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It will get closer and closer because this one happens to converge. Sometimes they do not and you have to pull out a different x and y.
If you look you will see that the two equations have been turned into two recurrences. They are just minus the subscripts.
Tell me what you get for x and y.
find_root is one command. Also
from both sides. You get
Multiply both sides by
Divide through by:
Now you have x on the right by itself. Do the same moves with the second equation to get a y by itself on the right.
Do you follow up to here?
It may be having since there are so many packages, I haven't used it yet.
Did you check for a numerical solver in Sage, all packages have them?
Fermat posed this problem with weights = 1 to Torricelli a long time ago. He solved it in a couple of ways. They were all geometric constructions. I feel the problem can be solved numerically but I can not find any analytical methods.
Is this still in active research or is there any good approximation algorithm?
The problem that I took this one from was so artificial that one of the vertices was the answer. I mean he could solve it using a small inequality. I know solutions like that make students happy and look great in books. The fact is in the real world there might be a couple of hundred points. Often you have to settle on a solution that is good but maybe not the best.