1. The Simpson Rule
- Posted by Neil Rigby <NRigbyking at AOL.COM> Dec 18, 1998
- 313 views
Regarding Simpson's Rule :- If we have an even number of strips i.e. on odd number of ordinates then Integral = strip_width/3 * ( ( first_ordinate_value + last_ordinate_value ) + 4.0*( sum of even_ordinate_values ) + 2.0*( sum of odd_ordinate_values(except first and last ) ) ) Therefore the factors 4.0 and 2.0 can be taken outside loops e.g. suppose we have 'n' points ( n = odd ) Integral = f(x_1) + f(x_n) sumeven = 0 for index = 2 to n-1 by 2 do sumeven = sumeven + f(x_index) end for sumodd = 0 for index = 3 to n-2 by 2 do sumodd = sumodd + f(x_index) end for integral = integral + 4.0 * sumeven + 2.0*sumodd integral = integral*strip_width/3 Simple code is usually fast code which makes Simpson's method attractive. If the cost of evaluating the function is high then there are other better methods such as Gaussian Quadrature that can give high accuracy with much fewer function evaluations. Lastly, Simpson's rule is based on representing the function over two intervals by a quadratic polynomial. Therefore Simpson's Rule should not be used on functions where this is not reasonable ( at least in the range of x of interest ) . For example Simpson's Rule should not be used to evaluate the integral of 1/x anywhere near x=0. Rational polynomials such as ( A + Px ) / ( B + Qx ) need careful consideration. I hope this is useful. -- Neil Rigby