Rearranging y(x) = a + bx^{-2} + cx^{-4} + dx^{-6} to Find x

[peterjaybee]

Hi, I have fitted a graph with a function of the form
$$y(x) = a + bx^{-2} + cx^{-4} + dx^{-6}$$

I now need to calculate x for a given y and have no idea how to go about rearranging this function.

I need to find x for a lot of y values, so I was hoping to write something in methematica, but I need to understand how to rearrange it before I can implement it in mathematica.

 

[stevenb]

Hi, I have fitted a graph with a function of the form
$$y(x) = a + bx^{-2} + cx^{-4} + dx^{-6}$$

I now need to calculate x for a given y and have no idea how to go about rearranging this function.

I need to find x for a lot of y values, so I was hoping to write something in methematica, but I need to understand how to rearrange it before I can implement it in mathematica.

Numerical techniques can be used, but if you want to solve it algebraically, then you can try the substitution u=1/x² and then use the usual cubic polynomial algebraic solution to solve for u.

There is a cubic solution cheat sheet at the end of this thread.

https://www.physicsforums.com/showthread.php?t=396973&highlight=cubic+solution

 

[peterjaybee]

Thanks for your reply. What numerical technique would you suggest?

 

[stevenb]

Thanks for your reply. What numerical technique would you suggest?

I usually start by plotting the function just to visually see how many real roots there might be, then I usually opt for Newton's method as a first try. The plot will allow you to generate some good initial guesses needed for Newton's method.

http://mathworld.wolfram.com/NewtonsMethod.html