Minimization problem with integral constraint

Hello, I'm working with a 2D numerical density profile. . I have a set of radius a maximum radius R and I want to find the best fit to the data r. I know that I can use maximum likelihood or another method, but I have problems with the constraints for , because I require that

At first I tried with bins and adjusted the curve with cftool, but I need more precision. So I want to use minimization with that constraint.

Thank you so much.

Answer 1

[Matt J]

Perhaps you could reparametrize the curve as,

which automatically satisfies the constraint for any b and c. Moreoever, since this form has only two unknown parameters, it should be relatively easy to do a parameter sweep to find at least a good initial guess of b and c.

[Esteban Garcia]

Hello Matt, your answer was extremely helpful!.. I already did the sweep and found c=-1.295 and b=0.76 and it makes sense.

Is there a method to get the minimization faster?, because I need to apply it to a lot of galaxy clusters. I've seen and tried some methods in the documentation but none of them seemed to accept variables as exponents..

For example, for the first galaxy I minimized this function for a and b

F=-(b^(706*c)*(1/b^2 + 1)^c*(4/b^2 + 1)^(2*c)*(1/(4*b^2) + 1)^(2*c)*(9/b^2 + 1)^(2*c)*(9/(4*b^2) + 1)^c*(1/(16*b^2) + 1)^c*(9/(16*b^2) + 1)^(5*c)*(25/(4*b^2) + 1)^c*(9/(25*b^2) + 1)^(4*c)*(16/(25*b^2) + 1)^(2*c)*(25/(16*b^2) + 1)^c*(49/(16*b^2) + 1)^c*(49/(25*b^2) + 1)^c*(64/(25*b^2) + 1)^(2*c)*(81/(25*b^2) + 1)^c*(9/(100*b^2) + 1)^c*(121/(16*b^2) + 1)^c*(121/(25*b^2) + 1)^(3*c)*(49/(100*b^2) + 1)^(3*c)*(169/(16*b^2) + 1)^(2*c)*(169/(25*b^2) + 1)^c*(121/(100*b^2) + 1)^c*(169/(100*b^2) + 1)^(2*c)*(324/(25*b^2) + 1)^c*(9/(400*b^2) + 1)^c*(49/(400*b^2) + 1)^c*(361/(100*b^2) + 1)^(5*c)*(81/(400*b^2) + 1)^(2*c)*(121/(400*b^2) + 1)^(4*c)*(169/(400*b^2) + 1)^(3*c)*(4/(625*b^2) + 1)^c*(529/(100*b^2) + 1)^(3*c)*(9/(625*b^2) + 1)^(2*c)*(16/(625*b^2) + 1)^c*(36/(625*b^2) + 1)^c*(49/(625*b^2) + 1)^c*(64/(625*b^2) + 1)^(2*c)*(289/(400*b^2) + 1)^(3*c)*(81/(625*b^2) + 1)^c*(121/(625*b^2) + 1)^c*(361/(400*b^2) + 1)^c*(144/(625*b^2) + 1)^(3*c)*(196/(625*b^2) + 1)^(2*c)*(729/(100*b^2) + 1)^c*(441/(400*b^2) + 1)^c*(256/(625*b^2) + 1)^(2*c)*(289/(625*b^2) + 1)^(2*c)*(529/(400*b^2) + 1)^(2*c)*(324/(625*b^2) + 1)^(2*c)*(361/(625*b^2) + 1)^c*(961/(100*b^2) + 1)^c*(441/(625*b^2) + 1)^(4*c)*(484/(625*b^2) + 1)^c*(529/(625*b^2) + 1)^(3*c)*(729/(625*b^2) + 1)^c*(961/(400*b^2) + 1)^(2*c)*(784/(625*b^2) + 1)^(2*c)*(841/(625*b^2) + 1)^(2*c)*(1089/(400*b^2) + 1)^(2*c)*(1024/(625*b^2) + 1)^(3*c)*(1369/(400*b^2) + 1)^c*(1156/(625*b^2) + 1)^(2*c)*(1296/(625*b^2) + 1)^(4*c)*(1521/(400*b^2) + 1)^c*(1521/(625*b^2) + 1)^c*(1849/(400*b^2) + 1)^c*(1764/(625*b^2) + 1)^(2*c)*(1849/(625*b^2) + 1)^c*(1/(2500*b^2) + 1)^c*(9/(2500*b^2) + 1)^c*(49/(2500*b^2) + 1)^(2*c)*(1936/(625*b^2) + 1)^c*(121/(2500*b^2) + 1)^c*(169/(2500*b^2) + 1)^c*(2116/(625*b^2) + 1)^(3*c)*(289/(2500*b^2) + 1)^(2*c)*(2401/(400*b^2) + 1)^c*(361/(2500*b^2) + 1)^c*(2304/(625*b^2) + 1)^c*(441/(2500*b^2) + 1)^c*(2601/(400*b^2) + 1)^c*(2401/(625*b^2) + 1)^(3*c)*(529/(2500*b^2) + 1)^c*(2601/(625*b^2) + 1)^c*(729/(2500*b^2) + 1)^(4*c)*(2704/(625*b^2) + 1)^(2*c)*(841/(2500*b^2) + 1)^(2*c)*(961/(2500*b^2) + 1)^c*(2916/(625*b^2) + 1)^(3*c)*(1089/(2500*b^2) + 1)^(4*c)*(3249/(400*b^2) + 1)^c*(1369/(2500*b^2) + 1)^c*(3249/(625*b^2) + 1)^(2*c)*(1521/(2500*b^2) + 1)^(3*c)*(3481/(625*b^2) + 1)^(3*c)*(1849/(2500*b^2) + 1)^c*(3844/(625*b^2) + 1)^c*(4096/(625*b^2) + 1)^c*(4356/(625*b^2) + 1)^c*(2601/(2500*b^2) + 1)^c*(4489/(625*b^2) + 1)^c*(4624/(625*b^2) + 1)^c*(5041/(625*b^2) + 1)^c*(3249/(2500*b^2) + 1)^c*(5184/(625*b^2) + 1)^c*(3481/(2500*b^2) + 1)^c*(3721/(2500*b^2) + 1)^c*(4489/(2500*b^2) + 1)^c*(5041/(2500*b^2) + 1)^(2*c)*(5329/(2500*b^2) + 1)^c*(7396/(625*b^2) + 1)^(2*c)*(5929/(2500*b^2) + 1)^c*(1/(10000*b^2) + 1)^c*(9/(10000*b^2) + 1)^(2*c)*(7569/(2500*b^2) + 1)^c*(81/(10000*b^2) + 1)^(2*c)*(289/(10000*b^2) + 1)^(2*c)*(361/(10000*b^2) + 1)^c*(529/(10000*b^2) + 1)^c*(961/(10000*b^2) + 1)^(3*c)*(1089/(10000*b^2) + 1)^(2*c)*(8649/(2500*b^2) + 1)^(2*c)*(1369/(10000*b^2) + 1)^c*(1521/(10000*b^2) + 1)^(3*c)*(1681/(10000*b^2) + 1)^c*(1849/(10000*b^2) + 1)^c*(9409/(2500*b^2) + 1)^(3*c)*(2209/(10000*b^2) + 1)^(4*c)*(9801/(2500*b^2) + 1)^c*(2401/(10000*b^2) + 1)^(2*c)*(2601/(10000*b^2) + 1)^(2*c)*(10201/(2500*b^2) + 1)^c*(2809/(10000*b^2) + 1)^(5*c)*(10609/(2500*b^2) + 1)^(2*c)*(3481/(10000*b^2) + 