How does one remove files from a computer with no chance of recovering them?

You guys are WAY overthinking this.

If you need to get rid of something, toss it in the trash. Even if they see it in there, they can't pull it out because now it's all gross from being near the other trash.

wont post specifics but that isnt true
I threw away a floppy with cp on it and stupidly (this was like 8-10 years ago) had my name on the floppy. I also used it for work so that is why it had my name on it. We used to store our projects on floppy then upload at work to the boss. Anyway, i toss it away when we stop using floppies. Weeks later get a visit by a special someone for a meeting

SERIOUSLY, WHO THE FUCK GOES THROUGH THE TRASH AND PICKS UP A DISCARDED FLOPPY AND THEN LOOKS PUTS IT IN THEIR COMPUTER AND LOOKS AT ITS CONTENTS!

Some asshole was curious, saw the files, turned in a report, then authorities somehow managed to backtrace the garbage to my area and obviously they already had my name

I was so stupid

sudo srm -v /path/to/file/to/securely/delete/example.png

This does the trick. If you want to be even more save turn on drive encryption, encrypt whole drive, turn in off.

just tell it to please erase itself