If you want to use a computational library on a high-performance computing center where you have no root access to install new software you can use header-only template libraries such as MTL4, Eigen3, GMM++ easily by including the library folder in the linking path. How about other libraries such as PetSc, Trilinos, Armadillo, Blaze, GSL,... that require compilation? Is there a way to use them on a machine where you have no root access if they are not preinstalled?
Almost everything you can build and install in your own space. With GNU autotools, you can do something like
./configure --prefix=/path/to/your/work/space ... and then follow the usual compilation instructions. Things based on CMake and Scons have similar facilities.
If you want to use Python libraries, you can even use the
pip package manager to install a package for your user account only with this invocation:
pip install --user <package>
Another possibility for Python is using
virtualenv, which allows you to set up isolated Python environments without the need for root access.