1)^c*(3721/(10000*b^2) + 1)^c*(3969/(10000*b^2) + 1)^(4*c)*(11881/(2500*b^2) + 1)^(6*c)*(4489/(10000*b^2) + 1)^c*(4761/(10000*b^2) + 1)^c*(12321/(2500*b^2) + 1)^(2*c)*(5041/(10000*b^2) + 1)^c*(12769/(2500*b^2) + 1)^(2*c)*(5329/(10000*b^2) + 1)^c*(13689/(2500*b^2) + 1)^c*(6561/(10000*b^2) + 1)^c*(6889/(10000*b^2) + 1)^c*(14641/(2500*b^2) + 1)^c*(7569/(10000*b^2) + 1)^(3*c)*(7921/(10000*b^2) + 1)^(3*c)*(8649/(10000*b^2) + 1)^c*(16641/(2500*b^2) + 1)^(2*c)*(9409/(10000*b^2) + 1)^c*(17161/(2500*b^2) + 1)^c*(17689/(2500*b^2) + 1)^c*(10201/(10000*b^2) + 1)^(2*c)*(10609/(10000*b^2) + 1)^(2*c)*(11449/(10000*b^2) + 1)^c*(11881/(10000*b^2) + 1)^c*(12321/(10000*b^2) + 1)^c*(12769/(10000*b^2) + 1)^c*(20449/(2500*b^2) + 1)^c*(13689/(10000*b^2) + 1)^c*(14161/(10000*b^2) + 1)^(3*c)*(14641/(10000*b^2) + 1)^(2*c)*(22801/(2500*b^2) + 1)^c*(16129/(10000*b^2) + 1)^c*(16641/(10000*b^2) + 1)^(2*c)*(17161/(10000*b^2) + 1)^(3*c)*(17689/(10000*b^2) + 1)^c*(19321/(10000*b^2) + 1)^c*(19881/(10000*b^2) + 1)^c*(20449/(10000*b^2) + 1)^(2*c)*(28561/(2500*b^2) + 1)^c*(21609/(10000*b^2) + 1)^(2*c)*(23409/(10000*b^2) + 1)^c*(24649/(10000*b^2) + 1)^c*(25281/(10000*b^2) + 1)^(3*c)*(25921/(10000*b^2) + 1)^c*(26569/(10000*b^2) + 1)^c*(27889/(10000*b^2) + 1)^(2*c)*(28561/(10000*b^2) + 1)^c*(29241/(10000*b^2) + 1)^c*(31329/(10000*b^2) + 1)^c*(32041/(10000*b^2) + 1)^c*(33489/(10000*b^2) + 1)^c*(35721/(10000*b^2) + 1)^c*(36481/(10000*b^2) + 1)^c*(37249/(10000*b^2) + 1)^c*(39601/(10000*b^2) + 1)^c*(40401/(10000*b^2) + 1)^(2*c)*(41209/(10000*b^2) + 1)^c*(42849/(10000*b^2) + 1)^(2*c)*(43681/(10000*b^2) + 1)^c*(44521/(10000*b^2) + 1)^c*(49729/(10000*b^2) + 1)^c*(51529/(10000*b^2) + 1)^(3*c)*(56169/(10000*b^2) + 1)^c*(57121/(10000*b^2) + 1)^c*(58081/(10000*b^2) + 1)^c*(59049/(10000*b^2) + 1)^(2*c)*(61009/(10000*b^2) + 1)^c*(63001/(10000*b^2) + 1)^(2*c)*(66049/(10000*b^2) + 1)^c*(69169/(10000*b^2) + 1)^c*(72361/(10000*b^2) + 1)^c*(73441/(10000*b^2) + 1)^c*(77841/(10000*b^2) + 1)^c*(80089/(10000*b^2) + 1)^(3*c)*(85849/(10000*b^2) + 1)^c*(95481/(10000*b^2) + 1)^c*(114921/(10000*b^2) + 1)^c*(116281/(10000*b^2) + 1)^c*(124609/(10000*b^2) + 1)^c*(137641/(10000*b^2) + 1)^c*(c + 1)^353)/(pi^353*((b^2 + 968562431735785/70368744177664)^(c + 1) - b^(2*c + 2))^353)

