First order corrections to a polynomial equation

In summary, Dan is looking for an approximation to the solutions of a polynomial equation in terms of a small perturbation. He has seen it done in a video but can no longer find it. He asks for help in finding first order corrections to the zeros of the polynomial equation. The solution involves using the first order Taylor expansion and a Newton-Raphson algorithm. Dan expresses his gratitude for the explanation.
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topsquark
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I've seen this done in a video but I can no longer find the video! :(

What I would like to do is approximate the solutions to a polynomial equation in terms of a small perturbation. For example, say we have y = f(x) and know the corresponding zeros exactly. How would I go about finding first order corrections to the zeros if we have y = f(x) + e (e is small)?

Any help would be appreciated.

-Dan
 
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  • #2
topsquark said:
I've seen this done in a video but I can no longer find the video! :(

What I would like to do is approximate the solutions to a polynomial equation in terms of a small perturbation. For example, say we have y = f(x) and know the corresponding zeros exactly. How would I go about finding first order corrections to the zeros if we have y = f(x) + e (e is small)?

Any help would be appreciated.

-Dan

Hey Dan!

Let g(x)=f(x)+e, and let x0 be a root of f(x).
Then we're looking for an x near x0 such that g(x)=0.
The first order Taylor expansion is: g(x) = g(x0) + (x-x0)g'(x0)
It follows at the first order that 0 = g(x) = f(x0) + e + (x-x0)f'(x0).
And therefore the first order correction is (x-x0) = -e / f'(x0).

This would actually be the first iteration of a Newton-Raphson algorithm, that will find the new zero unless f' is zero somewhere between the old and the new root.
In that case we can use f'' to see if the zero disappeared, or if there are 2 new roots at each side.

Is this perchance what you had in mind?
 
  • #3
I like Serena said:
Hey Dan!

Let g(x)=f(x)+e, and let x0 be a root of f(x).
Then we're looking for an x near x0 such that g(x)=0.
The first order Taylor expansion is: g(x) = g(x0) + (x-x0)g'(x0)
It follows at the first order that 0 = g(x) = f(x0) + e + (x-x0)f'(x0).
And therefore the first order correction is (x-x0) = -e / f'(x0).

This would actually be the first iteration of a Newton-Raphson algorithm, that will find the new zero unless f' is zero between the old and the new root.
In that case we can use f'' to see if the zero disappeared, or if there are 2 new roots at each side.

Is this perchance what you had in mind?
That's the one! Thanks. I knew Taylor was probably in there somewhere but I couldn't figure out how to apply it.

-Dan
 

FAQ: First order corrections to a polynomial equation

What are first order corrections to a polynomial equation?

First order corrections to a polynomial equation refer to the adjustments or modifications made to the coefficients of the equation in order to improve its accuracy or fit to the data.

Why are first order corrections necessary for polynomial equations?

First order corrections are necessary because polynomial equations often have a limited degree of accuracy and may not perfectly fit the data. These corrections help to improve the accuracy of the equation and make it a better representation of the data.

How are first order corrections calculated?

First order corrections are typically calculated using techniques such as least squares regression or gradient descent, which aim to minimize the error between the predicted values of the equation and the actual data points.

What is the significance of first order corrections in scientific research?

First order corrections are important in scientific research as they allow for more accurate and precise equations, which can lead to more reliable predictions and conclusions. They also help to identify any potential errors or limitations in the original equation.

Can first order corrections be applied to all types of polynomial equations?

Yes, first order corrections can be applied to any type of polynomial equation, regardless of its degree. However, the effectiveness of these corrections may vary depending on the complexity and accuracy of the original equation.

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