== Comments:
=== Q1:
There are three main points here:
1. construct a FPI with correct FP (the absolute majority has done this)
2. determine whether the constructed FPI converges near each root -
this is deretmined by the derivative (some of you only run some
numerical example)
3. determine the rate of convergence, i.e., how the error is decreased
near the roots. In the case of linear convergence this is
determined by the derivative again. Some of you had quadratically
convergent FPI (such as Newton's e.g.).
=== Q2:
a) This turned out to be basically a bonus question. There is just less than a
handful who made mistakes here.
b) Most of you have done this correctly, however when detemining the
max absolute value of some polynomial you only check critical points
and forget the endpoints of the interval. Still, many of you managed
to compute/estimate the third derivative incorrectly, or postulate
that max is assumed at some point (x=2 is a popular answer here)...
=== Q3:
a) Surprisingly many made all kinds of numerical mistakes here.
b) Many correct (or almost correct answers) here. Surprisingly many
also answer with something which has nothing to do with adaptivity
=== Q4:
This turned out to be probably the most difficult question in the exam.
a) Surprisingly many gave incorrect answers here
b) Many of you remember the formula for implicit Euler, but cannot
apply it (even when you had the correct ODE system in a)
c) A popular thing here was to reduce the system to 1 equation (since
one of the equations is linear). Many tried to run two Newton
iterations for each of the equations (which is a non-linear version of
Jacobi iteration and not the multi-variate Newton method).
=== Q5:
a) Some of you forgot the pivoting - I gave 50% for this. Some of you
simply wrote down the final matrices P, L, U without any intermidate
computations, which I cannot accept as a correct answer.
b) Some of you made silly numerical mistakes here, which I penalized
because one can easlily check the answer by substituting it back into
the linear system. Some of you used Gaussian elimination/row
operations again to solve this system, whcih in my opinion
demonstrates misunderstanding of the whole point behind LU
factorization - which basically does the Gaussian elimination/row
operations so that one does not have to do this again. I did
penalize for this as well.
c) Many of you computed the condition number. The error estimate
turned out to be much more difficult. I gave partial points for more
reasonable (and correct) versions of the estimate, e.g.
||x-x~||/||x|| <= ||A^{-1}|| ||b-b~||/||x||
== Preliminary averages:
Here is what I got after going through the exams once.
1a: 81
2a: 98
2b: 60
3a: 94
3b: 73
4a: 83
4b: 70
4c: 43
5a: 77
5b: 80
5c: 58
Total (exam): 74