Found a root method for your model number, but I'm not sure if it still works on the latest update to the OS.
Root new S4 mini model SGH-I257M - Page 7 - xda-developers
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No problem here with battery drain running JB 4.2.2, not anymore that is. ... It turn'd out to be involved with Nearby Services running for no good reason ... It is wise of you to be very cautious about rooting. There are too many ways for things to go fowl ...
Thanx UJ95x. I am already aware of that thread. I would need to do a lot of work to detail the actual list of softwares to download, which to the PC and which to the phone, what specific procedures to follow with each, in what sequence etc. I may do this and end with a good detail list that would be of use to others like myself who are new to rooting etc. But not just now.
So in the meantime i am following Wm7Long's advice, and being very cautious. It seems that is rooting not a simple process with a pre-determined result.