Engineering Optimization: Theory and Practice, Fourth Edition

5.12 Direct Root Methods 289

finite difference formulas as

f′(λi)=

f(λi+ λ)−f (λi− λ)

2 λ

(5.67)

f′′(λi)=

f(λi+ λ)− 2 f (λi) +f(λi− λ)

λ^2

(5.68)

whereλis a small step size. Substitution of Eqs. (5.67) and (5.68) into Eq. (5.65)
leads to

λi+ 1 =λi−

λ[f (λi+ λ)−f (λi− λ)]

2[f (λi+ λ)− 2 f (λi) +f(λi− λ)]

(5.69)

The iterative process indicated by Eq. (5.69) is known as thequasi-Newton method.
To test the convergence of the iterative process, the following criterion can be used:

|f′(λi+ 1 ) =|

∣

∣

∣

∣

f (λi+ 1 + λ)−f (λi+ 1 − λ)

2 λ

∣

∣

∣

∣≤ε (5.70)

where a central difference formula has been used for evaluating the derivative off
andεis a small quantity.

Remarks:

1.The central difference formulas have been used in Eqs. (5.69) and (5.70). How-

ever, the forward or backward difference formulas can also be used for this

purpose.

2.Equation (5.69) requires the evaluation of the function at the pointsλi+ λ

andλi− λin addition toλiin each iteration.

Example 5.13 Find the minimum of the function

f (λ)= 0. 65 −

0. 75

1 +λ^2

− 0. 65 λtan−^1

1

λ

using quasi-Newton method with the starting pointλ 1 = 0. 1 and the step sizeλ=
0 .01 in central difference formulas. Useε= 0 .01 in Eq. (5.70) for checking the con-
vergence.

SOLUTION

Iteration 1

λ 1 = 0. 1 , λ= 0. 01 , ε= 0. 01 , f 1 = f(λ 1 ) =− 0. 188197 ,

f 1 += f(λ 1 + λ)= − 0. 195512 , f 1 −= f(λ 1 − λ)= − 0. 180615

λ 2 =λ 1 −

λ(f 1 +−f 1 −)

2 (f 1 +− 2 f 1 +f 1 −)

= 0. 377882

Convergence check:

|f′(λ 2 ) =|

∣

∣

∣

∣

f 2 +−f 2 −

2 λ

∣

∣

∣

∣=^0.^137300 >ε