Newton constraint optimization convergence

[ptc-5019847]

Greetings,

This is a constraint optimization about finding the point on a sphere surface close as possible to the point (1,2,3)

So function to minimize :

f(x,y,z) = (x-1)² + (y-2)² + (z-3)²

Constraint to :

g(x,y,z) = x² + y² + z² - 1

System need to be solve using Newton, with a tolerance of 10^-7 and a maximum of 1000 iterations.

Two initial guess are required : (0.5,0.5,0.5,1.0) and (-0.5,-0.5,-0.5,1.0)

Determine the convergence or the non-convergence of the method. If method converge determine its order. If method diverge determine the type of divergence (oscillatory or explosion).

Since its been almost 7 years I havent been a student...I first used the built-in Mathcad minimization function before starting programming the Newton algorithm...

Now I need your help to complete my Mathcad sheet for evaluating the convergence/divergence part...as I remember on this forum we have very skillfull user...salutations à jmG and greetings to Sir Gutman!!!

Thanks,

Dany

[AlanStevens]

Looks like they both converge.

I'm not sure about the order, but I assume you do something like plotting error at iteration n+1 against error at iteration n on a log-log plot and take the slope of the best-fit straight line through it. This is based on the assumption that error(n+1) = lambda*error(n)^order. I guess the last column of your solutionA (and solutionB) matrix is the measure of error here.

I re-iterate, I'm guessing here! I have no expertise in this area.

Alan

PS Sadly, jmg and Tom Gutman no longer take part in this forum.