Advanced numerical analysis - numerical integration

In summary, when dealing with a discrete dataset where delta x is not a constant, there are various techniques available for integration. The trapezoid rule and Simpson's rule are options, but may not be suitable for all datasets. It is important to consider the characteristics of the data, such as randomness or aliasing, and use all available information to accurately fit a function through the data. Blindly following advanced techniques may not always give the best results.
  • #1
alpha754293
29
1
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.
 
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  • #2
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.
 
  • #3
alpha754293 said:
(*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.
 

1. What is numerical integration and why is it important in advanced numerical analysis?

Numerical integration is a numerical method used to approximate the definite integral of a function. It is important in advanced numerical analysis because it allows for the evaluation of integrals that cannot be solved analytically, and it is essential in solving differential equations and other mathematical problems.

2. What are the different types of numerical integration methods?

There are several types of numerical integration methods, including the trapezoidal rule, Simpson's rule, and Gaussian quadrature. These methods differ in their level of accuracy and the complexity of the function they can approximate.

3. How do you determine the accuracy of a numerical integration method?

The accuracy of a numerical integration method can be determined by calculating the error between the exact value of the integral and the approximation obtained using the method. A more accurate method will have a smaller error.

4. Can numerical integration be used for multidimensional integrals?

Yes, numerical integration can be extended to multidimensional integrals using techniques such as Monte Carlo methods and tensor product quadrature. These methods are more computationally complex but can accurately approximate integrals in higher dimensions.

5. What are some applications of advanced numerical integration?

Advanced numerical integration has numerous applications in fields such as physics, engineering, and finance. It is used to solve differential equations, evaluate complex integrals, and simulate physical systems. It is also essential in the development of computational models and simulations.

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