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significant digits in NMinimize

Yaj Bhattacharya
Posted 10 years ago

I am trying to get more significant digists (up to 8 or 10) for the following expression and for the value of $b$ where it is minimum,

NMinimize[{(-((4 $\alpha$b$^3$)/(2 b+$\mu$)^$2$)+(64 b$^5$ (6 b + 4 b$^3$ - 16 b$^5$+ (3 Sqrt[1/$b^2$] b $\pi$)/Sqrt[1-b$^2$]-(6 ArcSin[b])/Sqrt[1-$b^2$]))/((96 b$^5$-96 $b^7$) $\pi$))/(1+b$^2$), 0<=b<=0.99999999999},{b}, AccuracyGoal->20] /. { $\alpha$-> 0.05 , $\mu$-> 0.01}

The result is {1.99975, {b -> 0.0227689}}

How can I get some more significant digits?
POSTED BY: Yaj Bhattacharya
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Thank you Sander!
POSTED BY: Yaj Bhattacharya

Try to code it neatly, and first do the substitution to make it a lot easier. In addition, change the substitutions to exact fraction:

func=(-((4alpha b^3)/(2 b+mu)^2)+(64 b^5 (6 b+4 b^3-16 b^5+(3 Sqrt[1/b^2] b Pi)/Sqrt[1-b^2]-(6 ArcSin[b])/Sqrt[1-b^2]))/((96 b^5-96 b^7)(Pi)))/(1+b^2);
func=func/.{alpha->1/20,mu->1/100}//Simplify
NMinimize[{func,0<=b<1},{b},AccuracyGoal->40,WorkingPrecision->40]
POSTED BY: Sander Huisman

OK, WorkingPrecision works, but not AccuracyGoal.

What's the difference i.e. why would AccuracyGoal refuse to produce more significant digits, is it because WorkingPrecision is some sort of brute force?
POSTED BY: Yaj Bhattacharya

Tried formatting it for the forum, the actual input would be

NMinimize[{(-((4alphab^3)/(2 b + mu)^2) + (64 b^5 (6 b + 4 b^3 - 16 b^5 + (3 Sqrt[1/b^2] b [Pi])/ Sqrt[1 - b^2] - (6 ArcSin[b])/Sqrt[1 - b^2]))/((96 b^5 - 96 b^7) [Pi]))/(1 + b^2), 0 <= b <= 0.99999999999}, {b}, AccuracyGoal -> 20] /. {alpha -> 0.05 , mu -> 0.01}
POSTED BY: Yaj Bhattacharya
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