Here is a copy of the provisional notes for the course in
The exam is based on the course (hence my lectures) in addition to the homework sheets I produce which are used to prepare students for the type and range of questions I plan to use to assess the course.
The course will be delivered in person with 3 lectures a week over 10 weeks in two blocks of 5 weeks either side of the consolidation week. I will set homework each teaching week: 8 will be formative (not for credit) and 2 will be summative (assessed).
Homework set in weeks 4 and 7 (hand in dates are noon Monday 14th October and noon Monday 4th November) will assessed and count 5 per cent each to the final unit mark.
Past papers:
Lecture feedback
You can submit anonymous feedback on the lectures here and I will post my response below
Q: Can you just quote the Romberg iterates without deriving them in an
exam?
A: Yes, if the question does not state that you are required to derive them
Q: HW9, Q1c - what does it mean that ih ≪ 1? How can we find the error?
A: See 7.5.2 of the notes for a similar example.
Q: HW8, Q3 part (b) (i), isn't xj=cos(0)=1 another answer?
A: No it isn't because there's a denominator in the definition of U_n which for x_j=1 would give you a 0/0 limit. In the solutions, I assume the denominator is not zero and equation the numerator of U_n to zero. Also, the standardisation condition is U_n(1) = n+1 which shows that x=1 is not a zero. But not a silly question !
Q:
Would it be possible for you to produce a written example of applying
euler's method for higher order ODEs please.
A: page 1 and
page 2 of an example. But I'm sure this is also covered in the notes at some point and I'll probably do something like this in a problems class... a bit early in the ODE topic for problems class examples yet
Q:
In HW8, Q1 part b I find the approximation to I to be 48/61=0.78689.
Also in part c I am confused as to where the approximation to I being
3/4=0.75 is coming from.
A: You are right and I've corrected my wayward calculation. THANKS ! I've also
said a bit more in the solutions about where 0.75 comes from
Q: For the assessed HW 2 question part i) was it necessary to test the
bounds and if you didn't how many marks would you lose for getting
correct bounds but not testing them?
A: As long as the answer was clear and precise you would get full marks.
This might require that you test the bounds, depending upon how you
answered the question.
Q: for example 5.3.2, for equation 31, how is the last h term h^3?
A: Typo, should be h^6
Q: Do we need to know about 6.6.6 in the lecture notes?
A: Well, it's just a comment about quadrature schemes really so
that you are aware of different schemes for semi-infinite and
infinite intervals, which are not trivially dealt with using
Legendre scheme.
I don't cover these two in the notes, but Hermite are on the
HW8 sheet and Laguerre appear in exam questions (the online
recording of the Nov 11th problems class).
Q: In today's lecture Martin projected an example from what I think are
his original lecture notes onto the screen and I wondered if there was
a copy of it available.
A: I don't know what example this is and I don't have his old notes.
You should ask Martin I think, if he is going to use material beyond
what I've given him.
Q: Hi, I hope everything is doing okay , would it be possible to post
early solutions to HW8 please
A: Yes, will do on Thursday.
Q: In 6.1.1 of the lecture notes, how can we write the global truncation
error in powers of h?
A: I don't tell you so the simple answer is "Don't worry, it's not important". But if you want to know, The Euler-MacLaurin Formula is the answer. Go down to the section which says "Approximation of integrals". I have to pick and choose what I show you and this is a bit of a rabbit hole and not a place where I can ask you exam questions.
Comment: Hi just wanted to leave a note to say found the problem classes really
useful - thank you for these and for your help this year.
A: As for help, I've hardly been around. But thanks for the feedback on
the problems classes. I plan to record a load more in the last 3/4 weeks
of teaching because I don't want to run problems classes in the week
before exams.
Q: In chapter 5.1.1, for equation (19), I am confused about how you use
it to get that the coefficient for f(x_0) is (alpha+beta+gamma). It
looks like it should (-alpha-beta-gamma) if you expanded the brackets
above. I was hoping you could tell me where I've gone wrong, thanks
A: Yes, you get (-alpha-beta-gamma) = 0. But that's the same as
the equation I wrote.
Q: For the solutions to question 6 on PS4, I'm a bit confused where part
d)a) and b) have come from in the solutions as I can't seem to find
corresponding questions. Thanks
A: Arh ! I'd written a Q7 for PS4 but for some reason it didn't make it
to the final upload. This was confused by a question labelling problem
in Q6. I think all is resolved now and there is another Q on Sh4 for
you now. Thanks for spotting this
Q: There doesn’t seem to be any solutions for questions 1)b and 1)c on
problem sheet 5. I was just wondering if they could be added please.
A: Ooops. I rewrote and reorganised the course this year and I've been
having to move around questions and write new questions from what I
had last year. So this was an incomplete cut and paste job. But thanks
for alerting me to it, otherwise the solutions would have appeared
halfway through sheet 6 ! Should be all good now.
Q: I find x2 to be 0.261
A: Thanks so much... I checked my working and I don't know how the first version ever came to be, but it's now corrected and I get your answer. The solutions are updated.
Q: In Problem Sheet 4, Q5(c), are the solutions for x2, y2 correct?
...I find the same answer for y2, but not for x2
A: It's possible... This is a new Q this year. Perhaps I've misused
my calculator device. Can you let me know what
you think the answer is.
