CATAM News 1998-1999
Old News Headlines
1998 - 1999
23 Apr 99 - Part II: project 19.4 21 Apr 99 - Part II: minor clarification of project 17.2 19 Apr 99 - Electronic submission of project files 16 Apr 99 - Part IB: correction to earlier hint for project 1.2 13 Apr 99 - Part IB: clarification for project 2.1 (updated 19 Apr 99) 13 Apr 99 - Part II: Error in project 10.9 13 Apr 99 - Part II: clarification of question 1 in project 15.7 25 Mar 99 - Part II: clarification of project 14.2 23 Mar 99 - Part II: error in project 19.4 18 Mar 99 - Part II: minor correction to project 12.5 17 Mar 99 - Part II: minor correction to project 8.5 12 Mar 99 - Part II Marking Schemes 4 Mar 99 - Easter closing arrangements 4 Mar 99 - Part II: minor error in project 20.2 4 Mar 99 - Part II: clarification in project 16.1 25 Feb 99 - Part II: minor error in project 12.5 23 Feb 99 - Marking schemes 3 Dec 98 - Christmas Closing Arrangements 25 Nov 98 - Part II: Correction to project 3.9 17 Nov 98 - Part II: Correction to project 23.2 17 Nov 98 - Part II: Help for project 14.3 (and 14.2) 28 Oct 98 - Part II: Minor error in project 15.1 14 Oct 98 - CATAM PCs reserved Monday mornings 07 Oct 98 - Welcome back / Demonstrator availability 16 Sep 98 - Part II: Very minor typos in project 19.4 16 Sep 98 - Part II: Error in project 19.3 16 Sep 98 - Part IB: Hint for project 1.2 29 Jul 98 - Part IB: Minor error in project 1.1 29 Jul 98 - Part IB: Concerning project 1.2 reference books
23 Apr 99 - Part II: project 19.4
The definition of the feedback polynomial in the Introduction to project 19.4 is unfortunately not strictly consistent with some of the other statements made there. However, it is not believed that this will have a major impact on the project; at worst, your output for question 3 may not be exactly what you expect. At this late stage, any reasonable attempt using the definition given, together with an fair explanation of the output from Question 3, will be accepted.
Dr R. E. Hunt
CATAM Director
21 Apr 99 - Part II: minor clarification of project 17.2
In question 3 of this project, the sentence should end "the sum being over all leaves H of G". This is what the current wording is supposed to mean!
Dr R. E. Hunt
CATAM Director
19 Apr 99 - Electronic submission of project files
Click here for instructions on how to submit CATAM project files electronically.
16 Apr 99 - Part IB: correction to earlier hint for project 1.2
The CATAM News item of 16th September entitled "Hint for project 1.2" included a suggested function intmod(x,y) to replace the expression x mod y for negative x. If you did not use the suggestion given in that hint, you may ignore the rest of this correction.
The suggested function does not give the correct result (namely, zero) when x is a negative multiple of y. The following revised function should be used instead:
function intmod(x,y:integer):integer; begin intmod:=(y + x mod y) mod y; end;
If you used the original version of intmod but did not come across this problem, and obtained valid results, you will not be penalised in any way if you leave your programs as they are.
Dr R. E. Hunt
CATAM Director
13 Apr 99 - Part IB: clarification for project 2.1 (updated 19 Apr 99)
At the top of page 2 of project 2.1 it is suggested that students should use the CATSL routine Rkf45. Students wishing to attempt this project should note that the standard Runge-Kutta algorithm (RK4) is not likely to be suitable; it can be difficult to get convergent results using RK4. Howvever, successful attempts using RK4 will be perfectly acceptable.
Students who wish to attempt the project using a language other than Pascal may have to write a suitable RKf routine using a reference book such as "Numerical Recipes". This is not recommended, because of the extra effort involved, but anyone wishing to obtain more information should contact the CATAM Helpline.
Dr R. E. Hunt
CATAM Director
13 Apr 99 - Part II: Error in project 10.9
The data given in the appendix to this project contain an error: the average of 305.5 stated for 1997 should read 303.9. Apologies for any inconvenience that this may have caused.
Dr R. E. Hunt
CATAM Director
13 Apr 99 - Part II: clarification of question 1 in project 15.7
Several students have asked what is meant by "as soon as the next partial quotient can not be computed exactly" in question 1. The following explanation should clarify the meaning and give you a good idea of what your program is required to do. However, it must be emphasised that this explanation gives only a suggested method: ANY reasonable interpretation and implementation of the question will be accepted, and in particular, any reasonable attempt to understand how inaccuracies are generated by the computer will be accepted. (It is advisable, however, to read the following to ensure that you have not missed the point of the question.)
