Dev board communication over PCI

Hi, first of all, please forgive the noobishness, but what exactly is root-complex, and what is the difference between root-complex and peripheral boards? Just to put things in context, my experience of PCIe extends only to Tesla and Phi cards, both of which you can plug into a generic server board and they 'just work'.

Also, being primarily a C++ programmer, I'm curious as to whether it is possible to use C or C++ (or maybe something like mbeddr (http://mbeddr.com/)) to develop for FPGAs. Is it even feasible, do the FPGA IDEs support this notion? I understand that OpenCL is something in that ballpark, but being heavily involved with CUDA I've learned to hate OpenCL with a passion (it is really far too verbose). Also, I'm a heavy MATLAB user - do you know whether MATLAB is a feasible platform/substitute for HDL development?

And, on the subject, are FPGAs feasible for math? I mean, does a general-purpose PCIe development board have some chip on it to perform math calculations, or should it be part of the FPGA chip itself? Also, are there FPGA development boards that you can add (or pick and mix) components to?