FF505/FY505 – Computational Science
Week 6, Spring 2015 [pdf format]

MATLAB Exercises

Work in small groups at the computer to solve the following exercises.

  1. Solve the following systems of equations:
    x1+x2+x3=0
    x1+2x2+ =1
    2x1+ +2x3=−1
    		        
    x1+x2+x3=0
    x1+2x2+x3=1
    x1−x2+2x3=−1
  2. Solve the following system of equations:
    3x1+(2−i)x2+x3=4−i 
    (−2+3i)x1+(1−i)x2−x3=−2−2i 
    (2+i)x1+3x2+(3+i)x3=56+92i 
  3. Set n=200 and generate an n × n matrix and two vectors in ℝn, both having integer entries, by setting
    A= floor (10*rand (n)); 
    b= sum (A′)′; 
    z= ones (n,1)
    1. The exact solution of the system should be the vector z. Why?. Explain. One could compute the solution in MATLAB using the "\" operation or by computing A−1 and then multiplying A−1 times b. Let us compare these two computational methods for both speed and accuracy. One can use MATLAB’s tic and toc commands to measure the elapsed time for each computation. To do this, use the commands
       tic, x=A \ b; toc 
       tic, y=inv(A)*b; toc
      Which method is faster?

      To compare the accuracy of the two methods, we can measure how close the computed solutions x and y are to the exact solution z. Do this with the commands

       max(abs(x−z)) 
       max(abs(y−z))

      Which method produces the most accurate solution?

    2. Repeat part (a), using n=500 and n=1000.
  4. Find all solutions of the following system
    2x1−4x2+5x3=−4
    −4x1−2x2+3x3=4
    2x1+6x2−8x3=0
  5. Rank-1 Updates of Linesr Systems:
    1. Set A=round(10*rand(8)), b=round(10*rand(8,1)) and M=inv(A). Use the matrix M to solve the system Ax=b for x.
    2. Consider now the new system Cx=b, where C is constructed as follows:
      u=round(10*rand(8,1)), v=round(10*rand(8,1)), E=u*v′, C=A+E.
      Use MATLAB to verify that the rank of E is one. The matrices C and A differ by the rank 1 matrix E. Use MATLAB’s "\" operation to solve the system Cx=b and then compute the residual vector r=b−Cx.
    3. Let us now solve Cx=b by a new method that takes advantage of the fact that A and C differ by a rank-1 matrix. This new procedure is called rank-1 update method.
      Set z=M*u, c=v′*y, d=v′*z, e=c/(1+d) and then compute the solution x by x=y−e*z. Compute the residual vector b−Cx and compare it with the residual vector in part (b). This new method may seem more complicated, but it actually is much more computationally efficient.
    4. To see why the rank-1 update method works, use MATLAB to compute and compare Cy and b+cu. Also compute Cz and (1+d)u. Use the observed identities to prove Cx=b.
    5. Assuming that A is nonsingular, will the rank-1 update method always work? Under what conditions could it fail? Explain.
    6. Using the commands tic and toc, test which method is the fastest to solve the system of linear equations Ax=b among the three: with the INV function, with the \ operator and with the rank-1 update.
  6. Are magic squares nonsingular? Use the MATLAB command det(magic(n)) to compute the determinants of the magic squares matrices in the cases n=3,4,…, 10. What seems to be happening? Check the cases n=24 and n=25 to see if the pattern always holds.
  7. Use MATLAB to generate a matrix W and a vector x by setting W=triu(ones(5)) and x=[1:5]′. The columns of W form an ordered basis F={ w1, w2, …, w5}. Let L:ℝ5 → ℝ5 be a linear operator such that L(w1)=w2, L(w2)=w3, L(w3)=w4, and L(w4)=4w1+3w2+2w3+w4, L(w5)=w1+w2+w3+3w4+w5.
    1. Determine the matrix A representing L with respect to F, and enter it in MATLAB.
    2. Use MATLAB to compute the coordinate vector y=W−1x of x with respect to F.
    3. Use A to compute the coordinate vector z of L(x) with respect to F.
    4. W is the transition matrix from F to the standard basis for ℝ5. Use W to compute the coordinate vector of L(x) with respect to the standard basis.