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
Q: Last minute exam advice?
A: Write neat answers cos I gotta mark these exams before Xmas.
Q: Also for PS11 Q1cii, why is the approximation of y'' divided by 2? If
h is 1 and the approx in the solutions before was divided by h^2
wouldn't it be divided by 1?
A: You may have an old copy with a typo [questions below]
Q: on PS11, for Q1bii two questions: how does h=1 and why does x_0 =
1,2,3,4 rather than x_1, x_2 etc?
h=1 because the question asks you to use the mesh x = 1,2,3,4 and so the mesh step size is 1. The mesh is defined by x_i = i h. The previous formula for the second derivative using central difference was written in terms of x_0 and you need to shift this onto each mesh point to approximate the ODE at these points.
Q: in PS8 Q3bii what is the step to get the LHS and RHS in the solutions?
A: See question below but with sin's.
Q: how are the steps in the 2022 paper question 3)d) taken, I'm
struggling to see how to use the trig identity. Thanks
A: cos(A+B) + cos(A-B) = 2cos A cos B ? Similar question in lecture
notes on Chebychev polynomials.
Q: In PS5 q6 where does the equation 4e^{-x^2} (3-12x^2+4x^4) come from
please?
A: Maybe typo, should say f''''(x) = that expression. I guess, cos
I don't know what else it would be.
Q: Hi does 4 digit precision mean 4 digits after the decimal point or is
it like 4 significant figures?
A: I make the definition pretty clear in the first Chapter 1. x.xxx times 10^x is 4 digit precision. Chapter 1 is new this year and I tried to make sure the language throughout is consistent, but if you look at old questions you might see language that doesn't align so easily. I hope you are further through your revision than page 1.
Q: For the 2022 exam, Q2a)ii) how do you find the fixed point to be x*= 0
or x*=1, I got x*=-2+2root2 by setting g(x*)=x*, what method should I
be using?
A: I don't find a fixed point at 0 or 1. Read the question "Apply the fixed
point theorem..." for what you are supposed to do and see the solutions
for how to do it.
Q: Paper 2020 Q 2(a)ii, to show that the scheme has at least quadratic
convergence, if we calculate the first derivate and state that it is
zero at the root x*, is it enough ? or do we have to substitute x*=
-ln(2x*) in the derivative and show its zero?
A: See the solutions. I can't say it better than this.
Q: In Paper 2021, Q4 (a) why is the boundary yN=1?
A: Clearly a typo (next line is correct).
Q: Hi, for Q3c of 2022 Exam, i believe there is a mistake in the
solutions.
A: OK
Q: For multi-dimensional Newton’s method, why is finding y(m) from
J(x(m))y(m) = f(x(m)) useful as when doing so with G.E it is as easy
to find J^(-1)(x)? if valid, when is this method applicable? checking
the answers to the 2020 paper one can see this method is known in
2)b)i) and then ignored in 2)b)ii)
A: The point is that finding inverses is difficult but solving systems
of equations is relatively easy. So one avoids finding J^(-1) by
solving J y = f. That's the whole point of Chapter 1. Of course,
if you have a 2x2 or 3x3 J then finding inverses isn't difficult.
Q: Hi, I noticed that you have uploaded solutions to the 2024 exam, but
not the 2024 questions?
A: Someone asked for the solutions, so I thought the questions were
already available. I've removed them. There's already enough stuff
and I don't want to overwhelm people.
Q: I find that the 2023 past exam is quite a bit harder than the other
past exams we have access to. Could this be attributed to the fact
that in that year you could take 8 sides of notes in?
A: If you listen to the online recordings I say that 2023 Q1 and Q2
are a bit too hard. Nothing to do with notes. In 2022 (first year
I taught the course) the exam
marks came out high and I overcompensated the following year.
Q: Hi, for problem sheet 5 Q6, why do the e^-1 terms over h^2 not cancel
out? Thanks
A: Probably you want |a -2b +c| < |a| + 2 |b| + |c|
Q: In the 2021 exam, Q4 which parts do we need to be able to answer, does
no homework from PS11 imply this isn’t needed?
A: You can answer part (a) but the other parts are about shooting methods which I don't teach and so you can't answer.
Q: Hi, I have found this website really useful in studying, it's easier
to navigate than blackboard and I very much enjoy reading your
responses to the anonymous questions, its also very easy to ask
anything. I hope you keep this for any future courses.
On question 3b on the 2020 exam, please could you explain why you need
to expand the derivative in terms of all the partial derivatives?
A: You have a function f(t,y) - i.e. a function of 2 variables. So
derivatives of f will have to be partial derivatives. In the question
you want to expand f(t+c,y+d) where c and d are assumed to be small,
so you want to use multi-dimensional Taylor series ... you cover this
in MVC for example.
