I've worked on a few method development projects and I'm of the opinion that it is a good idea particularly if the author is well-known and well-liked (if they are a dick or you don't like them, that's a different story).
The obvious reason why is because when they happened to see your paper going out for review which claims a "massive improvement" over previous methods, they aren't going to get upset at you and pull strings to make your life miserable. Furthermore, they might have insight whether or not the new method is useful or merely enhancing a particular error which makes it look good.
The only instance where sharing the paper might be a bad idea is if they happened to be working on a similar improvement. Receiving the paper may motivate them to scoop you.