Numerical Linear Algebra
Math 706 -- Spring 2006

Professor Doug Meade
meade@math.sc.edu
Department of Mathematics
University of South Carolina

Homework Assignments

Date Assigned
 Date Due
 Assignment
 Comments
20 Jan 27 Jan
  1. Complete Exercises 1.4.52 and 1.4.54 on page 52.
  2. Write MATLAB functions (m-files) for forward and backward elimination using both row- and column-oriented implementations.
  3. Write a MATLAB function implementing the inner-product implementation of the Cholesky factorization.
  4. Use your functions to work Exercise 1.4.74 on page 54.
MATLAB functions should be e-mailed to meade@math.sc.edu no later than the beginning of class on Friday, January 27. All written work is also due at the beginning of class on January 27.
30 Jan
 3 Feb
  1. Complete Exercises 1.6.3 and 1.6.5 on pages 67-69.
  2. Complete Exercises 1.7.34, 1.7.35, and 1.7.36 on pages 85-86.
In addition to submitting your MATLAB files, please use MATLAB's diary command to prepare a hardcopy of functions and any computed results.
30 Jan
 10 Feb
  1. Complete Exercise 1.8.1 on page 94.
  2. Complete Exercise 1.8.7 on page 98.
  3. Complete Exercises 1.8.17 and 1.8.22 on pages 101-103.
  4. Complete Exercise 1.9.2* on page 109.
* For Exercise 1.9.2, your analysis should include a comparison of the number of non-zero entries in the LU factorization of the original matrix and the LU factorizations for different orderings of the rows.
10 Feb
 17 Feb
  1. Complete Exercises 2.1.23, 2.1.27, and 2.1.28.
  2. Complete Exercise 2.2.23.
  3. Complete Exercise 2.4.5.
  4. Complete Exercises 2.5.4 and 2.5.7.
  5. Complete Exercises 2.6.5 and 2.6.6.
  • This assignment is not as long as it might appear. There are quite a few computational examples that illustrate the main points made in lecture.
  • Use MATLAB diaries to report your results. Please edit the diaries to remove any errors in the input or nonessential output. You can insert your observations about these computations directly on the diary (either handwritten or typed into the diary).
  • Please do not hesitate to ask questions if you have any uncertainty about these problems.
17 Feb
 24 Feb
  1. Extend Exercise 2.6.6*.
  2. Complete Exercises 2.7.16(a), 2.7.26, and 2.7.27.
  3. Complete Exercises 2.9.15 and 2.9.16.
* Apply two iterations of the Iterative Refinement Algorithm to the solution found in Exercise 2.6.6. Compare the residual after each iteration with the original residual.
24 Feb
 3 Mar
  1. Complete Exercise 3.1.9.
  2. Complete Exercises 3.2.4 and 3.2.8.
  3. Complete Exercise 3.2.21.
  4. Complete Exercises 3.2.35, 3.2.39, and 3.2.41.
  5. Complete Exercise 3.3.10.
  6. Complete Exercises 3.4.22, 3.4.26, and 3.4.29.
 
9 Mar
 24 Mar
  1. Complete Exercises 4.1.11 and 4.1.15 on page 264.
  2. Complete Exercises 4.2.3, 4.2.5, 4.2.8, 4.2.10, 4.2.14, 4.2.19, 4.2.21, and 4.2.23 on pages 266--274.
  3. Complete Exercises 4.3.8, 4.3.9, and 4.3.11 on pages 279--280.
  • For 4.2.8, use matrices that are large enough for these computations to take enough time for MATLAB to record a positive execution time. Then, compare the three ways to compute the condition number.
  • There is a typographical error in 4.3.11. Find, with proof, the correct formula for the pseudoinverse.
6 Apr
 14 Apr
  1. Complete Exercise 5.2.20 on page 312.
  2. Complete Exercises 5.3.7 and 5.3.10-12 on pages 317 and 318.
  3. Complete Exercise 5.4.28 on page 342.
  4. Complete Exercise 5.5.3 on page 353.
  5. Complete Exercise 5.5.7 on page 355.
  6. Complete Exercise 5.7.19 on page 378.
  • Exercises 5.3.10-12 are very similar. Be sure to report the numeric results in an organized manner. Also, explain the reasons for the different behaviors among the three examples.
  • In Exercise 5.7.19, the real command is not needed for this example. It is a good idea in general to be sure there is no floating-point roundoff that creates a non-zero imaginary part.
24 Apr
  
  1. Complete Exercise 7.2.4 on page 532.
  2. Complete Exercise 7.2.12 on page 535.
  3. Complete Exercise 7.2.24 on page 542.
  4. Complete Exercise 7.3.22 on page 553.
  5. Complete Exercise 7.5.3 on page 572.
  6. Complete Exercise 7.6.4 on page 579.
  • These problems will not be collected for a grade. You should work these problems to help you solidify your understanding of these ideas, and to better understand the common ideas behind these algorithms.
  • I will be happy to discuss these questions with you.

Notes:

  • MATLAB scripts (.m files) should be downloaded to your local computer (I recommend creating a folder called, say, MATLAB-files.) Then, from within MATLAB, under the File menu, select Run Script ... and Browse to the location where you save the .m file; select the script and click OK. All output will appear in your MATLAB window.
  • Portable Document Format (PDF) files are viewable with acroread, a publicly available PDF viewer by Adobe.
  • PostScript (PS) files are viewable with ghostview, the public domain PS viewer.

    • If you have any questions, please send e-mail to meade@math.sc.edu
    Last modified: 27 February 2006