University of the Witwatersrand Computational Physics : 2004 Tutorial 1 : (40 Marks)
Elementary Mathematical Operations |

- In discussing computer representation of numbers and arithmetic we can
write :-

__true____stored__

The symbol represents any primary operation.- 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.
(5)

- Show in addition that error propogation in summing truncated
series is minimised by adding terms in ascending order of
magnitude.
(5)

- 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.
(5)
- Suppose
with .
- Consider Euler's method and derive a non-recursive formula
for the iteration in the numerical solution of the equation.
Show that in the critical case
stability depends on the choice of stepsize .
(5)

- Show that the exact solution occurs for the limit
,
.

( Hint: ) (5)

- What problems occur in practice if becomes too small and
what are the implications for the choice of ?
How would one optimise ?
(5)

- Consider Euler's method and derive a non-recursive formula
for the iteration in the numerical solution of the equation.
Show that in the critical case
stability depends on the choice of stepsize .
(5)

2/Page two...

- 3.
- Derive an analytic expression for the error in Simpson's Rule for one panel of integration.

(Hint : Expand the function in the integrand in a Taylor series, and compare to Simpson's Rule). (5)

- Comment on the result.
(3)

- Can one correct for this error using the fact that
an analytic expression is known for it.
(2)

- Derive an analytic expression for the error in Simpson's Rule for one panel of integration.
- 4.
- How would one integrate

[5]