Numerical Integration: Understanding Error, Big Oh and Taylor Expansion

In summary, the conversation is about numerical integration and the confusion around dealing with error and understanding the concept of order of method (Big Oh). The person is looking for clarification or resources to help understand this material. They also mention theta, but suggest that the listener can easily find information on it through a web search.
  • #1
Feldoh
1,342
3
We've covering numerical integration in one of my classes this semester, however I'm sort of at a loss as to how to deal with error in when using a taylor expansion for numerical integration, and also order of method (Big Oh?) and what exactly it means and was wondering if anyone could explain some of this to me or suggest some websites which covers this type of material?
 
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  • #2
Be more specific, big O means that in a neighbourhood of x->x0 f(x)=O(g(x)) iff there exists M>0 s.t |f(x)/g(x)|<=M.
I forgot what theta means, but searching the web shouldn't be a problem for you is it?
 

FAQ: Numerical Integration: Understanding Error, Big Oh and Taylor Expansion

1. What is numerical integration and why is it important?

Numerical integration is a method of approximating the definite integral of a function by dividing it into smaller intervals and using numerical techniques to calculate the area under the curve. It is important because it allows us to solve complex problems that cannot be solved analytically, and it also provides a way to numerically model and simulate real-world phenomena.

2. What is error in numerical integration and how is it measured?

Error in numerical integration refers to the difference between the actual value of the integral and the approximated value. It is measured using the error formula, which is the difference between the actual value and the approximate value divided by the actual value. It is also common to use the terms absolute error and relative error, which refer to the magnitude of the error and the error as a percentage, respectively.

3. What is Big Oh notation and how is it used in numerical integration?

Big Oh notation is a way of expressing the complexity or rate of growth of an algorithm or function. In numerical integration, it is used to describe the error of the approximation method in terms of the number of intervals used. For example, if an approximation method has a Big Oh of O(n^2), it means that the error decreases quadratically as the number of intervals used increases.

4. How does Taylor expansion help in numerical integration?

Taylor expansion is a mathematical technique used to approximate a function using its derivatives. In numerical integration, it can be used to improve the accuracy of the approximation by including higher-order terms in the calculation. By including more terms in the Taylor expansion, we can reduce the error and get a more precise estimate of the integral.

5. What are some common methods for numerical integration?

There are several methods for numerical integration, including the trapezoidal rule, Simpson's rule, and Gaussian quadrature. Each method has its own advantages and is suitable for different types of integrals. The trapezoidal rule is the simplest and most commonly used method, while Simpson's rule is more accurate and Gaussian quadrature is even more accurate for certain types of integrals. Choosing the appropriate method depends on the function being integrated and the desired level of accuracy.

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