Function Newton s method

A more sophisticated method for optimization of a single variable is Newton s method, which exploits first and second derivatives of the objective function. Newton s method starts by supposing that the following equation needs to be solved ... 

Owing to the special structure of the Cl energy function, Newton s method converges even faster than quadratically for Cl energies. According to Exercise 11.4.3, the third derivative at c = Cex with C at the origin becomes... 

The emphasis in this chapter has been on using the Newton method for finding roots of various functions. Newton s method is based upon a local Taylor series expansion of a function and approximating the function locally by the first two terms representing the value of the function and the first derivative as expressed by Eq. (3.17). But what about functions for which the first derivative is zero at the solution value that corresponds to zeros of the function From Eq. (3.19) one can see that the correction value beeomes ... 

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