Fixed Points for Nonlinear system of equations

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Numerical Solutions for Nonlinear Systems of Equations
Vector Norms
Definition 1: A vector norm on R^n is a function, ?.?, from R^n into R with the following properties:
?x??0 for all x?R^n,
?x?=0 if and only if x=0,
??x?=|?|?x? for all ??R and x?R^n,
?x+y?=?x?+?y? for all x,y ?R^n. ?

Definition 2: The l_1,l_2 and l_(? ) norms for the vector x=(x_1,x_2,…,x_n )^t are defined by:
?x?_1=?_(i=1)^n?|x_i |

?x?_2={?_(i=1)^n?x_i^2 }^(1?2)

?x?_?=max?(1?i?n)?|x_i | ?

The l_2 norm is called the Euclidean norm of the vector x because it represents the usual notion of distance from the origin in case x is in R^1?R,R^2,or R^3. For example, the l_2 norm of the vector x=(x_1,x_2,x_3 )^t gives the length of the straight line joining the points (0, 0, 0) and (x_1,x_2,x_3 ).

System of Nonlinear Equations
A system of nonlinear equations has the form: