Thread: Linearization of nonlinear equations with approximation

1. Linearization of nonlinear equations with approximation

Dear friends,

Can anybody help me to make an approximation to a nonlinear equation like (x-y)^0.5 in terms of x and y?

2. Use the Taylor formula

$\displaystyle f(x,y)\;\approx \;f(x_0,y_0)+\dfrac{\partial f}{\partial x}(x_0,y_0)\;(x-x_0)+\dfrac{\partial f}{\partial y}(x_0,y_0)\;(y-y_0)$

3. Thanks for your comment. Indeed, I need a parametric function in terms of x and y like what is written for (x-y)^2=x^2+y^2-2xy. However, I know there is not an exact extension for my equation. I think there is maybe a simplified form with a reasonable approximation.

4. Yes, an approximation is what you asked for and what FernandoRevilla gave you. Your function is f(x,y)= (x- y)^{0.5}, so its first order partial derivatives are f_x= (0.5)(x- y)^{-0.5} and f_y= -(0.5)(x- y)^{-0.5}. Evaluate those at some (x_0, y_0) and put into FernandoRevilla's formula. Of course, any approximation is not going to be equally accurate for all x and y. You need to decide where you want the approximation to be most accurate and pick x_0 and y_0 there.

5. Thank u HallsofIvy, I got your point. I will check it in my problem. Defining x0 and y0 is the central challenge here.

6. Originally Posted by ali1980
Thank u HallsofIvy, I got your point. I will check it in my problem. Defining x0 and y0 is the central challenge here.
There is another approach which minimises some function of the error in the linear approximation across some region of the x-y plane but from what you write I guess that would be beyond you at present.

CB

7. Thanks CaptainBlack, maybe an iterative solution can minimize the errors. But if you know a better approach, please let me know.