CAAM 353 Homework Assignments

HW 9: Due 4/17, Fri., 7pm

This is the last assignment consisting of two parts.
• Part 1: Exercises: 8.5 and 8.8.
• Part 2: (a) Write a Matlab function

        function d = my_eig(A)
  

  that computes all the eigenvalues (eigenvectors not required) of a symmetric matrix A without using Matlab functions such as eig or schur. Just do a simple implemention of the explicit single-shift QR algorithm as given in the textbook. Use hess to tridiagonalize the matrix first before QR. Choose the shift to be the diagonal element to the right of the largest lower-diagonal element at each iteration. Run this scritp test_eig.m to test your function. You can try on the instructor's pcode yz_eig.p.

  (b) Write a Matlab function

        function [A,B] = my_pca(M,r)
  

  to generate 2 low-rank matrices A and B so that A*B approximates M while the storage used by A and B is equal to r*100 percent of that used by M. We will test your function using an image M. You should use Matlab SVD function to do this "principle component analysis" on images. Run this scritp test_pca.m with different r less than 1 to test your function. You can try on the instructor's pcode yz_pca.p. The test image is cameraman2.jpg.

HW 8: Due 3/31, Tue., 7pm

• Implement the forward and backward Euler's method following the interface of the code ode23tx.m give in the textbook site. Then use your sovers to do Exercise 7.15 (a)-(c) (the predator-prey model). Also experiment with different step sizes with the two methods. Plot some results to demonstrate salient behavior and write a paragraph or two describing your observations.

HW 7: Due 3/17, Tue., 7pm

• Write a Matlab function with the following interface that solves the linear least squares (LLS) problem: min||Ax-b||²

        function x = mylls_qr(A,b)
  

  It should generate the same results as the Matlab command x = A\b without using the backslash. Use the QR factorization with column permutation.
  Run the test function test_lls_qr.m along with the instructor's code yzlls_qr.p. After debugging is done, try n=500 or larger if possible.
• Write a Matlab function with the following interface that solves the linear least squares (LLS) problem: min||Ax-b||²

        function x = mylls_svd(A,b)
  

  It should generate the same results as the Matlab command x = pinv(A)*b without using the pinv function. Use the singular value decomposition (SVD).
  Run the test function test_lls_svd.m along with the instructor's code yzlls_svd.p. After debugging is done, try n=500 or larger if possible.

HW 6: Due 3/10, Tue., 7pm

• Write a Matlab function with the following interface that solves the linear least squares (LLS) problem: min||Ax-b||²,

        function x = mylls_chol(A,b)
  

  You should use the sparse Cholesky factorization with proper ordering to solve the normal equations of the LLS problem.
• Download this zip file lls.zip and run the test function test_lls.m. It should run with or without your code.
• Test your code mylls_chol.m first with test_lls(1), and then with test_lls(2) which is a much larger problem.

HW 5: Due 2/24, 7pm

• Assignment: Use Newton's method to solve the 2 tent problems:
  

           min S(x,y,z), st. V(x,y,z) = V0, y + z = h0 
  	 max V(x,y,z), st. S(x,y,z) = S0, y + z = h0 
  

  where S is the surface area and V is the volume of the tent which has a square base of 2x by 2x, a height of y for the main body, and a height of z for the pyramid top. The tent does not have a bottom. Your matlab functions for the 2 problems should have the interfaces:
  

  	          [S,x,y,z,v] = mytent1(V0,h0); 
  	          [V,x,y,z,v] = mytent2(S0,h0); 
  

  where the lower case v is the Langrange multiplier vector. You may use finite difference approximation for the Jacobian of the Langrangian system.
• Requirements: Submit written models (formulas) and optimality conditions you use, your codes and the output from the test code test_tent.m.

  Without your codes, you can run the test with the instructor's pcodes pfiles.zip.

HW 4: Due 2/11, Wed., 7pm

• Exercises 3.4, 3.8, 3.11
  For 3.4, save your hand data and picture.
  For 3.8, write a function that evaluates T₅(x) using the 6 representations with the following interface:

  function y = myT5(x,flag)
  %
  % Evaluate the Chebychev polynomial of degree 5, 
  % T5(x), from 6 different formulas
  %
  % input:
  %    x -- a column vector
  %    flag -- indicates formula 1 to 6
  % output:
  %    y -- T5(x), same size as x
  

  Then use this test script to test your code testT5.m.
• Write a function myspline.m that has the same interface as splinetx.m. It obtains the natural cubic spline by solving the linear system given in class (which is different from what splinetx.m does). Repeat Exercise 3.4 using myspline.m instead of splinetx.m. Use this function to draw your hand with myhand.m. The formulas for natural cubic splines are given here.

  We will use this script to test your code myspline.m. In general, your code should generate a different spline than that from either pchiptx.m or splinetx.m, but it should generate the same one as that from this code.

HW 3: Due 2/3, Tue., 7pm

• Exercises 2.3, 2.7, 2.11, 2.18

HW 2: Due 1/27, Tue., 7pm

• Write a function mysolve3 to solve the folowing three systems:

      Ax = b;  A'*y = x; A*z = y;
  

  The input is (A,b) and the output is (x,y,z). You can implement the function anyway you want.
  Download this instructor's code yzsolve3.p. Run the test script test_mysolve3.m.
  Try a reasonably large n value, say n = 2000 or larger.
• Make the function bslashtx on page 10 of Chap. 2 fully functional by implementing all involved functions, and then test it.
  Replace the Matlab function chol in bslashtx with your own Cholesky factorization code mychol.m (implemented from scratch). Specifically, replace the line "[R,fail] = chol(A)" in bslashtx.m by

      if exist('mychol','file')
        [R,fail] = mychol(A);
      else
        [R,fail] = chol(A);
      end
  

  Test your code by runing the test script test_bslash.m for a reasonably large n value (say, n >= 2000).
  Here is an implementation yzchol.p that you may compare to (but you do not have to achieve this speed).
• Write a paragraph to describe what you have done in this assignment and summarize your results based on the output from your tests.
  Submit your writeup and output.