (due Friday, 3/23/07)
(Computer) : Write a matlab function named cubic_spline(t,y) that takes in a table of interpolation values and plots the natural cubic spline. You can use a modified version of this code for plotting the spline. Print the output of your code for three different interpolation tables (of various sizes) of your choice. Also try it for 9.2.32.
No problems will be assigned from Section 9.3 - Homework Set 10 (due Friday, 4/6/07)
Section 10.1 : 1., 2., 5.
Section 10.2 : 2., 3., 7., 9., 15., 18., 19., 21.
(Computer)-1 : Integrate 4th-order Runge-Kutta to solve_ode.m. Print and compare solutions of all three methods for h=2.5, 1, .5, .2, .1. Write your observations.
(Computer)-2 : Modify your code to solve the IVP given in computer problem 10.2.2. Plot the numerical solution of all three methods along with the analytical solution. Try for three other step-sizes, and hand in all four figures along with your code. - Homework Set 11 (due Wednesday, 4/18/07)
Section 10.3 : 9., 10., 11.
(Computer) : Implement the adaptive Runge-Kutta method in Matlab or Octave to solve 10.3.5. For simplicity, let the initial condition be x(1/2)=9/4, and check the error at t=3 (rather than the opposite). To check the error, simply take the difference between the numerical solution and the analytic solution at t=3. You may use the provided C++ code as a guideline, however I suggest that you write a simple code and not worry about details like robustness, error checking etc. Plot the numerical solution along with the actual solution on [0.5,3]. Create the same plot with three different tolerance values of your choice. Write down your observations. - Homework Set 12 (due Friday, 4/27/07)
Section 11.1 : 4.
Section 11.2 : 1., 4., 5., 6., 12.
(Computer) : Implement the following equation using 4th order Runge-Kutta method in Matlab.
