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- bobbym
- 2011-06-07 04:14:23
Hi gAr;
You are welcome!
- gAr
- 2011-06-07 04:09:44
Hi bobbym,
Okay. Thanks!
- bobbym
- 2011-06-07 04:02:27
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.
- gAr
- 2011-06-07 03:57:42
Hi bobbym,
After 30 iterations, I get (80.3416234530054, 86.8627875987953).
Wonderful!
- bobbym
- 2011-06-07 03:37:29
If you look you will see that the two equations have been turned into two recurrences. They are just minus the subscripts.
Start the process by putting x = 1.0 and y = 1.0
Now run the next two equations 1 time:
Tell me what you get for x and y.
- gAr
- 2011-06-07 03:31:28
Hi bobbym,
Yes, followed till there.
- bobbym
- 2011-06-07 02:45:53
find_root is one command. Also
sage: import scipy seems to load a bunch of root finders.
But let me see what I can do with those equations by hand. Please hold on.
Okay do this, start with the first equation:
Subtract
from both sides. You get
Multiply both sides by
You get.
Divide through by:
-2300
You get:
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?
- gAr
- 2011-06-07 02:41:19
It may be having since there are so many packages, I haven't used it yet.
- bobbym
- 2011-06-07 02:30:06
Did you check for a numerical solver in Sage, all packages have them?
- gAr
- 2011-06-07 02:19:35
Hi bobbym,
Ok, thanks.
Sage couldn't solve it. Is there any numerical method?
- bobbym
- 2011-06-07 01:30:26
Hi gAr;
These are the two partial derivatives set to 0. Mathematica can now solve them numerically. Sage must have a similar command.
- gAr
- 2011-06-07 01:16:29
Okay.
I guess numerical solution is good enough for this problem. After all, numbers is what we need.
But I don't know the numerical solution either. Which are the derivatives to be considered to solve?
- bobbym
- 2011-06-07 01:05:19
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.
I also do not know of any algorithm to solve it. I would form the distance equations. Take the partial derivatives and set them to 0 and solve the system of equations numerically.
- gAr
- 2011-06-07 00:54:42
Is this still in active research or is there any good approximation algorithm?
- bobbym
- 2011-06-07 00:46:28
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.