Q: Hi Richard, would it be possible for you to release solutions to 2024
Q3 and Q4 please? We have solutions to 1 and 2 already from the
assessed HW.
A: pffffft. I might if I'm feeling energetic enough which is not today.
You've got so much stuff already.
Q: In the 2019 exam, Q4b is about the linear shooting method, was this
not covered this year?
A: I've not covered it for 4 years, since I lectured the course.
Q: When can you release the solution to PS11, Q3?
A: You have an old copy of this sheet. I removed the last question
as it's for material not covered in the lectures
Q: A lot of practice questions rely on knowing hyperbolic functions,
exact integrals of different functions, different series expansions
etc. How important is this to know for the exam?
A: The practice questions are designed to give you an idea of the
types of question I might ask in an exam. There are pre-requisites
for the course for a reason and I expect students to know how to do things
that are taught in those courses (including A-level material).
I expect students to know
how to solve quadratic equations, use double angle formulae, how
to do partial fractions or how to integrate by parts. And how
to find solutions to differential equations, etc etc.
Q: In the 2018 Past Paper, Q3 part c, will we get full marks if we leave
our answer not simplified, as you left it in the recording?
A: Depends on the question. If it's a show that then you need to get somewhere. If you have done all the steps required and got an answer that works, then you've done what I asked for.
Q: Would we lose marks in the exam if we used the scaled partial pivoting
method from the problem sheet solutions?
A: No
Q: In PS9 Q2 why do we take f(t,y)=2ty and t=ih?
A: Check you have the latest version of this sheet. It's what the question askes for.
Q: in homework 1, q9, we are asked to employ scaled partial pivoting to
find the solution. Why do we divide all the row elements by their
respective scale factor? In the tutorial example, we just calculate
the division and use that to swap rows?
A: Yes, I'm sorry but the solutions have an old version and it's
not the correct scaled partial pivoting method.
Q: Does it matter that my calculator isn't programmable and can't map
graphs or anything, as we are allowed to have any calculator in the
exam but I only have a normal one
A: You will only need a basic calculator
Q: (2/12/25)
are there multiple ways to prove order of convergence? do we still get
the marks if we prove the order of convergence in a different way than
the mark scheme?
A: The order of convergence is just a concept and I give you its formal definition which is a useful way of determining it in practice. But there will be other ways of determining it for sure. In an exam I'm giving marks for clear arguments and accurate maths that are designed to answer the question. Unless I ask for a specific method to be applied, you can answer a question however you like, but you also need to be careful that you quote any results or state clearly anything you rely on.
Q: (2/12/25) HW 11 Q1 c(ii), why are the equations divided by 2
A: Hangover from a previous typo and you can see that the 2's disappear in the matrix eqn that follows.
Q: (2/12/25) For the i=1 case equation of the 10 mark BVP in the 28th Nov problems
class why do we not sub in y_1=1 into the equation?
A: Without replaying the video... y_i = y(x_i) and x_i = i/N. So y_0 is approx to y(0) and the question gives us that y(0) = -1 (I think). The mesh runs from i=0 to i=N and the equations hold on interior points i=1 to i=N-1 and we substitute boundary conditions for y_0 and y_N into the equations i=1 and i=N-1.
Q: (2/12/25) Hi, I noticed that in the 28 nov problems class recording, for
deriving the Taylor method of order 3, in the lecture notes you have
the third derivative to be different than the one that you calculated
in the video. Which one is correct.
A: They look the same to me.
Q: (2/12/25) the table In the example of 7.12 of the notes, I didn't understand how
to get to these values, can you write an example equation e.g. of how
the value was found for h=o.2 for AB2. thanks!!
A: As far as I remember it's quite tricky. I have to Euler step once to
get y_1 from y_0 and then I use AB2 from there because AB2 is a two-step scheme.
If you really want I can record a quick video, but I don't think it's
something you need to worry about.
Q: (2/12/25)So examinable material is up to section 8.2.3 ?
A: Section 8.3 is not included (SPECTRAL METHODS) nor is the other greyed
out section earlier (GRADIENT METHOD). I only examine taught material which
is the lectures, plus material from the problems sheet.
Q: (2/12/25) In HW11, 2c, I got the solutions to be different but when I go back to
check by substituting into the original equations, my solutions work.
However, your solutions also work when substituting back in. Why is
this?
A: You know as well as I do that the solutions to this are unique. And I know [cos I used wolframalpha) that my solutions are correct. So we can only infer that yours are not correct. Or we can fight.
Q: (2/12/25) in 2.3 in the lecture notes, we have epsilon = 10 ^-4, but we have 3
digit precision. In chapter 1 it has that if n digit precision is used
then epsilon + 10^-n. Why is this?