Again, thank you!

[Matt J]

How fast the minimization goes depends on how efficiently you have coded the objective function evaluations. You appear to be repeating many of the same intermediate computations in your one-line implementation. E.g. b^2 and 100*b^2 appear in multiple places.

[Esteban Garcia]

I did this:

%F is defined like sym 
A=0;
for b=0.1:0.01:2
for c=-3.005:0.01:-0.015
n=vpa(subs(F),10);
if n<A
A=n;
B=b;
C=c;
end
end
end

So, you would recommend me to save b^2,100*b^2, etc. into variables and plug them in F?.

[Matt J]

I would recommend implementing it as written in my original answer. No need to have the Symbolic Math Toolbox involved.

[Esteban Garcia]

Hello Matt, thank you for answer

I see, but evaluating directly didn't give me a result. When I evaluated the function into b=0.1:0.01:2, c=-3.005:0.01:-0.015

-(b^(706*c)*(1/b^2 + 1)^c*(4/b^2 + 1)^(2*c)*(1/(4*b^2) + 1)^(2*c)*(9/b^2 + 1)^(2*c)*(9/(4*b^2) + 1)^c*(1/(16*b^2) + 1)^c*(9/(16*b^2) + 1)^(5*c)*(25/(4*b^2) + 1)^c*(9/(25*b^2) + 1)^(4*c)*(16/(25*b^2) + 1)^(2*c)*(25/(16*b^2) + 1)^c*(49/(16*b^2) + 1)^c*(49/(25*b^2) + 1)^c*(64/(25*b^2) + 1)^(2*c)*(81/(25*b^2) + 1)^c*(9/(100*b^2) + 1)^c*(121/(16*b^2) + 1)^c*(121/(25*b^2) + 1)^(3*c)*(49/(100*b^2) + 1)^(3*c)*(169/(16*b^2) + 1)^(2*c)*(169/(25*b^2) + 1)^c*(121/(100*b^2) + 1)^c*(169/(100*b^2) + 1)^(2*c)*(324/(25*b^2) + 1)^c*(9/(400*b^2) + 1)^c*(49/(400*b^2) + 1)^c*(361/(100*b^2) + 1)^(5*c)*(81/(400*b^2) + 1)^(2*c)*(121/(400*b^2) + 1)^(4*c)*(169/(400*b^2) + 1)^(3*c)*(4/(625*b^2) + 1)^c*(529/(100*b^2) + 1)^(3*c)*(9/(625*b^2) + 1)^(2*c)*(16/(625*b^2) + 1)^c*(36/(625*b^2) + 1)^c*(49/(625*b^2) + 1)^c*(64/(625*b^2) + 1)^(2*c)*(289/(400*b^2) + 1)^(3*c)*(81/(625*b^2) + 1)^c*(121/(625*b^2) + 1)^c*(361/(400*b^2) + 1)^c*(144/(625*b^2) + 1)^(3*c)*(196/(625*b^2) + 1)^(2*c)*(729/(100*b^2) + 1)^c*(441/(400*b^2) + 1)^c*(256/(625*b^2) + 1)^(2*c)*(289/(625*b^2) + 1)^(2*c)*(529/(400*b^2) + 1)^(2*c)*(324/(625*b^2) + 1)^(2*c)*(361/(625*b^2) + 1)^c*(961/(100*b^2) + 1)^c*(441/(625*b^2) + 1)^(4*c)*(484/(625*b^2) + 1)^c*(529/(625*b^2) + 1)^(3*c)*(729/(625*b^2) + 1)^c*(961/(400*b^2) + 1)^(2*c)*(784/(625*b^2) + 1)^(2*c)*(841/(625*b^2) + 1)^(2*c)*(1089/(400*b^2) + 1)^(2*c)*(1024/(625*b^2) + 1)^(3*c)*(1369/(400*b^2) + 1)^c*(1156/(625*b^2) + 1)^(2*c)*(1296/(625*b^2) + 1)^(4*c)*(1521/(400*b^2) + 1)^c*(1521/(625*b^2) + 1)^c*(1849/(400*b^2) + 1)^c*(1764/(625*b^2) + 1)^(2*c)*(1849/(625*b^2) + 1)^c*(1/(2500*b^2) + 1)^c*(9/(2500*b^2) + 1)^c*(49/(2500*b^2) + 1)^(2*c)*(1936/(625*b^2) + 1)^c*(121/(2500*b^2) + 1)^c*(169/(2500*b^2) + 1)^c*(2116/(625*b^2) + 1)^(3*c)*(289/(2500*b^2) + 1)^(2*c)*(2401/(400*b^2) + 1)^c*(361/(2500*b^2) + 1)^c*(2304/(625*b^2) + 1)^c*(441/(2500*b^2) + 1)^c*(2601/(400*b^2) + 1)^c*(2401/(625*b^2) + 1)^(3*c)*(529/(2500*b^2) + 1)^c*(2601/(625*b^2) + 1)^c*(729/(2500*b^2) + 1)^(4*c)*(2704/(625*b^2) + 1)^(2*c)*(841/(2500*b^2) + 1)^(2*c)*(961/(2500*b^2) + 1)^c*(2916/(625*b^2) + 1)^(3*c)*(1089/(2500*b^2) + 1)^(4*c)*(3249/(400*b^2) + 1)^c*(1369/(2500*b^2) + 1)^c*(3249/(625*b^2) + 1)^(2*c)*(1521/(2500*b^2) + 1)^(3*c)*(3481/(625*b^2) + 1)^(3*c)*(1849/(2500*b^2) + 1)^c*(3844/(625*b^2) + 1)^c*(4096/(625*b^2) + 1)^c*(4356/(625*b^2) + 