# Optimizing Polynomial Approximations for Even Functions on Symmetric Intervals

• EvLer
In summary, the conversation discussed finding the closest function to x^2 on the interval [-1,1] using the standard inner product. The attempt involved using a matrix to find the coefficients a and b, but the result seemed suspicious. Suggestions were given to check the work and consider adding part of an odd function to the approximation.
EvLer

## Homework Statement

find closest function a+bx3 to x2 on the iterval [-1,1]
(we consider standard inner product (f,g) = integral(-1 to 1):fgdx

So, here is my attempt, but I got a suspicious result:

[(1,1) (1,x3)] [a]
[(1,x3) (x3,x3)]
=
[(1,x2)]
[x3, x2)]

then after I've done all the integration I came up with this:
[2 0] [a]
[0 2/7]
=
[1/3]
[0]
which means that x3 term disappears...

I can't help you fix it. Because you are completely correct. Why do you think that is suspicious? Find ways to check yourself.

You are approximating an even function over a symmetric interval. Can adding part of an odd function help you?

## 1. What is the purpose of approximating polynomials?

The purpose of approximating polynomials is to find a simpler polynomial that closely matches a given polynomial. This can be useful in various fields such as engineering, physics, and computer science where complex polynomials need to be simplified for easier calculations.

## 2. How is a polynomial approximated?

A polynomial can be approximated using various methods such as Taylor series, Lagrange interpolation, or least squares approximation. These methods involve using known points or values of the polynomial to find a simpler polynomial that closely fits the data.

## 3. What is the difference between interpolation and extrapolation?

Interpolation involves approximating a polynomial within the given data points, while extrapolation involves extending the polynomial outside of the given data points. Interpolation is generally considered more accurate, while extrapolation can be more prone to errors.

## 4. Can all polynomials be approximated?

No, not all polynomials can be accurately approximated. The accuracy of the approximation depends on the complexity of the original polynomial and the method used for approximation. Some polynomials may require a higher degree of approximation to achieve a close match.

## 5. Are there any limitations to approximating polynomials?

Yes, there are limitations to approximating polynomials. One limitation is that the approximation may not be accurate for all values of the independent variable. Additionally, the accuracy of the approximation may decrease as the degree of the polynomial increases.

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