Q:

Difference between rounding modes in computational science?

Are there any instances of scientific numerical problems where the choice of rounding mode matters?
There are usually a number of different rounding modes available: to $0$, away from $0$, to $\pm\infty$, nearest ties to even, nearest ties down. Apart from various currency-manipulation problems where rounding is mandated by problem details, does anyone know an example of where choice of rounding mode might actually matter?
A numerically stable algorithm would be insensitive to round-off errors and hence to choice of rounding mode, but are there any other issues that might be relevant?

A:

Using the rounding modes, it's possible to implement interval arithmetic. 
Suppose that two positive quantities $x$ and $y$ are represented by the intervals $[x_l,x_u]$ and $[y_l,y_u]$. The product is represented in interval arithmetic by $[x_l \otimes y_l, x_u \otimes y_u]$, where $x_l\otimes y_l$ should be rounded down and $x_u\otimes y_u$ should be rounded up. Control over rounding modes makes it easy to do this.
Varying the rounding mode and rerunning your code is also a quick test of your algorithm- if the answers change a lot then you know that the algorithm is quite sensitive to round-off errors (and that you probably need to rewrite it!)