1)^c*(2601/(2500*b^2) + 1)^c*(4489/(625*b^2) + 1)^c*(4624/(625*b^2) + 1)^c*(5041/(625*b^2) + 1)^c*(3249/(2500*b^2) + 1)^c*(5184/(625*b^2) + 1)^c*(3481/(2500*b^2) + 1)^c*(3721/(2500*b^2) + 1)^c*(4489/(2500*b^2) + 1)^c*(5041/(2500*b^2) + 1)^(2*c)*(5329/(2500*b^2) + 1)^c*(7396/(625*b^2) + 1)^(2*c)*(5929/(2500*b^2) + 1)^c*(1/(10000*b^2) + 1)^c*(9/(10000*b^2) + 1)^(2*c)*(7569/(2500*b^2) + 1)^c*(81/(10000*b^2) + 1)^(2*c)*(289/(10000*b^2) + 1)^(2*c)*(361/(10000*b^2) + 1)^c*(529/(10000*b^2) + 1)^c*(961/(10000*b^2) + 1)^(3*c)*(1089/(10000*b^2) + 1)^(2*c)*(8649/(2500*b^2) + 1)^(2*c)*(1369/(10000*b^2) + 1)^c*(1521/(10000*b^2) + 1)^(3*c)*(1681/(10000*b^2) + 1)^c*(1849/(10000*b^2) + 1)^c*(9409/(2500*b^2) + 1)^(3*c)*(2209/(10000*b^2) + 1)^(4*c)*(9801/(2500*b^2) + 1)^c*(2401/(10000*b^2) + 1)^(2*c)*(2601/(10000*b^2) + 1)^(2*c)*(10201/(2500*b^2) + 1)^c*(2809/(10000*b^2) + 1)^(5*c)*(10609/(2500*b^2) + 1)^(2*c)*(3481/(10000*b^2) + 1)^c*(3721/(10000*b^2) + 1)^c*(3969/(10000*b^2) + 1)^(4*c)*(11881/(2500*b^2) + 1)^(6*c)*(4489/(10000*b^2) + 1)^c*(4761/(10000*b^2) + 1)^c*(12321/(2500*b^2) + 1)^(2*c)*(5041/(10000*b^2) + 1)^c*(12769/(2500*b^2) + 1)^(2*c)*(5329/(10000*b^2) + 1)^c*(13689/(2500*b^2) + 1)^c*(6561/(10000*b^2) + 1)^c*(6889/(10000*b^2) + 1)^c*(14641/(2500*b^2) + 1)^c*(7569/(10000*b^2) + 1)^(3*c)*(7921/(10000*b^2) + 1)^(3*c)*(8649/(10000*b^2) + 1)^c*(16641/(2500*b^2) + 1)^(2*c)*(9409/(10000*b^2) + 1)^c*(17161/(2500*b^2) + 1)^c*(17689/(2500*b^2) + 1)^c*(10201/(10000*b^2) + 1)^(2*c)*(10609/(10000*b^2) + 1)^(2*c)*(11449/(10000*b^2) + 1)^c*(11881/(10000*b^2) + 1)^c*(12321/(10000*b^2) + 1)^c*(12769/(10000*b^2) + 1)^c*(20449/(2500*b^2) + 1)^c*(13689/(10000*b^2) + 1)^c*(14161/(10000*b^2) + 1)^(3*c)*(14641/(10000*b^2) + 1)^(2*c)*(22801/(2500*b^2) + 1)^c*(16129/(10000*b^2) + 1)^c*(16641/(10000*b^2) + 1)^(2*c)*(17161/(10000*b^2) + 1)^(3*c)*(17689/(10000*b^2) + 1)^c*(19321/(10000*b^2) + 1)^c*(19881/(10000*b^2) + 1)^c*(20449/(10000*b^2) + 1)^(2*c)*(28561/(2500*b^2) + 1)^c*(21609/(10000*b^2) + 1)^(2*c)*(23409/(10000*b^2) + 1)^c*(24649/(10000*b^2) + 1)^c*(25281/(10000*b^2) + 1)^(3*c)*(25921/(10000*b^2) + 1)^c*(26569/(10000*b^2) + 1)^c*(27889/(10000*b^2) + 1)^(2*c)*(28561/(10000*b^2) + 1)^c*(29241/(10000*b^2) + 1)^c*(31329/(10000*b^2) + 1)^c*(32041/(10000*b^2) + 1)^c*(33489/(10000*b^2) + 1)^c*(35721/(10000*b^2) + 1)^c*(36481/(10000*b^2) + 1)^c*(37249/(10000*b^2) + 1)^c*(39601/(10000*b^2) + 1)^c*(40401/(10000*b^2) + 1)^(2*c)*(41209/(10000*b^2) + 1)^c*(42849/(10000*b^2) + 1)^(2*c)*(43681/(10000*b^2) + 1)^c*(44521/(10000*b^2) + 1)^c*(49729/(10000*b^2) + 1)^c*(51529/(10000*b^2) + 1)^(3*c)*(56169/(10000*b^2) + 1)^c*(57121/(10000*b^2) + 1)^c*(58081/(10000*b^2) + 1)^c*(59049/(10000*b^2) + 1)^(2*c)*(61009/(10000*b^2) + 1)^c*(63001/(10000*b^2) + 1)^(2*c)*(66049/(10000*b^2) + 1)^c*(69169/(10000*b^2) + 1)^c*(72361/(10000*b^2) + 1)^c*(73441/(10000*b^2) + 1)^c*(77841/(10000*b^2) + 1)^c*(80089/(10000*b^2) + 1)^(3*c)*(85849/(10000*b^2) + 1)^c*(95481/(10000*b^2) + 1)^c*(114921/(10000*b^2) + 1)^c*(116281/(10000*b^2) + 1)^c*(124609/(10000*b^2) + 1)^c*(137641/(10000*b^2) + 1)^c*(c + 1)^353)/(pi^353*((b^2 + 968562431735785/70368744177664)^(c + 1) - b^(2*c + 2))^353)

