6.1 Introduction 307the second-order Taylor’s series approximation of a general nonlinear function at the
design vectorXican be expressed as
f(X)=c+BTX+^12 XT[A]X (6.12)where
c=f (Xi) (6.13)B=
∂f
∂x 1∣
∣
∣
∣
Xi
..
.
∂f
∂xn∣
∣
∣
∣
Xi
(6.14)
[A]=
∂^2 f
∂x 12∣
∣
∣
∣
Xi·· ·
∂^2 f
∂x 1 ∂xn∣
∣
∣
∣
Xi
..
...
.
∂^2 f
∂xn∂x 1∣
∣
∣
∣
Xi·· ·
∂^2 f
∂x^2 n∣
∣
∣
∣
Xi
(6.15)
The transformations indicated by Eqs. (6.7) and (6.9) can be applied to the matrix [A]
given by Eq. (6.15). The procedure of scaling the design variables is illustrated with
the following example.
Example 6.2 Find a suitable scaling (or transformation) of variables to reduce the
condition number of the Hessian matrix of the following function to 1:
f (x 1 , x 2 )= 6 x^21 − 6 x 1 x 2 + 2 x 22 −x 1 − 2 x 2 (E 1 )SOLUTION The quadratic function can be expressed as
f (X)=BTX+^12 XT[A]X (E 2 )where
X={
x 1
x 2}
, B=
{
− 1
− 2
}
, and [A]=[
12 − 6
−6 4
]
As indicated above, the desired scaling of variables can be accomplished in two
stages.
Stage 1: Reducing [A] to a Diagonal Form, [A ̃]
The eigenvectors of the matrix [A] can be found by solving the eigenvalue problem
[[A]−λi[ ]I] ui= 0 (E 3 )