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Finding Root(s) of Non Linear Equation / Non Linear System of Equations using Newton Raphson Method

1. One or more Root(s) of any Non Linear Equation of One Variable can be found out using the Newton Raphson Method using the following steps
   1. Calculate the Derivative of the Equation. For example given a Non Linear Equation of a Variable \(x\), \(F(x)=0\), find it's Derivative Equation \(F'(x)=0\) .
   2. Select any Arbitrary Value \(v\) as Initial Value for Variable \(x\) and repeatedly evaluate the following expression
      
      \(v = v - {\Large \frac{F(v)}{F'(v)}}\)   ...(1)
      
      where \(F(v)\) and \(F'(v)\) are the Values of the Function \(F(x)\) and \(F'(x)\) respectively for \(x=v\), till \(F(v)\) evaluates to \(0\) upto the required precision. The corresponding value of \(v\) is the Root of the Equation.
      
      Please note that Root of the Non Linear Equation calculated using this method may be Different for Different Initial Value chosen for Variable \(x\).
      
      Also note that if anytime during repeated evaluation of expression (1) above the Value of \(F'(v)\) becomes 0, it means that the Chosen Intitial Value of Variable \(x\) Cannot be used for finding the Root of the Given Equation and a Different Initial Value of Variable \(x\) Must be used .
2. Given \(N\) Variables \(x_1, x_2, ..., x_N\) and associated \(N\) number of Non Linear System of Equations \(F_1(x_1, x_2, x_3, ..., x_N)=0\), \(F_2(x_1, x_2, x_3, ..., x_N)=0\), ..., \(F_N(x_1, x_2, x_3, ..., x_N)=0\), the corresponding One or more Set(s) of Roots can be found out using the Newton Raphson Method using the following steps
   1. Arrange the Functions \(F_1, F_2, ..., F_N\) in form of a Vector \(\vec{F}\) as follows
      
      \(\vec{F} = \begin{bmatrix}F_1(x_1, x_2, ..., x_N) \\ F_2(x_1, x_2, ..., x_N) \\ \vdots \\ F_N(x_1, x_2, ..., x_N)\end{bmatrix}\)
   2. Calculate the Jacobian Matrix corresponding to Functions \(F_1, F_2, ..., F_N\) as follows
      
      \(J=\begin{bmatrix}{\Large \frac{\partial F_1}{\partial x_1}} & {\Large \frac{\partial F_1}{\partial x_2}} & \cdots & {\Large \frac{\partial F_1}{\partial x_N}} \\ {\Large \frac{\partial F_2}{\partial x_1}} & {\Large \frac{\partial F_2}{\partial x_2}} & \cdots & {\Large \frac{\partial F_2}{\partial x_N}} \\ \vdots & \vdots & \ddots & \vdots \\ {\Large \frac{\partial F_N}{\partial x_1}} & {\Large \frac{\partial F_N}{\partial x_2}} & \cdots & {\Large \frac{\partial F_N}{\partial x_N}} \end{bmatrix}\)
   3. Select any Arbitrary Set of Values \(v_1, v_2, \cdots, v_N\) as Initial Values for Variables \(x_1, x_2, ..., x_N\) and repeatedly evaluate the following expression
      
      \(\vec{v} = \vec{v} - {J(v)}^{-1}\vec{F(v)}\)   ...(2)
      
      where \(\vec{v} = \begin{bmatrix}v_1\\ v_2 \\ \vdots \\ v_N\end{bmatrix}\), \(\vec{F(v)} = \begin{bmatrix}F_1(v_1, v_2, ..., v_N) \\ F_2(v_1, v_2, ..., v_N) \\ \vdots \\ F_N(v_1, v_2, ..., v_N)\end{bmatrix}\), and \(J(v)\) is the Jacobian Matrix evaluated for Variables \(v_1, v_2, \cdots, v_N\) , till \(\vec{F(v)}\) evaluates to a Null Vector upto the required precision. The corresponding values of \(v_1, v_2, ..., v_N\) in Vector \(\vec{v}\) are the Roots of Given Set of the Equations.
      
      Please note that Roots calculated using this method may be Different for Different Set of Initial Values chosen for Variables \(x_1, x_2, ..., x_N\).
      
      Also note that if anytime during repeated evaluation of expression (2) above the Value of the Determinant of \(J(v)\) becomes 0, it means that the Chosen Set of Intitial Values of Variables \(x_1, x_2, ..., x_N\) Cannot be used for finding the Roots of the Given System of Equation Equation and a Different Initial Value of Altleast One Variable Must be used .