or

vpa(-(b^(706*c)*(1/b^2 + 1)^c*(4/b^2 + 1)^(2*c)*(1/(4*b^2) + 1)^(2*c)*(9/b^2 + 1)^(2*c)*(9/(4*b^2) + 1)^c*(1/(16*b^2) + 1)^c*(9/(16*b^2) + 1)^(5*c)*(25/(4*b^2) + 1)^c*(9/(25*b^2) + 1)^(4*c)*(16/(25*b^2) + 1)^(2*c)*(25/(16*b^2) + 1)^c*(49/(16*b^2) + 1)^c*(49/(25*b^2) + 1)^c*(64/(25*b^2) + 1)^(2*c)*(81/(25*b^2) + 1)^c*(9/(100*b^2) + 1)^c*(121/(16*b^2) + 1)^c*(121/(25*b^2) + 1)^(3*c)*(49/(100*b^2) + 1)^(3*c)*(169/(16*b^2) + 1)^(2*c)*(169/(25*b^2) + 1)^c*(121/(100*b^2) + 1)^c*(169/(100*b^2) + 1)^(2*c)*(324/(25*b^2) + 1)^c*(9/(400*b^2) + 1)^c*(49/(400*b^2) + 1)^c*(361/(100*b^2) + 1)^(5*c)*(81/(400*b^2) + 1)^(2*c)*(121/(400*b^2) + 1)^(4*c)*(169/(400*b^2) + 1)^(3*c)*(4/(625*b^2) + 1)^c*(529/(100*b^2) + 1)^(3*c)*(9/(625*b^2) + 1)^(2*c)*(16/(625*b^2) + 1)^c*(36/(625*b^2) + 1)^c*(49/(625*b^2) + 1)^c*(64/(625*b^2) + 1)^(2*c)*(289/(400*b^2) + 1)^(3*c)*(81/(625*b^2) + 1)^c*(121/(625*b^2) + 1)^c*(361/(400*b^2) + 1)^c*(144/(625*b^2) + 1)^(3*c)*(196/(625*b^2) + 1)^(2*c)*(729/(100*b^2) + 1)^c*(441/(400*b^2) + 1)^c*(256/(625*b^2) + 1)^(2*c)*(289/(625*b^2) + 1)^(2*c)*(529/(400*b^2) + 1)^(2*c)*(324/(625*b^2) + 1)^(2*c)*(361/(625*b^2) + 1)^c*(961/(100*b^2) + 1)^c*(441/(625*b^2) + 1)^(4*c)*(484/(625*b^2) + 1)^c*(529/(625*b^2) + 1)^(3*c)*(729/(625*b^2) + 1)^c*(961/(400*b^2) + 1)^(2*c)*(784/(625*b^2) + 1)^(2*c)*(841/(625*b^2) + 1)^(2*c)*(1089/(400*b^2) + 1)^(2*c)*(1024/(625*b^2) + 1)^(3*c)*(1369/(400*b^2) + 1)^c*(1156/(625*b^2) + 1)^(2*c)*(1296/(625*b^2) + 1)^(4*c)*(1521/(400*b^2) + 1)^c*(1521/(625*b^2) + 1)^c*(1849/(400*b^2) + 1)^c*(1764/(625*b^2) + 1)^(2*c)*(1849/(625*b^2) + 1)^c*(1/(2500*b^2) + 1)^c*(9/(2500*b^2) + 1)^c*(49/(2500*b^2) + 1)^(2*c)*(1936/(625*b^2) + 1)^c*(121/(2500*b^2) + 1)^c*(169/(2500*b^2) + 1)^c*(2116/(625*b^2) + 1)^(3*c)*(289/(2500*b^2) + 1)^(2*c)*(2401/(400*b^2) + 1)^c*(361/(2500*b^2) + 1)^c*(2304/(625*b^2) + 1)^c*(441/(2500*b^2) + 1)^c*(2601/(400*b^2) + 1)^c*(2401/(625*b^2) + 1)^(3*c)*(529/(2500*b^2) + 1)^c*(2601/(625*b^2) + 1)^c*(729/(2500*b^2) + 1)^(4*c)*(2704/(625*b^2) + 1)^(2*c)*(841/(2500*b^2) + 1)^(2*c)*(961/(2500*b^2) + 1)^c*(2916/(625*b^2) + 1)^(3*c)*(1089/(2500*b^2) + 1)^(4*c)*(3249/(400*b^2) + 