TITLE: Current Research in Numeric Mathematics
QUESTION [16 upvotes]: To me, as an non-expert in the field, it seems as if numeric mathematics should have lost its importance because nowadays symbolic calculations or calculations with unlimited precision are generally available.
So, just out of curiosity, I would like to know, whether my impression is wrong and what current hot research-topics of practical relevance in numeric mathematics are.

REPLY [9 votes]: Most of the problems to which mathematics is applied nowadays cannot be solved symbolically; think of flow in an oil reservoir or air flow past a vehicle.  These problems are modelled by complicated systems of nonlinear PDEs and have highly irregular boundaries, all of which make analytic solution impossible.  Numerical approximations are the only reasonable approach, and they require billions, trillions (or more) arithmetic operations.  If those operations were performed in "unlimited precision" (i.e., rational arithmetic) the size of the denominators required would be similarly large and it would quickly become impractical to store the solution.  It would also be millions of times slower (compared to doing it in double precision floating point).  Even in floating point, there are plenty of problems (e.g. turbulence modeling) that we cannot solve to acceptable accuracy in reasonable time on the largest supercomputers.
One other note: rounding errors are only one source of error in numerical computations; discretization error is frequently more significant.  Even if you could use infinite precision for these computations, the stability of the numerical discretizations would continue to be an important area of research.