Free (as in freedom) drivers for GPUs GEFORCE 7300GT and AMD Radeon HD 4650

For Nvidia cards you can view Nouveau's (free-software Nvidia driver) feature matrix : https://nouveau.freedesktop.org/wiki/FeatureMatrix/

The 7300GT is classified as an NV40, which is supported. You can see which features are available by looking at its corresponding column.

AMD gpus require proprietary firmware for video acceleration and get very hot in the summer. An Intel gpu seems the best option, OP.

I tested much GPUs with free software (only). All >> "ALL" ! AMD/ATI GPUs need non-free firmware data blob, it is not an runnable, just data, but trully non-free. This blob is about 2kb or about 70 lines of hex code in old graphics cards, today (for new cards) is about 500kb or even more, it is much harder to try reverse engineer it. Some (mostly) old ATI cards can work (really NOT well) without that blob, but forget about 3d (even 2d) acceleration, I can say that it is almost not usable without that blob, and some ATI/AMD card will not show/detect the right monitor's EDID and resolution. So if you try to use AMD/ATI card without a proper blob (all blobs included in standard non-free linux in /lib/firmware/radeon/, obviously you will not find those in linux-libre or one deblobbed linux), the main processor(CPU) will do all software rendering (for 2d for example), and it is a bit slow and buggy), very unpleasant. AMD/ATI GPUs can not to work properly without this blob, just forget those GPUs forever if you want use a computer with free software only. Also some chips can overheat if here is no blob loaded.

For NVIDIA card you will have no problems. Non-free blob (microcode-firmware as in ATI/AMD GPUs) is burned into SPI flash or rom-epprom memory, which is soldered on the GPU itself (no need non-free microcode-firmware in your OS, outside of GPU). You will have no automatic reclock* for 2d/3d rendering, so your GPUs will run idle core and memory clocks. YOU CAN DO manual reclock (core and memory) for KEPLER (example: gt730,gtx770,gtx780ti), and probably MAXWELL v1 (example: gtx750ti), you will get a decent 3d performance. For more info ask in IRC #nouveau channel, they always stay with updated info. After MAXWELL v1, next gen is MAXWELL v2 (example: gtx970), all Nvidia GPUs need for non-free firmware-microcode (some Nvidia's genius idea for introduce an singning non-free firmware) to work, so forget paying a cent for new generations.

I don't know, because I did not tested and do not have: probably old Voodoo and Banshee will work fine, I don't know. It is too ancient GPUs for too ancient hardware, with a few (8-16mb) of memory. Forget ancient Matrox or SiS, they run non-free and never supported GNULinux.

The best and stong performace you can get without running non-free is Intel integrated GPUs. I did not tested latest Skylake and Kabylake hardware, I do not know if those need for some non-free microcode. But previous generations you can get something strong as HD4000 or even better, some Intel Iris integrated GPU.

If you have no problem for manual reclock (every time you need it), you can get some gt730 with 2gb gDDR5, or gtx770 with 4gb. I have both cards.
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look at Nvidias codes:
https://nouveau.freedesktop.org/wiki/CodeNames/

reclock: get you video card to run high core and memory frequencies for 2d/3d performance, Nvidia drivers will do it automatically when any software demanding it. Do not confuse with downclock or overclock. Reclock is something basic in all GPUs.

I have HD4670 and this card can not run without blobs. As ALL AMD/ATI graphics hardware.

I am going to play devils advocate here, with a pinch of genuine ignorance.

Drivers aside, principles aside, what actual harm can non free gpu firmware do? What's the difference than say the software on my calculator? Could it not just be considered some sort of "soft circuit"? I understand the security risks on ethernet, wifi or CPU, but what about GPU?

Wow, fast responses. Thank you.