Let eps be the smallest positive real number such that, when calculated by the computer, the expression 1+eps is not equal to 1. (Incidentally, eps is known as the machine epsilon, as described on page 4 of the CATSL booklet.) You should assume that any real number which you type into the computer is only stored to within a relative (fractional) error of eps; so that a real number whose precise value is x might be stored internally by the computer anywhere within the interval (x-eps*x, x+eps*x). Thus the epsilon of question 1 is equal to eps*x.
Your program may either take x as input, or it may take an interval (a,b) within which x is known to lie.
You should assume that whenever you use the computer's inbuilt operations (x+y, x/y, etc.) and functions (exp(x), sqrt(x), etc.), the results are only accurate to within a relative error of eps. Thus, for example, assuming for the moment that some real number y has been stored precisely, after the statement
z := exp(y);
you should assume that the TRUE value of exp(y) lies in the range (z-eps*z, z+eps*z).
In fact, of course, y will not have been stored precisely; instead, you will know the range of values within which its true value lies. The exp function will therefore compound the imprecision, and you should be able to keep track of this. When the imprecision results in your program being unable to be sure of the correct value of the next partial quotient, it should stop.
Dr R. E. Hunt
CATAM Director
25 Mar 99 - Part II: clarification of project 14.2
Some confusion has arisen over the "Programming Note" for this project. To clarify the situation: students may use ANY of Maple's tensor routines, including ones such as "Riemann" which are specifically related to tensors in General Relativity, with no penalty. These routines are available only in the full version of Maple (as available on the PWF), and it is highly recommended that you use this version.
Yours,
Dr R. E. Hunt
CATAM Director
23 Mar 99 - Part II: error in project 19.4
Sequence (1), as printed in the project, is incorrect. The final zero should be changed to a one. Full marks will, however, be given for either of the following approaches in your write-up:
(i) Computations using the corrected version of sequence (1), together with sequences (2) and (3);
(ii) Computations using only sequences (2) and (3), together with a full explanation of why sequence (1) as printed in the manual is not usable as an example.
Furthermore, there is a small error in the definition of the feedback polynomial in the first paragraph of the project: C^d (where ^ represents a power) should, of course, read Z^d.
Dr R. E. Hunt
CATAM Director
18 Mar 99 - Part II: minor correction to project 12.5
A small error was introduced in project 12.5 when the mark schemes were published. If you have not yet downloaded the version of the project which includes the mark scheme, then you may ignore this correction.
On the second page, "Include some typical results from your program" should read "Print out the results generated by your program".
Dr R. E. Hunt
CATAM Director
17 Mar 99 - Part II: minor correction to project 8.5
At the start of section 2, in the definitions of m_n and M_n, the min and max respectively are to be taken over the whole expression; i.e.
m_n = min [V_n(i,j) - V_{n-1}(i,j)]
and similarly for M_n. (Of course, no other interpretation of the definitions is sensible!)
Dr R. E. Hunt
CATAM Director
12 Mar 99 - Part II Marking Schemes
Marking schemes for Part II CATAM projects are now available. The schemes have been incorporated into the text of the projects themselves. In order to see a marking scheme, you will need to download the project text from the CATAM web site. Follow the "Part II Computational Projects Manual" link from the bottom of the CATAM home page. Then find the project you are interested in, and click on the "ps" link next to the title. What happens next depends on how you are accessing the web page: if you are using the CATAM PWF, the project will appear in a separate window. Select "Print" from this new window's menus to print out a copy of the project. If you are not using the CATAM PWF, all sorts of things might happen depending on the set-up of the computer you're using; for example, the project might be saved to disk, and you could then send the file to your printer.
If you have any trouble printing out the projects, please ask a demonstrator to show you.
If the version of the project which appears does not contain a marking scheme, then you are looking at an out-of-date copy, probably because your computer is using a cache or a proxy. If you are unable to solve this problem yourself, the demonstrators can help.
In the marking schemes, the total mark for each project is 18. There are 2 further marks for each project, not explicitly mentioned in the marking scheme, which may be awarded by the Examiners for overall quality, including mathematical insight, ideas and presentation; the maximum mark for each project is therefore 20 (as mentioned on the Contents page of the blue booklet).
Some marking schemes are more detailed than others, because of the different nature of the projects; in some cases marks can accurately be allocated to individual questions asked in the project whereas in others marks are awarded for sensible answers to longer, more investigative problems.
