Page 277 - MATLAB an introduction with applications

262 ——— MATLAB: An Introduction with Applications

Differentiating Eq.(5.1) w.r.t. x gives
i
∂ F =− b + A x
i ∑
x ∂ i j ij j
or in vector notation

∇= − b + Ax ...(5.2)
F
where ∇F is the gradient of F.
The gradient along u when the motion takes place along the line x = x + su, where s is the distance moved
0
is given by
∇
F 0 x + su = − b + A x + su ) = ∇ F 0 x + s Au
( 0
If the change in the gradient sAu is perpendicular to a vector V, then
VAu = 0 ...(5.2A)
T
The directions of u and V are said to be mutually conjugate.

5.3 NEWTON’S METHOD

Newton’s method is a gradient method and can be conveniently used to optimize functions with several
parameters. Let the function to be optimized be ()Ux , where x is the vector of parameters x , x , …, x .
1
2
n
The function Ux can be expanded in the Taylor’s series about a point x as
()
*
n ∂ Ux * )
(
Ux = (x * ) + ∑ (x − x * i )
() U
i
i= 1 ∂ X i
Newton’s method uses only two terms in the series. Expressing in concise form, the above series can be
written as
( ) Ux +
Ux = ( * ) g − T (x * )(x − x * ) ...(5.3)
where (gx * ) is the vector of first derivatives given by
 ∂ Ux * ) ∂ Ux * )  T
(
(
( gx * ) =  , ,.... 
 x ∂ 1 x ∂ 2 
*
The minimum ()Ux obtained by setting
∂ U = 0
x ∂ i

In Eq.(5.3) which yields the set of equations
+
( gx * ) J (x * )(x − x * ) = 0 ...(5.4)