Newton s Method

Newton s Method
In Newton s method the linear approximation of the function can be done by using Taylor s series, so
x_(k+1)=x_k-f(x_k )/(f^ (x_k ) ). (4)

The properties of Newton s method
1. Newton s method has a quadratic convergence when the chosen point is close enough to zero. If the derivative is zero at the root, it has only local quadratic convergence.
2. Numerical difficulties occur when the first-derivative is zero.
3. If a poor starting point is chosen, then the method may fail to converge or diverge.
4. If it is difficult to find the derivation, secant method may be used.
For a smooth continuous function when proper starting point is chosen, Newton s method can be real fast.
Example: Find the (convergence or divergence) of f(x)=x/(1+x^2 ) with x_0=0.5, x_0=0.75 and ?=?10?^(-4) .
k
0
1
2
3 x_k
0.5=x_0 ?f(x_0 )=0.4
-0.333333
0.083333
-0.001166 x_(k+1)
-0.333333=x_1 ?f(x_1 )=-0.299999
0.083333=x_2 ?f(x_2 )= 0.082758
-0.001166=x_3 ?f(x_3 )=-0.001165
0 =x_4 ?f(x_4 )= 0

Newton s method converging with x_0=0.5
At x_0=0.75
k
0
1
2
3
4
?
20 x_k
0.75=x_0
-1.928571
-5.275529
-10.944297
-22.072876
?
x_(k+1)
-1.928571 =x_1
-5.275529 =x_2
-10.944297=x_3
-22.072876=x_4
-44.236547=x_5
?
-725265.55757

To find the minimum, Newton s method uses the first derivative of the approximation and equaling it to zero.
f^ (x_(k+1) )=f(x_k )+f^ (x_k)(x_(k+1)-x_k)
Now, if we take the second derivative we get
f^ (x_(k+1) )=f^ (x_k )+f^ (x_k)(x_(k+1)-x_k)