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. 



Current Research in Numeric Mathematics
No, research in numerical mathematics is still very relevant today.


One of the main challenges is **big data**: scaling the usual algorithms up to larger dimensions. Today's linear systems may involve sparse matrices of dimensions 100k or 1M, for instance. Using traditional methods such as Gaussian elimination will take ages even on modern computers, and require way more RAM than they have. To obtain faster methods, one has to understand the structure of the problem and exploit, for instance, knowledge of the behaviour of similar smaller problems to construct better approximations. There is a lot of research on algorithms that can scale up to huge sizes and their mathematical properties. To mention just one example, designing randomized numerical algorithms and proving their effectiveness requires deep mathematical results.


For another extreme case, in some quantum chemistry applications, one has to compute approximate eigenpairs in a context in which a single *vector* of the dimension wouldn't fit in RAM using normal storage. $n=10^{30}$ isn't out of the question, for instance.


Another observation is that **most problems aren't solved with infinite precision** even on today's computers. High precision or exact rational arithmetic is a cause of major slowdowns of a very large factor on modern architectures, and doesn't really solve the issue: if you are using an unstable algorithm, things will go wrong even if you throw 500 digits of precision in it. For instance, the error in your measurements and input data will be amplified to a factor to which you have no more significant digits. People agree that the solution is designing more stable algorithms, not raising the precision. For instance, by exploiting symmetries and hidden structures in the data.


