Engineering Optimization: Theory and Practice, Fourth Edition

5.11 Cubic Interpolation Method 281

is used to approximate the functionf (λ)between pointsAandB, we need to find the
valuesfA= f(λ=A), fA′= df/dλ(λ=A), fB= f(λ=B), andfB′= df/dλ(λ=
B)in order to evaluate the constants,a, b, c, anddin Eq. (5.45). By assuming that
A
=0, we can derive a general formula forλ ̃∗. From Eq. (5.45) we have

fA= a+bA+cA^2 + dA^3

fB= a+bB+cB^2 + dB^3

fA′ = b+ 2 cA+ 3 dA^2

fB′ = b+ 2 cB+ 3 dB^2 (5.46)

Equations(5.46) can be solved to find the constants as

a=fA− bA−cA^2 − dA^3 (5.47)

with

b=

1

(A−B)^2

(B^2 fA′+A^2 fB′+ 2 ABZ) (5.48)

c= −

1

(A−B)^2

[ A( +B)Z+BfA′+ AfB′] (5.49)

and

d=

1

3 (A−B)^2

( 2 Z+fA′+fB′) (5.50)

where

Z=

3 (fA−fB)

B−A

+fA′+fB′ (5.51)

The necessary condition for the minimum ofh(λ)given by Eq. (5.45) is that

dh

dλ

=b+ 2 cλ+ 3 dλ^2 = 0

thatis,

λ ̃∗=−c±(c

(^2) − 3 bd) 1 / 2
3 d

(5.52)

The application of the sufficiency condition for the minimum ofh(λ)leads to the
relation

d^2 h

dλ^2

∣

∣

∣

∣ ̃

λ∗

= 2 c+ 6 dλ ̃∗> 0 (5.53)

By substituting the expressions forb,c, andd given by Eqs. (5.48) to (5.50) into
Eqs. (5.52) and (5.53), we obtain

λ ̃∗=A+ f

′

A+Z±Q

fA′+fB′+ 2 Z

(B−A) (5.54)