Thus, the initial bicrystal interface structures are obtained using molecular statics calculations which utilize an energy minimization technique. Although the secant method requires the evaluation of f( x) at both x n and x n−1 during each iteration, it can be implemented in such a way that previous evaluations of f( x n−1) are kept which greatly speeds up the program. Even when an explicit expression for the derivative is known, the evaluation of these two distinct functions during each iteration is computationally expensive. However, as we have previously noted, the Newton-Raphson method requires an evaluation of both f( x n) and f′( x n) during each iteration.
While it is possible to prove this with an analysis of the convergence speeds, this is beyond the scope of this book and we take this fact from observations of the above examples. Four roots are found over the full interval −5,5, x r ≃ ± x 5 1 = ± 1.26027 and x r ≃ ± x 6 2 = ± 2.67672. Equation ( 13.7) can then be applied for both sets of initial data, as shown in Table 13.12.
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