Newton extended formula

If a dose Newton-Cotes formula is iteratively applied to adjacent intervak, the extended Newton-Cotes formulae are obtained and they exploit the points shared by the adjacent intervak. 

A simple way to estimate the error with an algorithm based on the extended Newton—Cotes formulae is to compare the results by doubling the integration step. 

A general code segment for implementing Newton s method is shown in Listing 3.4. The main iterative loop extends from line 6 through line 15 with a maximum iteration number of NTT = 200. Newton s formula is implemented on lines 10 and 11 with the correction value on line 11. Lines 7 through 11 are used to perform a numerical derivative for use in Newton s method and a simple first-order numerical approximation is used in the form ... 

The use of Halley s method requires an evaluation of the second derivative of a function in addition to the first derivative. If the function is relatively simple in mathematical form an analytical derivative can sometimes be obtained for use in the expression and a customized version of flie method can be used for a specific function. For a general root finding method the approach with Newton s method has been to use a numerically evaluated first derivative. Extending fliis approach to Halley s method requires a numerically evaluated second derivative and this requires an evaluation of flie function at three points as opposed to two function evaluations for flie first derivative. Thus some (or perhaps most) of flie advantage of such a formula will be lost in terms of computational time by the extra fimction evaluation. 

---