Answer to Quiz 6 Fri 15 Feb 2002

1. Consider the matrix
 
        [ 9.5013 10^11     6.0684 10^(-1) ]
    A = [                                 ] 
        [ 2.3114 10^(-1)   4.8598 10^(-1) ]

   with its inverse


             [  1.0525 10^(-12)  -1.3142 10^(-12) ]
    A^(-1) = [                                    ]
             [ -5.0057 10^(-13)   2.0577          ]

Compute the condition number of A using the norm ||.||_1.


   ||A||_1 = 9.5013 10^11 + 2.3114 10^(-1) = 9.5013 10^11

   ||A^(-1)||_1 = 2.0577 + | -1.3142 10^(-12) | = 2.0577

   cond(A) = ||A||_1 ||A^(-1)||_1 = 1.9551 10^12


2. Suppose we wish to solve the system A x = b,
   using the matrix A from above.  Assuming a relative error of
   10^(-16) on the coefficients of the matrix, what is the
   bound on the relative error of the solution?

   Denote the computed vector by xx and AA the vector with the
   relative errors of 10^(-16):

    ||x - xx||            ||A - AA||
    ---------- <= cond(A) ---------- = 1.9551 10^12 10^(-16)
      ||x||                  ||A||

               <= 1.9551 10^(-4)