2D Milne's rule composite integration

In summary, the conversation discusses various methods of approximation for integrals, including Simpson's rule and Milne's rule. The composite versions of these rules involve repeating the pattern of numbers in a matrix. In the case of Milne's rule, the pattern is 2 -1 2, and in the case of Simpson's rule it is 1 4 1. For a 2D integral, the matrix would have 2n+1 rows, and the same rule is applied to each row. The specific method of applying the rule to the rows is unclear, but it could involve multiplying or adding the numbers in the rule. The conversation also suggests a potential single rule for a 2D integral, which involves the numbers
  • #1
presto
23
0
http://en.wikipedia.org/wiki/Newton–Cotes_formulas

Simpson rule: 1 4 1, and the composite version: 1 4 2 4 2 4 2 4 ...4 1
in a double integral we just compute rows, and next columns, a this gives a matrix:
http://mathfaculty.fullerton.edu/mathews/n2003/SimpsonsRule2DMod.html

Milne's rule is: 2 -1 2, thus a composite version should be probably:
2 -1 4 -1 4 -1 ... -1 2for 2D we have 2n+1 such rows, then I must apply the same rule to the rows,
but I don't konow how do that...

in a first column there are 2 only, in a second -1, ext.

then I apply the 2 -1 2 scheme to these 2s, but how: adding or multipling by these numbers: 2 -1 2 ?
 
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  • #2
4 -2 8 -2 8 -2 8 -2 4
-2 1 -4 1 -4 1 ...
8 -4 16 -4 16 -4
-2 1 -4 1 -4 1
8 -4 16 -4 16 -4
-2 1 -4 1 -4 1
4 -2 8 -2 8 -2

probably something like this... and a single rule in 2D:

4 -2 4
-2 1 -2
4 -2 4

4*4 - 4*2 + 1 = 9
 
Last edited:
  • #3
The OP has been banned, so closing this thread.
 

1. What is 2D Milne's rule composite integration?

2D Milne's rule composite integration is a numerical method used to approximate the definite integral of a two-dimensional function over a specific region. It involves dividing the region into smaller subregions and using a combination of Simpson's rule and trapezoidal rule to calculate the integral.

2. How does 2D Milne's rule composite integration work?

The method works by dividing the region into smaller subregions and approximating the integral over each subregion using Simpson's rule and trapezoidal rule. These approximations are then combined to give an overall approximation of the integral over the entire region.

3. What are the advantages of using 2D Milne's rule composite integration?

One advantage is that it can provide a more accurate approximation compared to using a single integration method. It also allows for flexibility in choosing the number of subregions, which can improve the accuracy of the approximation. Additionally, it can handle integrals with irregularly shaped regions.

4. Are there any limitations to 2D Milne's rule composite integration?

Like any numerical method, the accuracy of the approximation depends on the number of subregions used. Using too few subregions may result in a less accurate approximation, while using too many subregions can be computationally expensive. Additionally, it may not be suitable for integrals with discontinuities or singularities within the region.

5. In what applications is 2D Milne's rule composite integration commonly used?

This method is commonly used in the fields of engineering, physics, and mathematics for approximating the integrals of two-dimensional functions. It can be applied in various contexts, such as calculating the area under a curve, finding the volume of a solid, or solving differential equations.

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