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

 

[CellCoree]

I understand your need to calculate x for a given y value and to be able to implement it in a program like Mathematica. In order to rearrange this function to solve for x, you can follow these steps:

1. Start by subtracting "a" from both sides of the equation, so that you have: y(x) - a = bx^{-2} + cx^{-4} + dx^{-6}

2. Next, multiply both sides by x^6 to eliminate the negative exponents: x^6(y(x) - a) = bx^4 + cx^2 + d

3. Now, you can rearrange the equation to have all the terms on one side: bx^4 + cx^2 + d - x^6(y(x) - a) = 0

4. This is now a polynomial equation, which you can solve using the quadratic formula or other methods to find the roots (or solutions) for x.

5. Once you have the roots, you can plug in the given y value to find the corresponding x value.

I hope this helps you understand how to rearrange the function and solve for x. Good luck with your calculations!