Newtons Divided Difference First Derivative

In summary, The conversation is about finding the minimum of a function represented by a NDDP. The person is having trouble differentiating the function due to the presence of pi operators and sigma summations, which they have never dealt with before. They ask if the first derivative for a nth order NDDP is known. Someone suggests looking at 2.2 for help, but the person is not familiar with the recurrence relation version. They eventually find a pattern for the derivative, which is shown as a summation in the summary.
  • #1
NotASmurf
150
2
Hey all, for a function approximation program t run fast enough i need to solve for where the function (represented by a NDDP) is at a minimum (necessary trust me), althogh I have no idea how to go about differentiating it, i tried to break it up from its's general formula (the pi operators and the sigma summations make the differentiation difficult for me as i have never had to differentiate a pi operator before), but that seems to make things worst is the first derivative for a nth order NDDP known? Any help apreciated.
 
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  • #2
2.2 here perhaps ? Or did you find that already ?
 
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Likes NotASmurf
  • #3
Thanks :D
 
  • #4
Wasnt familiar with the reccurance relation version so that paper didn't help too much, however found a nice pattern, turns out the derivative is

$$ \sum_{k=0}^{k<=n}\sum_{i=0}^{k} \Pi_{j=0, j \neq i}^{k-1} (x-x_j) $$

You can see it in action here for 4th order poly
 

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1. What is Newton's Divided Difference First Derivative?

Newton's Divided Difference First Derivative is a numerical method used in calculus to approximate the derivative of a given function at a specific point. It is based on the idea of using divided differences to calculate the slope of a curve.

2. How is Newton's Divided Difference First Derivative calculated?

To calculate Newton's Divided Difference First Derivative, you need to first determine the divided differences of the given function at the point of interest. Then, these divided differences are used to construct a polynomial, known as the Newton interpolation polynomial, which can be used to approximate the derivative at the desired point.

3. When is Newton's Divided Difference First Derivative used?

Newton's Divided Difference First Derivative is often used when the function is only known at specific data points, and the derivative needs to be approximated at a point in between those data points. It is also commonly used in numerical analysis and scientific computing.

4. What are the advantages of using Newton's Divided Difference First Derivative?

One of the main advantages of using Newton's Divided Difference First Derivative is that it is a simple and efficient numerical method for approximating derivatives. It also allows for the use of higher order divided differences, which can lead to more accurate results.

5. Are there any limitations to using Newton's Divided Difference First Derivative?

One limitation of Newton's Divided Difference First Derivative is that it can only approximate the derivative at a single point, and cannot give a general expression for the derivative of the function. It also requires the function to be differentiable at the point of interest.

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