Solving Interpolation Error w/ ≤ 5*10-8 Precision

In summary, interpolation error refers to the difference between the actual value and the interpolated value of a function at a specific point and is caused by approximating the function using a finite number of data points. It is typically measured using the maximum absolute error or root mean square error, and it is important to solve it with high precision in scientific and engineering applications. A precision of ≤ 5*10-8 is considered very high and can be achieved by using higher-order interpolation methods, increasing the number of data points, and carefully selecting appropriate techniques for the given problem.
  • #1
peripatein
880
0
Hi,

Homework Statement


Interpolation for the function f(x)=cos(x) for evenly distributed values of x in [0,π] (h=xi+1-xi, xi=ih, i=0,1,...,πh) is carried out. Also known are f(xi).
I am asked to determine the value of h so that the interpolation's error is ≤ 5*10-8.

Homework Equations




The Attempt at a Solution


I have found f'''(x) for x in [0,π] to be ≤ 1.
Now I am stuck with evaluating |(x-xi+1)(x-xi)(x-xi-1)|.
I'd truly appreciate some assistance. Thanks in advance!
 
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  • #2
I have managed on my own with this one. Thank you in any case :).
 

1. What is interpolation error?

Interpolation error is the difference between the actual value and the interpolated value of a function at a particular point. It is caused by the approximation of the function using a finite number of data points.

2. How is interpolation error measured?

Interpolation error is typically measured using the maximum absolute error, which is the largest difference between the actual value and the interpolated value over a range of input values. It can also be measured using the root mean square error, which takes into account the average of the squared errors.

3. Why is it important to solve interpolation error with high precision?

High precision in solving interpolation error is important because it ensures that the interpolated values are as close to the actual values as possible. This is crucial in scientific and engineering applications where accurate data is required for making informed decisions and predictions.

4. What is the significance of a precision of ≤ 5*10-8 in solving interpolation error?

A precision of ≤ 5*10-8 means that the interpolated values will have a maximum absolute error of 5*10-8. This level of precision is considered very high and is often required in scientific and engineering calculations. It ensures that the errors are minimal and have little impact on the accuracy of the results.

5. How can interpolation error with a precision of ≤ 5*10-8 be achieved?

To achieve a precision of ≤ 5*10-8, various techniques can be used such as using higher-order interpolation methods, increasing the number of data points, and using more advanced algorithms. It is also important to carefully analyze the data and choose appropriate interpolation techniques for the given problem.

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