    Four pages - Page one University of the Witwatersrand Advanced Techniques in Physics : 2002 Examination : June 2002

Instructions
1. There are six questions in total, grouped into two sections. The two sections must be answered in separate books.
2. Section 1 comprises of questions 1, 2 and 3.
3. Section 2 comprises of questions 4 and 5. Section two also contains a take - home question. This will be question 6 which does not appear in this paper and will not be answered in the answer book. You will receive question 6 on Wednesday 8:30 27 June. There will be 24 hours to answer question 6.
4. Answer all questions, 1 to 5. Questions can be attempted in any order. Start each new question on a new page.
 Time: Questions 1 to 5 2 hours (exam conditions) Total Marks (1-5) = 130 Question 6 One day (take home conditions) Total Marks (6) = 70

Questions 1-3 not included.

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PHYS400
PHYSICS HONOURS
June 2002 Advanced Techniques in Physics

4.
Consider the ordinary differential equation where initial values are specified, and may have either sign.
(a)
Discuss the condition (ill- or otherwise) of an attempted numerical integration of this O.D.E. by, for example, a Runge-Kutta method.
(6)
(b)
Relate the above discussion to the condition of numerical generation of special functions by recurrence relations. As a specific example consider recurrence relation for the cylindrical Bessel functions  and are replaced by or for the regular or irregular solution respectively. Demonstrate (using the recurrence relation) why forward recurrence is stable for generation of the but not for generation of the unless .
(6)
(c)
Describe an alternative method for generation of the when .
(3)

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PHYS400
PHYSICS HONOURS
June 2002 Advanced Techniques in Physics

5.
In discussing computer representation of numbers and arithmetic we can write :-
 true stored      The symbol represents any primary operation.

a)
Find the relative error in addition to first order and show that if and differ in sign with then there is a disasterous loss of accuracy.
(8)
b)
Show in addition that error propogation in summing truncated series is minimised by adding terms in ascending order of magnitude.
(7)

6.
To be a take home component on Friday 21 June.
(70)

Total Question 1-5 : [130 marks]   