Note that you will not be penalised if you provide answers or insights in a different order to that expected by the Examiner. For instance, if questions 1 and 2 are both worth 2 marks, but in your answer to question 2 you provide the required explanation of some aspect of your answer to question 1, you will still gain full marks for question 1 (even though your answer to question 1 is strictly speaking incomplete).
No other textual changes have been made to the projects, other than to insert paragraph breaks or add new headings in order to make the marking schemes clearer.
Dr R. E. Hunt
CATAM Director
4 Mar 99 - Easter closing arrangements
The Mill Lane Lecture Rooms building will be closed over the Easter period as follows:
- Thursday 1 April to Monday 5th April inclusive
The Mill Lane Lecture Rooms building is closed on Saturdays outside Full Term.
When the building is closed, the PWF will be available for use by MLLR keyholders, but a continuous service cannot be guaranteed during the Easter weekend and faults are unlikely to be fixed with the usual efficiency! The Mill Lane PWF will operate normally at all other times.
Please note that the demonstrators will be available during the week after Lent Full Term and the week before Easter Full Term.
4 Mar 99 - Part II: minor error in project 20.2
In the last paragraph of the first page of project 20.2, the path
x_0 -> x_2 -> ... -> x_n (where _ denotes a subscript)
should have x_2 replaced by x_1.
Dr R. E. Hunt
CATAM Director
4 Mar 99 - Part II: clarification in project 16.1
At the end of the second paragraph of the Introduction to project 16.1, which states that the decomposition group is isomorphic to a subgroup of the Galois group, the following further explanation should be added. [This information is standard material, and may be assumed without proof.]
"Furthermore, it is isomorphic in a way which preserves cycle types, so the cycle type of the generator of the decomposition group will also occur in the Galois group."
Dr R. E. Hunt
CATAM Director
25 Feb 99 - Part II: minor error in project 12.5
The last term in each of equations (7) and (8) of project 12.5 should read O(e^{n+2}) rather than O(epsilon^{n+2}).
Dr R. E. Hunt
CATAM Director
23 Feb 99 - Marking schemes
Marking schemes for Part IB CATAM projects are now available. Part II marking schemes will be available later this term.
The schemes have been incorporated into the text of the projects themselves. In order to see a marking scheme, you will need to download the project text from the CATAM web site. Follow the "Part IB Computational Projects Manual" link from the bottom of the CATAM home page. Then find the project you are interested in, and click on the "ps" link next to the title. What happens next depends on how you are accessing the web page: if you are using the CATAM PWF, the project will appear in a separate window. Select "Print" from this new window's menus to print out a copy of the project. If you are not using the CATAM PWF, all sorts of things might happen depending on the set-up of the computer you're using; for example, the project might be saved to disk, and you could then send the file to your printer.
If you have any trouble printing out the projects, please ask a demonstrator to show you.
If the version of the project which appears does not contain a marking scheme, then you are looking at an out-of-date copy, probably because your computer is using a cache or a proxy. If you are unable to solve this problem yourself, the demonstrators can help.
In the marking schemes, the total mark for each project is 18. There are 2 further marks for each project, not explicitly mentioned in the marking scheme, which may be awarded by the Examiners for overall quality, including mathematical insight, ideas and presentation; the maximum mark for each project is therefore 20 (as mentioned on the Contents page of the blue booklet). Your marks for projects 1.1 and 1.2 are first multiplied by 1.5; then your total marks for all five projects are scaled to be equivalent to a 16-lecture course (again, as described on the Contents page) before being added to your examination marks.
Note that you will not be penalised if you provide answers or insights in a different order to that expected by the Examiner. For instance, if questions 1 and 2 are both worth 2 marks, but in your answer to question 2 you provide the required explanation of some aspect of your answer to question 1, you will still gain full marks for question 1 (even though your answer to question 1 is strictly speaking incomplete).
No other textual changes have been made to the projects, other than to add two headings to project 1.1 (in order to make the marking scheme clearer).
Dr R. E. Hunt
CATAM Director
3 Dec 98 - Christmas Closing Arrangements
The Mill Lane Lecture Rooms building is closed on Saturdays outside Full Term.
The Mill Lane Lecture Rooms building will be closed over the Christmas period from Thursday 24 December to Sunday 3rd January inclusive.
When the building is closed, the PWF will be available for use by MLLR keyholders, but a continuous service cannot be guaranteed during the Christmas holiday period.
