Use the inverse function theorem to estimate the change in the roots

[ianchenmu]

Homework Statement

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

Homework Equations

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The Attempt at a Solution

How can I use the inverse function theorem to estimate?

 

[mighty2000]

The inverse function theorem can be used to estimate the change in the roots of a polynomial when the coefficients are changed. In this case, we can use the following steps:

1. First, we need to find the derivative of the polynomial p(\lambda) with respect to the coefficients a_i. This can be done by expanding the polynomial and taking the derivative of each term.

2. Next, we need to calculate the inverse of the derivative matrix. This can be done by using the formula: (d/dx)^-1 = (1/det(dx/dx))*adj(dx/dx), where adj(dx/dx) is the adjugate matrix of the derivative.

3. Once we have the inverse of the derivative matrix, we can use it to estimate the change in the roots of the polynomial. The formula for this is given by: \Delta x = -(d/dx)^-1 * \Delta a, where \Delta a is the change in the coefficients.

4. Finally, we can use the estimated change in the roots to find the new values of the roots. This can be done by adding the estimated change to the original values of the roots.

In this case, we have a cubic polynomial with three roots, so the estimated change in the roots would be a 3x1 matrix. By following the above steps, we can estimate the change in the roots and find the new values for x_1, x_2, and x_3 when the coefficients change by 0.01 times their original values.