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Solution Procedures 45
initialize
evaluate f0 at v0, f1 at v1 = v0 +Δv, f2 at v2 = v1 +Δv
iteration
calculate derivative f ′
secant: from f0 and f1
false position: from f0, and f1 or f2 (opposite sign from f1)
calculate gain: C = λ/f ′
increment solution: δv = −Cf
shift: f2 = f1, f1 = f0
evaluate f
test convergence
Figure 5-5. Outline of secant method or method of false position.
initialize
evaluate f0 at x0, f1 at x1 = x0 +Δx, f2 at x2 = x1 +Δx
bracket maximum: while not f1 ≥ f0,f2
if f2 >f0, then x3 = x2 +(x2 − x1); 1,2,3 → 0,1,2
if f0 >f2, then x3 = x0 − (x1 − x0); 3,0,1 → 0,1,2
iteration (search)
if x2 − x1 >x1 − x0, then x3 = x1 + W (x2 − x1)
if f3 f1, then 1,3,2 → 0,1,2
if x1 − x0 >x2 − x1, then x3 = x1 − W (x1 − x0)
if f3 f1, then 0,3,1 → 0,1,2
test convergence
Figure 5-6. Outline of golden-section search.
using k = n − 1 or k = n − 2 such that f(xn) and f(xk) have opposite signs. The convergence is
slower (roughly linear) than for the secant method, but by keeping the solution bracketed convergence
is guaranteed. The process for the method of false position is shown in Figure 5-5.
5-2.5 Golden-Section Search
The golden-section search method can be used to find the solution x that maximizes f(x). The
problem of maximizing f(x) can be attacked by applying the secant method or method of false position to
the derivative f ′ (x) =0, but that approach is often not satisfactory as it depends on numerical evaluation
of the second derivative. The golden-section search method begins with a set of three values x0