A: No no no... I've used epsilon in this example as a completely different parameter to the definition of machine accuracy.
Q: (2/12/25)
Is the linear shooting method mentioned in Question 4(b) of the 2019
exam paper within the scope of our syllabus and study materials
A: I will only examine things I (or others taking my place) have taught. I have changed what I teach since taking over the course (which is completely
natural) and shooting methods are not something I teach [I do mention
them briefly so that you are aware, but I do this a lot]
Q: (2/12/25)
in Q4c of sheet 10, the second quotient error has a mistake I believe
since I don't get the answer we were given
A: OK, I've updated to 6.25 from 6.18. Is that better ?
Q: in 5.1.1 in the lecture notes, when we derive the truncation error,
when solving for alpha, beta, gamma, do we include h in the bracket
that we then solve or do we keep the h term next to the f(x) term?
A: I'm not sure what you are asking. We are trying to eliminate
as many terms as possible so that the order of the error is as
small as possible (in terms of h). So we track coefficients of
f, f' and f'' and so on and whatever those coefficients are, we
set them to zero. The h's are in there because we are taylor
expanding. But they don't magically become something else.
if ah + bh = 0 then a+b = 0 and vice versa.
Q: for example 8.2.1, for the discrete representation I'm slightly
confused why y_0 = -1 and y_n = 5 rather than y_0 = 1, as -1 is
outside of the interval [0,6}. Thanks
A: y_0 is approx to y(x_0) = y(0) and y_n is approx to y(x_n) = y(6).
These are the values of the function at the end points of the interval
(0,6). In the problem these values are given as -1 and 5.
Q: Hi Richard, is there a solution for PS11 Q3 please?
A: It's a HW problem. I'll provide it on Monday. Dont @ me... this
is the UoB's idea of cramming exams right up against our teaching.
Q: For PS11 Q1 c)ii) why is it over 2? Should it be 1 because h^2=1 for
h=1?
A: Typo, refresh page.
Q: In PS11 Q1a, where does the h^2/2 disappear to when working out the
coefficient of y''?
A: Typo
Q: Are there any other past exams we can have access to? Obviously we've
done a couple of questions from 2016 and 2018 and wondered if the rest
of the paper is available.
A: I've made a lot of exams available and done most of the problems classes from other relevant questions, so you've got a lot already.
Q: HW 6 Q1 (iv), to show that the convergence is at least second order,
do we have to calculate g''(x)? and then how do we show that it is not
zero for x*
A: At least means more than first order, so you only need to show that g'(x*) = 0.
Q: In 1a of sheet 11, how is the error O(h^4)? I've got that the
coefficient of y'''' is -(h^2)/12 so would that not make the error
O(h^2)?
A: Yes, this is a typo.
Q: for hw sheet 10, q4b how is the order of accuracy p = 2 when the local
truncation error contains a y^(4) term?
A: I don't understand your problem: the leading order term is O(h^3)
for most values of eta and so the error is O(h^2).
Q: Hi professor, in HW sheet 5 Q3(b) I find different answers for f'(1)
for h=10^(-n) n=2,3,4,5. and so I find the best result is obtained for
h^(-5) which does not agree with the answer to part (c). Are your
values wrong? Otherwise I'm making a mistake.
A: You should be using 4 decimal place accuracy. So, for e.g.
for n=2, exp(1.01) = 2.7456 and exp(1) = 2.7183 and the difference
is 0.0273 and dividing by 0.01 gives 2.7300.
Q: I like the way you organised the lecture notes thank you
A: Thanks
Q: Since we only have two lectures left and we haven't finished stiff
ODE's yet, will anything we don't manage to finish in lectures be
non-examinable?
A: Only lecture material is examinable. Between me and Martin we are
currently planning on finishing half-way through Chapter 8, doing
finite differences but not spectral methods.
Q: HW3 Q3 (d), I found different values for the table. e.g. x3=1.17513,
x4=1.21079.... and so I find the next column different as well
A: Thanks for letting me know. One of us has made a mistake.
Q: For PS5 are there solutions to Q1 part b and c please?
A: You may have not refreshed the webpage... they are there for me
Q: For question 4c on problem sheet 10 where does the approximation
error(h)≈ ch^p in the solutions come from please?
A: This is just "us" proposing that the error is O(h^p) and hence
proportional to h^p and so exactly C h^p where C is the (unknown)
coefficient of proportionalilty. If we look at the ratio of
error for different step sizes we eliminate C. This is something
we've been doing without always being explicit all course. Go back to where
the error was decreasing by a half as the step size is decreased by
a half and we say this is linear (p=1) convergence. Is that OK ?
