# What is an easy way to calculate numerical integration uncertainty/error

• jephthah
In summary, to integrate the area under a waveform (pulse), you can use one of the basic numeric integration methods such as Trapezoidal, Simpson's, or Boole's. To estimate the uncertainty/error of the numeric integration compared to the "true" value, a fairly easy method is to use Simpson's Rule and calculate the error bound using the formula \frac{1}{90}(\frac{b-a}{2})^5 |f''''(z)| where z is the number between a and b that maximizes |f''''(x)| between a and b. This information can be found on Wikipedia. While there may be other methods, taking a number of derivatives is necessary to determine the error, as these methods
jephthah
i'm measuring a waveform (pulse) that i want to integrate the area under.

i take a bunch of samples and use one of the basic numeric integration methods (Trapezoidal, Simpson's, Boole's)

what is a fairly easy method to estimate the uncertainty/error of the numeric integration compared to the "true" value?

thanks

For Simpson's Rule, the error bound is:

$$\frac{1}{90}(\frac{b-a}{2})^5 |f''''(z)|$$ where z is the number between a and b that maximizes |f''''(x)| between a and b.

I got this from:

http://en.wikipedia.org/wiki/Simpson's_rule

You can find the other's on wiki as well. As far as an easier method of determining the error, I'm not aware of any. Since these methods are in some way, shape or form derived from Taylor approximations, I'm pretty sure you can't rid of taking a number of derivatives.

## 1. What is numerical integration uncertainty/error?

Numerical integration uncertainty/error refers to the amount of inaccuracy or deviation from the true value in a numerical integration calculation. It can be caused by various factors such as rounding errors, truncation errors, or the use of an approximation method.

## 2. Why is it important to calculate numerical integration uncertainty/error?

Calculating numerical integration uncertainty/error is important because it allows us to assess the reliability and accuracy of our numerical integration results. It also helps us understand the potential sources of error and improve our methods for future calculations.

## 3. What is an easy way to calculate numerical integration uncertainty/error?

An easy way to calculate numerical integration uncertainty/error is by using the trapezoidal rule. This method involves dividing the integration interval into smaller segments and calculating the area under the curve using the trapezoidal formula. The difference between the actual value and the calculated value can then be used to determine the uncertainty/error.

## 4. Can numerical integration uncertainty/error be reduced?

Yes, numerical integration uncertainty/error can be reduced by using more accurate integration methods such as Simpson's rule or Gaussian quadrature. Additionally, increasing the number of intervals or using a smaller step size can also help reduce the uncertainty/error.

## 5. How can I determine the acceptable level of uncertainty/error in a numerical integration calculation?

The acceptable level of uncertainty/error in a numerical integration calculation can vary depending on the application and the desired level of accuracy. In some cases, a small amount of uncertainty/error may be acceptable, while in others it may need to be minimized as much as possible. It is important to consider the purpose of the calculation and the potential impact of the uncertainty/error when determining an acceptable level.

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