# Thread: Use the inverse function theorem to estimate the change in the roots

1. ## Use the inverse function theorem to estimate the change in the roots

Let $\displaystyle p(\lambda )=\lambda^3+a_2\lambda^2+a_1\lambda+a_0=(\lambda-x_1)(\lambda-x_2)(\lambda-x_3)$ be a cubic polynomial in 1 variable $\displaystyle \lambda$. Use the inverse function theorem to estimate the change in the roots $\displaystyle 0<x_1<x_2<x_3$ if $\displaystyle a=(a_2,a_1,a_0)=(-6,11,-6)$ and $\displaystyle a$ changes by $\displaystyle \Delta a=0.01a$.

How can I use the inverse function theorem to estimate?

2. ## Re: Use the inverse function theorem to estimate the change in the roots

Hey ianchenmu.

If you have a root, then the inverse function would set x = 0 and y = the root. Now if you can find where this happens with the inverse function theorem and do an approximate taylor expansion with a linear component, then you can estimate the changes in the roots.

3. ## Re: Use the inverse function theorem to estimate the change in the roots

What $\displaystyle \Delta a$ means? Can you give me a more complete answer? Thank you.

4. ## Re: Use the inverse function theorem to estimate the change in the roots

Originally Posted by chiro
Hey ianchenmu.

If you have a root, then the inverse function would set x = 0 and y = the root. Now if you can find where this happens with the inverse function theorem and do an approximate taylor expansion with a linear component, then you can estimate the changes in the roots.
What $\displaystyle \Delta a$ means? Can you give me a more complete answer? Thank you.