Taylor expansion of an unknown function

In summary, the conversation discussed finding an analytical estimate for an unknown function using available information such as its exact value and first derivative, and whether the obtained quadratic function is a result of Taylor series expansion or polynomial regression. The appropriate term for this process was debated, with suggestions such as truncated Taylor series curve fitting, quadratic polynomial fit, and polynomial interpolation. Ultimately, it was concluded that the process is best described as quadratic Hermite interpolation.
  • #1
Alex_F
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TL;DR Summary
Taylor series expansion or polynomial regression which is the correct term here for finding an analytical estimate for an unknown function?
Hello,

I have a question regarding the Taylor expansion of an unknown function and I would be tanksful to have your comments on that.

Suppose we want to find an analytical estimate for an unknown function. The available information for this function is; its exact value at x0 (f0) and first derivative at x0 (f0'), and its exact value at x=x1 (f1) which is far from x0. This means that we only need to find the second derivative through solving f1 = f0 + f0'*(x1-x0) + (f"0/2)*(x1-x0)^2 to form the truncated Taylor series for f.

My question is that whether the obtained quadratic function is the result of Taylor series expansion or polynomial regression?
 
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  • #2
If you are sure that the function does not have any higher order term components, e.g. ##x^3, x^4##, it is Taylor series, otherwise not. Though I do not know much about polynomial regression, it is often used for statistical data so I am not sure if it applies here.
 
  • #3
Alex_F said:
Suppose we want to find an analytical estimate for an unknown function. The available information for this function is; its exact value at x0 (f0) and first derivative at x0 (f0'), and its exact value at x=x1 (f1) which is far from x0. This means that we only need to find the second derivative through solving f1 = f0 + f0'*(x1-x0) + (f"0/2)*(x1-x0)^2 to form the truncated Taylor series for f.
OK, nothing unusual here, we are just talking about ## s = ut + \frac12 at^2 ##.

Alex_F said:
My question is that whether the obtained quadratic function is the result of Taylor series expansion or polynomial regression?
Well it's certainly not polynomial regression; I wouldn't personally use the term 'Taylor series expansion' either - perhaps 'truncated Taylor series curve fitting'? Why do you need a name anyway? I'd just call it "solving for constant f''(x)", or for the example I gave "solving for acceleration".
 
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  • #4
pbuk said:
OK, nothing unusual here, we are just talking about s=ut+12at2.Well it's certainly not polynomial regression; I wouldn't personally use the term 'Taylor series expansion' either - perhaps 'truncated Taylor series curve fitting'? Why do you need a name anyway? I'd just call it "solving for constant f''(x)", or for the example I gave "solving for acceleration".

Thanks for your answer. You are right, it is definitely not a Taylor expansion. I think we can call it quadratic polynomial fit as the results are identical when you fit a second-order polynomial to the available data. I checked this for couple of functions and it should be also possible to prove it analytically.

I needed a name, first to better know what I was doing and second to write in a report about it. However, I got rid of this as it was just a curve fitting using very few data. Moreover, this type of fitting would also give me misleading information on the function behavior around x0.
 
  • #5
I would call that polynomial interpolation. If I recall correctly Hermite type. Given the value of a function and/or its some derivatives we can look for a polynomial that has the same values. If as in your example whenever we know a derivative we know all lower derivative and including the 0th (function value) this can always be done uniquely. If some derivatives are missing (Birkhoff type) we may have no solution or multiple solutions.
 
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  • #6
lurflurf said:
I would call that polynomial interpolation. If I recall correctly Hermite type.
Yes, (quadratic) Hermite interpolation is the best description.
 

1. What is the Taylor expansion of an unknown function?

The Taylor expansion of an unknown function is a mathematical technique used to approximate a function with a polynomial. It is based on the idea that any smooth function can be approximated by a polynomial of increasing degree.

2. Why is the Taylor expansion useful?

The Taylor expansion is useful because it allows us to approximate a function with a simpler polynomial, making it easier to analyze and manipulate. It also helps us understand the behavior of a function around a specific point.

3. How is the Taylor expansion calculated?

The Taylor expansion is calculated by taking the derivatives of the function at a specific point and using them to construct a polynomial. The more derivatives we include, the more accurate the approximation will be.

4. What is the difference between a Taylor series and a Taylor polynomial?

A Taylor series is an infinite sum of terms, while a Taylor polynomial is a finite sum of terms. The Taylor series gives the exact representation of a function, while the Taylor polynomial is an approximation.

5. Can the Taylor expansion be used for any function?

No, the Taylor expansion can only be used for functions that are infinitely differentiable, meaning they have derivatives of all orders. If a function is not infinitely differentiable, the Taylor expansion will not be accurate.

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