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If a Galaxy-wide Civilization like Star Wars were to build a Jupiter brain or Matrioksha brain, would it probably cost more to build then the Death Star? In Star Wars the Darth Star was over a trillion credits and estimated as a few quadrillion dollars on Earth and octillion dollars to run it for one day
Also, would it even be worth it to build a planetary computer. Would it be a great boon to society or a waste of money?
It would depend mostly on what you mean by "cost". In Star Wars, construction occurs pretty much the way it does right now on earth - hulls are laid down, shipyard crews install additional components, eventually you have a functioning starship. Materials have to be manufactured and transported. Fuel costs are incurred, salaries are paid. In the case of the Death Star, Wookies were enslaved as construction workers, which seems inefficient, but does make the point that the Empire Is Bad.
If you were building a Matrioshka Brain or Jupiter Brain, first of all, normal construction rules probably wouldn't apply. Your best bet would be to create a Von Neumann machine whose directives specify a given number of replications, construction of the computer via disassembly of available materials, and then self-disassembly as the final stage in the construction. No logistics, no salaries. So no "traditional" cost.
However, you're going to use up the mass of a gas giant (or, in the matrioshka case, the mass of a solar system) and that mass can't be used for anything else. So depending on how that's priced, it could be "expensive". It's hard to imagine that a galaxy-spanning civilization wouldn't have an uninhabitable solar system to spare, however.
As for its utility, that would largely depend on the speed of communication. If you're limited by speed-of-light, the utility of such a construct would be sharply curtailed, because any calculation/simulation it could perform would be badly out of date by the time you found out about it. However, speed-of-light limitations would also make it difficult or nigh-impossible to maintain a galaxy-spanning civilization. So if we assume that you have FTL communication capacity, there are all sorts of applications to which an astonishing amount of computational power could be put.
Most notably, simulation. For things like protein folding and DNA modelling here on earth, we use massively parallel operations to speed the process along. Such a processor could do that trivially. Since you're clearly in the business of remodelling solar systems, you might need to do an orbital simulation with thousands or millions of orbiting bodies. No problem!
Additionally, assuming that true artificial intelligence is possible, you could simulate millions or billions of sophonts to do social simulations in compressed time, or to do actual original research. As we've discovered with big data, in some cases, computational quantity is a quality all its own.
A Jupiter sized computer would be bjllions times more expensive than a Death Star since it is also thousands of times larger. The Death Star is probably the size of the moon, yet 3 Earths could fit in just the Red Spot storm on Jupiter! So, there is no possible way a Death Star would be more expensive than a Jupiter-sized computer.
Some disambiguation is in order - a Jupiter Brain is the size of a Jupiter, whereas a matrioshka brain is meant to be the size of a smallish star. A red dwarf star contains as much mass as about 80 Jupiters, and is much more energetic.
The matrioshka brain is so named because of nested Dyson-type spheres, like the Russian nesting dolls. If you could build it the way it was intended, it would have as many layers as it could until another outer layer couldn't extract energy from the waste heat. In terms of material expense and work done, it would be untold orders of magnitude more expensive than a Jupiter Brain.
Because of signal propogation, a highly optimized Jupiter Brain should be able to simulate smaller (still very large) things much much faster than a matrioshka brain, but a matrioshka brain could iterate through unimaginably vast simulations in impossibly high detail. With decent communications, the two would complement each other quite nicely, like the interplay between a high end computer workstation and a massively integrated database farm.
Now to actually answer your question - these interplanetary mega structures are so powerful, and so expensive, and so dangerous, that they can't really be built until it becomes necessary to build them. Even now human governments prioritize the spending on life-saving electronic traffic controls based on how many people have died at a particular location, and how frequently it occurs. If it is infrequent enough, the controls aren't installed, regardless of a recent fatality.
Once it does become necessary, there won't really be a price tag anymore, and it's uses (whatever they are) will be evident.
Now answering your question regarding the utility of a stellar engine primarily, its uses are factually infinite. And reckoning its infinitesimal functionality, it wouldn't be fictitious to assume that a megastructure with the computational power like that of a Matrioshka brain or a Jupiter Brain would be more than enough to upload the consciousness of the entire human race in it. The mentioned megastructures are categorized as Class-B stellar engines, and to be brutally honest, their creation is far beyond the humanity's engineering capacity in the present era.
The only feasible options in this case for the construction of a solar megastructure aimed towards the sustenance of galaxy spanning civilizations, are self replicating autonomous machines, such as the von Neumann models and Dyson replicating systems.
Cost is irrelevant. Cost depends on who is building it, not what is being built.
Building giant computer would most likely be done by self replicating robots so it's not a matter of cost but where do the raw materials come from?
As for a use Shlock Mercenary (A Space Opera) had a Matrioksha brain called the All Star. It was used to store the minds of the galaxy's citizens digitally in a giant VR simulation. When an event happen that would wipe out civilization, the All Star would upload everyone. Whilst their bodies would die, they would effectively be immortal inside the All Star.
The All Star would hide from said event and life would evolve from scratch all over again.
