ST790R -- Fall 2010

  Homework #1 -- due Thursday, 02 September 2010

  R Exercises:

  1) Let u = ( 1, 3, 5, 1, 3, 5)'.  
   a) Compute U = I - (2/d) u u' where d = u'u.  
   (This type of matrix is known as an 'elementary
    reflector' or a 'Householder transformation.')
   b) Find the largest and smallest off-diagonal elements of U*U.
   c) Find the largest and smallest diagonal elements of U*U.
   d) Compute U*u.
   e) Compute max_j sum_i abs(U(i,j))
   f) Print the second row of U.
   g) Print the elements of the second column below the diagonal. 
   h) Let A be the first three columns of U.  Compute P = AA'.
   i) Show that P is idempotent by recomputing (e) with P*P-P.
   j) Let B be the last three columns of U.  Compute Q = BB'.
   k) Show that Q is idempotent by recomputing (e) with Q*Q-Q.
   l) Compute P+Q.

  2) Read in the matrix in the file 'oringp.dat' on the failure of
   O-rings leading to the Challenger disaster.  The columns are 
   flight number, date, number of O-rings, number failed, and 
   temperature at launch.  Compute the correlation between number
   of failures and temperature at launch, deleting the last, missing
   observation (the disaster).

  3) Let the n*n matrix A have elements A(i,j)=1/(|i-j|+1).
   a) Compute and print A for n=10.
   b) Compute and print the Cholesky factorization for A for n=10.
   c) Compute the Cholesky factorization for n=20.  Does it fail?
   If not, find the determinant.