Advanced numerical analysis - numerical integration

[alpha754293]

If I have a x,y table of discrete datapoints with a discrete dataset, such that delta x is not a constant, what are some of the more advanced techiques that I can use to integrate this?

I remember that there were Trapezoidal rules and Simpson's rule where delta x IS a constant (and there are additional requirements for Simpson's rule, for example), but my data set doesn't fit (and can't be made to fit) those requirements.

Is the trapezoid rule the only option that I have to integrate numerical data sets?

The y=f(x) is a pretty random curve (there's no rhyme or rhythm to it) and for all practical intents and purposes, it's pretty much random.* (*The reality is a little more complicated than that, but I don't want to get into the complication issues right now, because I want to focus on what are the options that are available to me for numerical analysis.)

Help/suggestion/advice would be greatly appreciated.

 

[lurflurf]

So what you would usually do is fit a function (ie polynomial, exponential, trigonometric, or Bessel function) with parameters to the data. The question becomes how do we fit it? We could use exact agreement, sum of absolute value, sum of squares, or mini-max as our metric. It will depend how accurate our data is. There is the problem of aliasing. That basically means that a pattern in our data might mislead us. For example if our function were sin(pi*x) and all our x values were integers were would believe the function is 0. Evenly spaced data can have bad aliasing, but there might be some with our unevenly spaced data as well.

 

[AlephZero]

(*The reality is a little more complicated than that, but I don't want to get into the complication issues right now, because I want to focus on what are the options that are available to me for numerical analysis.)

That is usually a bad strategy. To get a good solution to a problem, you need to use all the information you have about it.

For example if the data points are "noisy" measurements of something, the best way to "fit a smooth curve" through them is to use a function that you know represents the right type of response. For example if you are measuring the temperature of something cooling down, it is likely to be an exponential function of time, because of Newton's law of cooling. Or if you are measuring the response of a mechanical system to a force, it is likely to be some sort of damped oscillation. Using information like that will usually give better results than blindly following an "advanced" recipe from a numerical analysis cookbook.