But I'm not sure I'd recommend going into the field. There's something demoralizing about doing research on something which already has dozens of valid and successful methods, of which you are trying to create a slightly more optimized version.
I am so into optimizing fast polynomial multiplication I assure you there is nothing that will demoralize me from creating a slightly more optimized version.
I should probably qualify my statement more. There are certainly new and interesting problems in numerics. And even going from say O(n) convergence to O(n log n) convergence can lead to whole new classes of problems you can solve. I don't want to discourage anyone who loves numerics.
What I was trying to say was that as a graduate student you might be given a problem that already has many really good and smart solutions and be essentially told to find a better solution than all of these. How this goes will depend a lot on the specific problem, your advisor, etc.
