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METHODS
Simpson Method

The Simpson approach states that the "integration range [a,b]" can always be divided into n (n must be even) strips of width h = (b - a) / n. Applying the Simpson's rule to two adjacent strips:

 xi+2∫xi f(x)dx = h/3 [f(xi) + 4 f(x(i+1) + f(xi+2)]

Then the integral can be obtained by the sum:

 I = ∫ba f(x)dx = ∑ni=0,2,4,... ∫x i+2xi f(x)dx = h/3 ∑ ni=0,2,4,... [f(xi>) + 4 f(xi+1) + f(xi+2)]

It must also be stated that the Simpson's 1/3 rule requires the number of strips n to be even. If this condition is not satisfied, it is posible to integrate over the first (or last) three strips by using Simpson's 3/8 rule (which is another condition):

 I = 3h/8 [f(x0) + 3 f(x1) + 3 f(x2) + f (x3)]

and then use the Simpson's 1/3 rule for the rest of strips. Although a bit confusing it is really very simple to implement this rule.

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EIGENVALUE
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