Evolution and Optimum Seeking

One Dimensional Strategies 35
It is inconvenient when dealing with minimization problems that the derivatives of
the function are required. If the slopes are obtained from function values by a di erence
method, di culties can arise from the nite accuracy of such a process. For this reason
Brent (1973) combines regula falsi iteration with division according to the golden section.
Further variations can be found in Schmidt and Trinkaus (1966), Dowell and Jarratt
(1972), King (1973), and Anderson and Bjorck (1973).
3.1.2.3.2 Newton-Raphson Iteration. Newton's interpolation formula for improving
an approximate solution x (k) to the equation F (x) = 0 (see for example Madsen,
1973)
x (k+1) = x (k) ; F (x(k) )
Fx(x (k) )
uses only one argument value, but requires the value of the derivative of the function
as well as the function itself. If F (x) is linear in x, the zero is correctly predicted here,
otherwise an improved approximation is obtained at best, and the process must be repeated.
Like regula falsi, Newton's recursion formula can also be applied to determining
Fx(x) = 0, with of course the reservations already stated. The so-called Newton-Raphson
rule is then
x (k+1) = x (k) ; Fx(x (k) )
Fxx(x (k) )
(3.15)
If F (x) is not quadratic, the necessary number of iterations must be made until a termination
criterion is satis ed. Dixon (1972a) for example uses the condition jx (k+1) ; x (k) j