The Mill Lane PWF (PWF-MATHS) network will not be available on Monday 14 December (equipment upgrades).
PWF-MATHS will operate normally at all other times.
25 Nov 98 - Part II: Correction to project 3.9
In section 3, the formula given for approximating the Jacobian matrix:
J_{ij} = F_i(H_1, ..., H_j+delta, ..., H_N)/delta should be changed to:
J_{ij} = (F_i(H_1, ..., H_j+delta, ..., H_N) - F_i(H_1, ..., H_j, ..., H_N))/delta. (Here underline denotes a subscript.)
Dr R. E. Hunt
CATAM Director
17 Nov 98 - Part II: Correction to project 23.2
In Question 3, the right hand side of the equation defining h(e) should be divided by (1-e^2)^(5/2), where ^ denotes a power. Furthermore, the second sentence of Question 5 should start "This determines the initial value of a but not of e". The copy of the project on the CATAM web pages has been updated.
Dr R. E. Hunt
CATAM Director
17 Nov 98 - Part II: Help for project 14.3 (and 14.2)
Any students intending to attempt project 14.3 (Petrov classification of a gravitational field) may be interested in some tutorial notes explaining how to use Maple to perform tensor calculations as needed for the project. The notes include some actual Maple routines to perform specific tasks with such tensors. Students attempting project 14.2 may also find the notes useful.
These notes may be obtained by e-mailing catam@maths.cam.ac.uk ; they will be sent to you via the University mail.
Dr R. E. Hunt
CATAM Director
28 Oct 98 - Part II: Minor error in project 15.1
In the paragraph following Question 5, second line, the expression "1 <= j < r" should read "0 <= j < r" instead.
Dr R. E. Hunt
CATAM Director
14 Oct 98 - CATAM PCs reserved Monday mornings
Please note that the PCs in the Mill Lane CATAM Room are reserved for a Statistics Class every Monday this term, 10am to noon.
07 Oct 98 - Welcome back / Demonstrator availability
Welcome back to Cambridge after the summer. Good luck with the CATAM projects: we hope everything will go as smoothly as possible.
Demonstrators are available in the CATAM Room this term on Mondays to Fridays between 2pm and 4pm. This will continue until (and including) the week after the end of Full Term, when many students stay up to make serious headway on their projects. Please remember that the demonstrators are available to sort out any problems you might have with CATAM: you can ask them to help with difficult pieces of your programs, to help debug your code, and for clarification of any of the projects, including mathematical aspects.
If the demonstrators are unable to solve your problem, do feel free to ask the CATAM Helpline by e-mail: catam@maths.cam.ac.uk.
This is also the way to report broken machines in the CATAM Room and potential errors in the projects.
Dr R. E. Hunt
CATAM Director
16 Sep 98 - Part II: Very minor typos in project 19.4
The algorithm discussed in section 2 of this project is the "Berlekamp-Massey algorithm" (an extra "k" mistakenly made its way into the project booklet). In the final paragraph of section 1, "filed" should read "field".
16 Sep 98 - Part II: Error in project 19.3
In the definition of W(C;x,y), the references to w(C) in the exponents of x and y should read w(c) instead.
16 Sep 98 - Part IB: Hint for project 1.2
You may find it helpful to know the following information about the "mod" operator in Borland Pascal. Let x and y be integers, with y positive. For positive x, the expression x mod y will be a non-negative integer (in the range 0 to y-1). But for negative x, it will instead be a non-positive integer; in fact,
x mod y = -((-x) mod y)
which is not suitable for some of the calculations in this project. There are several ways around this; you could replace all occurances of x mod y by
(y + x mod y) mod y
or you could define a function
function intmod(x,y:integer):integer; begin if x<0 then intmod:=y + x mod y else intmod:=x mod y; end;
and replace all occurances of x mod y by intmod(x,y).
Dr R. E. Hunt
CATAM Director
29 Jul 98 - Part IB: Minor error in project 1.1
In the first paragraph of section 3 of project 1.1, the text "equations (2) and (3)" should just read "equation (3)".
29 Jul 98 - Part IB: Concerning project 1.2 reference books
Project 1.2 refers to two textbooks, Norman and Cohn. Note that the definition of echelon form in Norman is slightly more stringent than the one given in the project, in that it also requires that m_{kl(i)} = 0 if k < i.
In response to student queries, you can ignore this requirement and the row operations which are needed to make it true.
Here _ refers to a subscript; i.e., m_{kl(i)} refers to the entry in the k'th row and l(i)'th column of M.