Q:
when deriving the local truncation error, how do we know when to stop
the series expansion? or is it an intuition thing?
A: Yes, the latter. Go as far as you think you need, test and go again
if you need to.
Q:
In HW2, Q5, looking at the solution, we know that the diagonals of L1
are 1, and also diagonals of InverseL2 should also be 1. But how do we
know that the lower off diagonal terms are zero, and so multiplying
them yields the identity matrix.
A: You are missing the point that L_1 L_2^-1 is lower triangular
and it equals U_1^{-1} U_2 whichi is upper triangular. The only way this
can happen is if both are equal to a diagonal matrix. The rest of
the argument is made to show that this diagonal must be 1s and hence
the diagonal matrix is the Identity.
Q: are the questions in the HW sheets marked with [HARD], considered
examinable ?
A: I'm unlikely to set a question straight from the HW sheet, but I
may put hard questions (in small doses) in the exam. Maybe what you
are asking is should you worry about whether you can do the HARD
questions and, no, you shouldn't.
Q: Hi, hope you enjoyed the cinema :), As the exam is getting closer, I
was wondering if you have any tips for success...
A: The best part was that I met an old friend in the toilets afterwards, who was annoyed I shook him by the hand before I'd washed my hands. Understandably.
Practice the questions on the homework sheets;
work from 9-6 but take time off to unwind; remember there are 4 questions
so make sure you know the last part as well as the first part. Be happy
and calm and try to look forward to the exam and enjoy it.
Q: There doesn't appear to be a solution on PS7 for question 1(e). Just
wondering if you could upload it please. Thanks
A: Oh dear. I've added one now. Thanks for spotting all these gaffes.
I'm off to Showcase Cinema now to watch a film called Heretic
since it's grim outside and Mrs P (or, more accurately, Dr P)
wants to see it as she's in love with Hugh Grant. Wish me luck.
Q: in the multistep methods, is the number before h our step size or is
It the beta_0,.. beta_k . for example am2 is h/12, so is h/12 the step
size?
A: No, h is always called the step size if t_i = t_0 + i h and y_i = y(t_i).
That is, the time step is h; it is the difference in time between two
succesive approximations to the solution y(t). In multistep methods
you generally define y_{i+1} (i.e. the approximation to the next time step)
in terms of a weighted blend of y_i, y_{i-1}, y_{i-2} etc and f_i, f_{i-1}
etc. So the coefficients represent the weighting given to a contribution
from the solution from a previous time step. Hope that makes sense.
Q: if we were to show that the AB2 scheme is stable as h goes to zero,
could we use the root condition theorem?
A: Yes, this is exactly what you use.
Q: Hi for the main result in 6.6.2 the theorem stating the integral
wf(x)dx roughly equals sum w_jf(x_j), does it work for any interval
(a,b)? because in q problem sheet 8 1b it uses the mapping so
wondering if this theorem requires that interval or its something
specific about the question?
A: The theorem about it being exact for a polynomial of degree (2n-1)
or less applies to the original integral. If a mapping is used, the
integrand is transformed and may not be a polynomial of degree (2n-1)
under the transformation. BUT if it is, then the approximation is
indeed exact. Hope that makes sense.
Q: On problem sheet 3, Q5 refers to a graphical method of Newton raphson,
but I can't find anything in the lecture notes about a graphical
method. Is this something we could get asked to do in an exam?
A: I sketched out the graphical interpretation in the lectures.
It's not that important.
Q: For the solutions to PS9 Q2 is it using a different problem to the one
in Q1a that it asks for?
Q: In the solutions for sheet 9, question 2 uses f(t,y)=2ty but I thought
f(t,y) = y-e^(-t) as this is what dy/dt is in 1a - is this a mistake?
A: Oh yes... I've been moving questions around a bit this year and this
is a sync problem. I've updated the question sheet.
Q: With partial pivoting do you pivot after each step of Gaussian
elimination is it just for the first step?
A: Yes, you do it after each step. The notes show an example where you
pivot at the initial step and the second step.
Q: Could you please do an example of scaled partial pivoting because the
solutions for problem sheets do it differently to how you did it in
lectures I think.
A: You'll have to give me a bit of time for this, but yes I'll do something.
Q: in the 21st October problems class, for question 1c, how does x_1 = 48
should it be x_1 = 49?
A: I haven't looked again, but from memory I said something about there
not being a lot to choose between 48 and 49. I might have been confused.
In general I'm quite relaxed [when it comes to marking] about these decimal
calculations, as it sometimes depends on how you order the individual
arithmetic steps; I'm more interested in students following the right
process.
Q: Could you please record a tutorial on solving first order difference equations if you get chance. A: Incoming...
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.
