Newton s Method for Nonlinear System of equations

Newton s Method for Non-linear system
Jacobian Matrix
Define the Jacobian matrix J(x) by
J(x)=[?((?f_1)/(?x_1 ) (x)&(?f_1)/(?x_2 ) (x)&?(?&(?f_1)/(?x_n ) (x) )@(?f_2)/(?x_1 ) (x)&(?f_2)/(?x_2 ) (x)&?(?&(?f_2)/(?x_n ) (x) )@?(?@(?f_n)/(?x_1 ) (x) )&?(?@(?f_n)/(?x_2 ) (x) )&?(?&?(?@(?f_n)/(?x_n ) (x) )))]
The function G is defined by:
G(x)=x-?J(x)?^(-1) F(x)
x^((k) )=G(x^((k-1) ) )-?J(x^((k-1) ) )?^(-1) F(x^((k-1) ) )
This is called Newton’s method for nonlinear systems. A weakness in Newton’s method arises from the need to compute and inverse of the matrix J(x) at each step.
In this method we firstly start with an initial approximation vector x^((0) ), then we find the Jacobian matrix J(x^((0) ) ) and F(x^((0) ) ) and solve the system

J(x^((0) ) ) y^((0) )=-F(x^((0) ) )
y^((0) )=-?J(x^((0) ) )?^(-1) F(x^((0) ) )
x^((1) )=x^((0) )+y^((0) )
To generalize the situation for every k?1
y^((k-1) )=-?J(x^((k-1) ) )?^(-1) F(x^((k-1) ) )
x^((k) )=x^((k-1) )+y^((k-1) )
J(x^((k-1) ) )=[?((?f_1)/(?x_1 ) x^((k-1) )&(?f_1)/(?x_2 ) (x^((k-1) ) )&?(?&(?f_1)/(?x_n ) (x^((k-1) ) ) )@(?f_2)/(?x_1 ) (x)&(?f_2)/(?x_2 ) (x)&?(?&(?f_2)/(?x_n ) (x^((k-1) ) ) )@?(?@(?f_n)/(?x_1 ) (x^((k-1) ) ) )&?(?@(?f_n)/(?x_2 ) (x^((k-1) ) ) )&?(?&?(?@(?f_n)/(?x_n ) (x^((k-1) ) ) )))]

[?(x_1^k@?(x_2^k@?)@x_n^k )]=[?(x_1^((k-1) )@?(x_2^((k-1) )@?)@x_n^((k-1) ) )]+[?(y_1^((k-1) )@?(y_2^((k-1) )@?)@y_n^((k-1) ) )]


Example 1: The nonlinear system:
3x_1-cos(x_2 x_3 )-1/2=0
x_1^2-81(x_2+0.1)^2+sin??x_3 ?+1.06=0
e^(-x_1 x_2 )+20x_3+(10?-3)/3=0
Apply Newton s method to this problem with x^((0) )=(0.1,0.1,-0.1)^t.