Marking Scheme for the 2019 Midterm Test
		========================================


Q1:

     2 pts for writing down the fact Delta_{-} = I - exp(-hD).

     1 pt for Taylor expansion of exp(-hD) until O(h^4).

     2 pts for correctly expanding 
          Delta_{-}^s = [hD - (hD)^2/2! + (hD)^3/3! + O(h^4)]^s 
     and getting the correct C_s.


Q2:

(a) 3 pts for the correct A and 2 pts for the correct b.

     Some students forget to mention A_{i,j} = 0 when | i - j | > 1. 
     In this case, I took off 1 pt.


(b) 3 points for pointing out the discretization is second order for f'' 
    and first order for f'.

     2 points for making correct conclusion that this is first order of 
     consistency.


(c) 1 pts for non-singular conclusion.

     4 pts for the correct proof.

     Some students used Gersgorin circle theorem to show that A does not 
     have a zero eigenvalue, which is correct.

     However, they didn't show the Gersgorin circles corresponding to 
     the 1st row and last rows.

     In these two rows, the off-diagonal entries are slightly different. 
     They need to be discussed separately.

     In this case, I took off 1 pt.