About upgrading a software, an application, or a system, in Esperanto, do we say,
or something better?
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For computer related terms Komputeko is a good place to begin with. You can search there by English terms.
There are two things worth noting here. Firstly what is understood by an upgrade can vary from company to company. Secondly to avoid cultural-based misunderstandings Esperanto uses often quite concrete expressions, much more than English. Given these you have to little decipher, what an "upgrade" in your context means.
Usually an update, ĝisdatigo or aktualigo, improves existing features or add new ones without an extra charge.
For an upgrade you usually have to pay. An upgrade can be just a prolongation of some period (say, from a one year service plan to a three years plan), so it's essentially daŭrigo.
If you get extra features, whether these are better service (e.g. you get a faster response) or new features (e.g. additional filters in an image manipulation application), you get a new, upgraded edition of the software. So you get promocian eldonon or plibonigan eldonon with the nuance difference, that the former is more suitable for something non-tangible (e.g. service) and the later for something more concrete (e.g. new features).