1)^c*(1369/(2500*b^2) + 1)^c*(3249/(625*b^2) + 1)^(2*c)*(1521/(2500*b^2) + 1)^(3*c)*(3481/(625*b^2) + 1)^(3*c)*(1849/(2500*b^2) + 1)^c*(3844/(625*b^2) + 1)^c*(4096/(625*b^2) + 1)^c*(4356/(625*b^2) + 1)^c*(2601/(2500*b^2) + 1)^c*(4489/(625*b^2) + 1)^c*(4624/(625*b^2) + 1)^c*(5041/(625*b^2) + 1)^c*(3249/(2500*b^2) + 1)^c*(5184/(625*b^2) + 1)^c*(3481/(2500*b^2) + 1)^c*(3721/(2500*b^2) + 1)^c*(4489/(2500*b^2) + 1)^c*(5041/(2500*b^2) + 1)^(2*c)*(5329/(2500*b^2) + 1)^c*(7396/(625*b^2) + 1)^(2*c)*(5929/(2500*b^2) + 1)^c*(1/(10000*b^2) + 1)^c*(9/(10000*b^2) + 1)^(2*c)*(7569/(2500*b^2) + 1)^c*(81/(10000*b^2) + 1)^(2*c)*(289/(10000*b^2) + 1)^(2*c)*(361/(10000*b^2) + 1)^c*(529/(10000*b^2) + 1)^c*(961/(10000*b^2) + 1)^(3*c)*(1089/(10000*b^2) + 1)^(2*c)*(8649/(2500*b^2) + 1)^(2*c)*(1369/(10000*b^2) + 1)^c*(1521/(10000*b^2) + 1)^(3*c)*(1681/(10000*b^2) + 1)^c*(1849/(10000*b^2) + 1)^c*(9409/(2500*b^2) + 1)^(3*c)*(2209/(10000*b^2) + 1)^(4*c)*(9801/(2500*b^2) + 1)^c*(2401/(10000*b^2) + 1)^(2*c)*(2601/(10000*b^2) + 1)^(2*c)*(10201/(2500*b^2) + 1)^c*(2809/(10000*b^2) + 1)^(5*c)*(10609/(2500*b^2) + 1)^(2*c)*(3481/(10000*b^2) + 1)^c*(3721/(10000*b^2) + 1)^c*(3969/(10000*b^2) + 1)^(4*c)*(11881/(2500*b^2) + 1)^(6*c)*(4489/(10000*b^2) + 1)^c*(4761/(10000*b^2) + 1)^c*(12321/(2500*b^2) + 1)^(2*c)*(5041/(10000*b^2) + 1)^c*(12769/(2500*b^2) + 1)^(2*c)*(5329/(10000*b^2) + 1)^c*(13689/(2500*b^2) + 1)^c*(6561/(10000*b^2) + 1)^c*(6889/(10000*b^2) + 1)^c*(14641/(2500*b^2) + 1)^c*(7569/(10000*b^2) + 1)^(3*c)*(7921/(10000*b^2) + 1)^(3*c)*(8649/(10000*b^2) + 1)^c*(16641/(2500*b^2) + 1)^(2*c)*(9409/(10000*b^2) + 1)^c*(17161/(2500*b^2) + 1)^c*(17689/(2500*b^2) + 1)^c*(10201/(10000*b^2) + 1)^(2*c)*(10609/(10000*b^2) + 1)^(2*c)*(11449/(10000*b^2) + 1)^c*(11881/(10000*b^2) + 1)^c*(12321/(10000*b^2) + 1)^c*(12769/(10000*b^2) + 1)^c*(20449/(2500*b^2) + 1)^c*(13689/(10000*b^2) + 1)^c*(14161/(10000*b^2) + 1)^(3*c)*(14641/(10000*b^2) + 1)^(2*c)*(22801/(2500*b^2) + 1)^c*(16129/(10000*b^2) + 1)^c*(16641/(10000*b^2) + 1)^(2*c)*(17161/(10000*b^2) + 1)^(3*c)*(17689/(10000*b^2) + 1)^c*(19321/(10000*b^2) + 1)^c*(19881/(10000*b^2) + 1)^c*(20449/(10000*b^2) + 1)^(2*c)*(28561/(2500*b^2) + 