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## Topic review (newest first)

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.