I would like to express (x) as function of (A) and (n) in a closed form. For real values of (n) and (A), given that 1.5 < n < 6, and 2^{1-n} < A < 1
How to obtain an approximated solution, if exact solution is not available ?
First, the lhs is monotone increasing, and its infimum is 2 for 0
Dear Dr. Hermann, Many thanks for your valuable response. It is really helpful.
I would like to express (x) as function of (A) and (n) in a closed form. For real values of (n) and (A), 1.5 < n < 6, and 2^{1-n} < A < 1. It there any approximation method to do so ? other than surface fitting.
Dear Akram Hourani,
why do you want to express x(A,n) in a closed form? I did not realize that you also wanted the expression for non-integer n. For those no closed form exists.
Solving for fixed A,n the equation is likely almost as fast as evaluating one of the explicit expressions that exist for integer n. By implicit differentiation you can get gradients and even Hessians for x(A,n) from the equation at every fixed A,n. So you can even effectively solve optimization problems.
Yes, but the point is that you cannot find closed expressions for arbitrary n (even rational), and the various expressions are not suitable to define a closed form depending on A and n.
Thanks for the suggestion Aldo Dall'Osso, I have already obtained the closed form expression for n=2, n=3, n=4.
Is there ant mathematical method to "approximate" the solution in a closed form for arbitrary values of 1.5 < n < 6. I am trying to solve an engineering problem which can tolerate some inaccuracy. However forming (x) as function of (n) and (A) is quite important for building further tractable approach in my problem.
Akram,
again: what exactly do you want to do with that function x(A,n)?
If you could tell us that more specifically, we could help you with something like a closed form expression.
You can always construct an arbitrarily close approximation by precomputing x(A,n) on a number of points (e.g. on a grid) and use NURBS-Patches or easier Bezier-patches to interpolate these points. The resulting 2D-Spline is then something like a closed-form approximation of x(A,n). The formulae you get from Mathematica look great but you have to be very careful since they are (as expressed by Mathematica) numerically unstable and in certain situations the roundoff errors can become very big.
The given equation is symmetric about 0 as replacing x by -x gives the same equation. Therefore, it may be solved for positive x only. Now at x=0, A = 2^{1-n} and x=1, A=1.Since a solution is sought for 2^{1-n} < A < 1, that is, you are seeking solution for 0
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