1)^c*(21609/(10000*b^2) + 1)^(2*c)*(23409/(10000*b^2) + 1)^c*(24649/(10000*b^2) + 1)^c*(25281/(10000*b^2) + 1)^(3*c)*(25921/(10000*b^2) + 1)^c*(26569/(10000*b^2) + 1)^c*(27889/(10000*b^2) + 1)^(2*c)*(28561/(10000*b^2) + 1)^c*(29241/(10000*b^2) + 1)^c*(31329/(10000*b^2) + 1)^c*(32041/(10000*b^2) + 1)^c*(33489/(10000*b^2) + 1)^c*(35721/(10000*b^2) + 1)^c*(36481/(10000*b^2) + 1)^c*(37249/(10000*b^2) + 1)^c*(39601/(10000*b^2) + 1)^c*(40401/(10000*b^2) + 1)^(2*c)*(41209/(10000*b^2) + 1)^c*(42849/(10000*b^2) + 1)^(2*c)*(43681/(10000*b^2) + 1)^c*(44521/(10000*b^2) + 1)^c*(49729/(10000*b^2) + 1)^c*(51529/(10000*b^2) + 1)^(3*c)*(56169/(10000*b^2) + 1)^c*(57121/(10000*b^2) + 1)^c*(58081/(10000*b^2) + 1)^c*(59049/(10000*b^2) + 1)^(2*c)*(61009/(10000*b^2) + 1)^c*(63001/(10000*b^2) + 1)^(2*c)*(66049/(10000*b^2) + 1)^c*(69169/(10000*b^2) + 1)^c*(72361/(10000*b^2) + 1)^c*(73441/(10000*b^2) + 1)^c*(77841/(10000*b^2) + 1)^c*(80089/(10000*b^2) + 1)^(3*c)*(85849/(10000*b^2) + 1)^c*(95481/(10000*b^2) + 1)^c*(114921/(10000*b^2) + 1)^c*(116281/(10000*b^2) + 1)^c*(124609/(10000*b^2) + 1)^c*(137641/(10000*b^2) + 1)^c*(c + 1)^353)/(pi^353*((b^2 + 968562431735785/70368744177664)^(c + 1) - b^(2*c + 2))^353)) 

The result is always -inf...

But using the symbolyc function I got numbers like,

-2.966948035e-508

I think that is because the precision

[Matt J]

Maybe fitting log(sigma) will behave better.

[Esteban Garcia]

Hello Matt it is much faster now without symbolyc and with log. Thank you

N=length(Rproj);
R=max(Rproj);
A=1000000
B=0
C=0
syms b c
for j=1:N
F(j)=log(2*Rproj(j)*(1+(Rproj(j)/b)^2)^c/(((b^2+R^2)^(c+1)-b^(2*c+2))/(b^(2*c)*(c+1))));
end
E=-sum(F);
fstr=string(E);
fstr=replace(fstr,'b','b(k)');
fstr=replace(fstr,'c','c(j)');
b=0.01:0.001:0.8
c=-1.405:0.001:-0.705
for k=1:length(b)
for j=1:length(c)
n=eval(fstr);
if n<A
A=n;
B=b(k);
C=c(j);
end
end
end

Answer 2

[Mitch Lautigar]

My suggestion is to use a smaller step size for <a,b,c> if you know what they are. Typically when you are trying to fix the curve, the only thing you can do is try to add in more datapoints. If you can provide some code, I can provide more feedback.