Q: Should the coefficient of epsilon^7 be 3 not 12 in the assessed homework Q1a)iii)? A: The question is accurate.
Q: In section 5.3.1 , about halfway through page 41 of the notes, should
4 x (27) - 25 also be divided by 3?
A: No, on the left there is 4N - N and then it is defined below this as N_3 with the division by 3 having been made
Q: Hi, Hope you are well. On the FPT questions, does the word
"monotonically" imply continuity?
A: Continuity is assumed in everything we do. Sometimes I make it
explicit in a theorem etc, something it's just assumed. So yes.
Q: Please could explain what Richardson's extrapolation formula actually
is and how we can then apply it, I'm confused by the notes and how to
actually apply it to a question
A: I thought I did a good job of explaining it in the lectures, so it's unlikely I can do better in a tweet. It tells you that if you have a method of approximating something, say N(h), then if you use two different values of h (in notes h/2) to get two approximation N(h) and N(h/2), you can combine these two approximations to get a new approximation N_2(h) which is more accurate than either N(h) or N(h/2). See course text or wookiepedia
Q: In example 3.3.4 please could I ask how the table shows linear
convergence? thank you
A: As n increases the ratio of the error at successive steps tends to 1/2.
I.e. the error is reducing by about 1/2 after each iteration. This is linear.
Or ... the formula for the order of convergence is |e_n|/|e_{n-1}|^alpha to \lambda as n tends to infinity. So alpha = 1 and lambda = 0.5 here.
Q: I'm wondering how you get to the form of f'(x_0) at the start of
example 5.3.2. For the infinite series I got that it would be divided
by (2k+1)! not 2(2k+1)! and was wondering where I went wrong. Thanks
A: I'm looking at the notes I used for the lectures and there isn't a
factor of 2 in the denominator there. So you've picked up a typo in the
notes that I'll fix. Thanks.
Q: Hi I'm really struggling with the non-assessed questions on PS3. I have attended the support session but am not finding it very helpful. Is it possible for you to go over a few examples similar to the style of questions on this sheet either in a lecture or for a problems class please. Particularly Q2,4,5,6b. I don't feel like there are many concrete examples in the notes. A: There are solutions available from today and they might help. Also, see Burden and Faires book, pages 63 to 66 for some illustrations and examples. I also have office hours you can use. But, in any case, I will be using problems classes in week 6 to go over more examples of this part of the course.
Q: A question about the assessed HW
A: I don't want to respond to questions about the assessed HW unless it is because there is an error or ambiguity. There was an error but I think the rest of the questions can be answered as they are written. If I ask for something and you give me the answer then you've got the marks.
Q: How much attention should we pay to proofs in this module?
A: I don't know how to answer this. I hope that everything I choose to tell you about is important and has a reason to be there and helps you understand the topic. So I hope you pay the same amount of attention to proofs as everything else.
Q: for example in 2.4.2, why we multiply the equation both side by P(23)
to make P=P(23)P(13) ?
A: Because otherwise we don't have an L matrix. Alternatively, you
can think of P as recording all the required row swaps that have to be
made in advance of an LU decomposition which requires no partial
pivoting.
Q: Like your lectures!
A: Thanks... I don't feel like I've got off to a very good start this year.
Q: When we doing the partial pivoting, are we only compare the first
column magnitude and why?
A: For partial pivoting we are going to divide through by a_{11} (in step 1), the pivot element. If we do row swaps, only the entries in the leading column will replace a_{11} under a row swap so it is only their magnitude we are interested in
Q: When we doing the probability on rounding errors of GE, should I do
the Scaled partial pivoting first, then Partial pivoting?
A: I don't understand the meaning of "probability" here. But if I
understand the rest of the question... you try partial pivoting if you
want to make the computation less prone to rounding errors. If
that fails, you try scaled partial pivoting.
Q: for example in 2.4.2, why we multiply the equation both side by P(23)
to make P=P(23)P(13) ?
A: I haven't got this far in the lectures yet. I explain this today
Q: In today lecture 2.4.1 Advantage of LU decomposition iii) why we let b
be identity and how to make it? To find A^-1 can we do from L^-1 and
U^-1 ?
A: We do not let b be the identity. We solve Ax_i = b_i n times for i=1,...,n with different b_i, which are the n columns of the identity matrix. In that way, x_i are the n columns of the inverse matrix.
Q: in 1.1 Binary storage, why the largest Next number is replacing the
last digit by 1,and for the smallest next one is replacing the rest
all by 1. It seems like the magnitude of replacing all by 1 is large
than on replacing the last digit?
A: Maybe a simple example: in integer binary numbers the number 12 is represented by 00001100 and the next biggest integer is this plus 00000001, i.e. 00001101.The next smallest number is 00001100 - 00000001 which is 00001011. Or 32 is 00010000 and 33= 00010001 and 31 is 00001111. The nth bit represents whether 2^n is "on" of "off". In the notes, the fractional part works in the same way but the bits represent whether 1/2^n are turned on or off.
Q: IS the general method the k-1 terms a(ij) should be a(kk)? Also for
the first example a(ij) replaced by a(11)?
A: No, I don't see what you are saying here and I've checked the notes
and they look OK to me.
