MHB How to Solve Non-Linear Equations of 3 Variables Using Newton-Raphson Method?

[ra_forever8]

The three non-linear equations are given by
\begin{equation}
c[(6.7 * 10^8) + (1.2 * 10^8)s+(1-q)(2.6*10^8)]-0.00114532=0
\end{equation}
\begin{equation}
s[2.001 *c + 835(1-q)]-2.001*c =0
\end{equation}
\begin{equation}
q[2.73 + (5.98*10^{10})c]-(5.98 *10^{10})c =0
\end{equation}
Using the Newton-Raphson Method solve these equations in terms of $c$,$s$ and $q$.=> It is really difficult question for me because i don't know very much about the Newton-Raphson Method and also these non-linear equations contain 3 variables.I have try by applying the Newton-Raphson method to each equations:-
\begin{equation}
f(c,s,q)=0= c[(6.7 * 10^8) + (1.2 * 10^8)s+(1-q)(2.6*10^8)]-0.00114532
\end{equation}
\begin{equation}
g(c,s,q)=0= s[2.001 *c + 835(1-q)]-2.001*c
\end{equation}
\begin{equation}
h(c,s,q)=0= q[2.73 + (5.98*10^{10})c]-(5.98 *10^{10})c
\end{equation}
now i guess i need to work out $f'(c,s,q), g'(c,s,q), h'(c,s,q)$ but i don't know how?and after working out $f'(c,s,q), g'(c,s,q), h'(c,s,q)$ . After that i think i need to use Newton-raphson iteration:$c_{n+1}= c_n - \frac{f(c,s,q)}{f'(c,s,q)}$ but the $f(c,s,q)$ and $f'(c,s,q)$ contains the $s$ and $q$.

Similarly, for $s_{n+1}= s_n - \frac{g(c,s,q)}{g'(c,s,q)}$ will have $g(c,s,q)$ and $g'(c,s,q)$ containing the $c$ and $q$.$q_{n+1}= q_n - \frac{h(c,s,q)}{h'(c,s,q)}$ will have $h(c,s,q)$ and $h'(c,s,q)$ containing the $c$.so am i not sure what to do please help me. to find the values of $c,s,q$.

 

[zzephod]

The three non-linear equations are given by
\begin{equation}
c[(6.7 * 10^8) + (1.2 * 10^8)s+(1-q)(2.6*10^8)]-0.00114532=0
\end{equation}
\begin{equation}
s[2.001 *c + 835(1-q)]-2.001*c =0
\end{equation}
\begin{equation}
q[2.73 + (5.98*10^{10})c]-(5.98 *10^{10})c =0
\end{equation}
Using the Newton-Raphson Method solve these equations in terms of $c$,$s$ and $q$.=> It is really difficult question for me because i don't know very much about the Newton-Raphson Method and also these non-linear equations contain 3 variables.I have try by applying the Newton-Raphson method to each equations:-
\begin{equation}
f(c,s,q)=0= c[(6.7 * 10^8) + (1.2 * 10^8)s+(1-q)(2.6*10^8)]-0.00114532
\end{equation}
\begin{equation}
g(c,s,q)=0= s[2.001 *c + 835(1-q)]-2.001*c
\end{equation}
\begin{equation}
h(c,s,q)=0= q[2.73 + (5.98*10^{10})c]-(5.98 *10^{10})c
\end{equation}
now i guess i need to work out $f'(c,s,q), g'(c,s,q), h'(c,s,q)$ but i don't know how?and after working out $f'(c,s,q), g'(c,s,q), h'(c,s,q)$ . After that i think i need to use Newton-raphson iteration:$c_{n+1}= c_n - \frac{f(c,s,q)}{f'(c,s,q)}$ but the $f(c,s,q)$ and $f'(c,s,q)$ contains the $s$ and $q$.

Similarly, for $s_{n+1}= s_n - \frac{g(c,s,q)}{g'(c,s,q)}$ will have $g(c,s,q)$ and $g'(c,s,q)$ containing the $c$ and $q$.$q_{n+1}= q_n - \frac{h(c,s,q)}{h'(c,s,q)}$ will have $h(c,s,q)$ and $h'(c,s,q)$ containing the $c$.so am i not sure what to do please help me. to find the values of $c,s,q$.

Write your system as a vector equation in a vector variable, you want a solution of the system:
$${\bf{f}}({\bf{x}})={\bf{0}}$$

The NR iteration for this is:

$${\bf{x}}_{n+1}={\bf{x}}_n-[{\rm{J}}({\bf{x}}_n)]^{-1}{\bf{f}}({\bf{x_n}})$$

where ${\rm{J}}({\bf{x}})$ is the matrix with $i.j$ th element equal to $ \dfrac{\partial {\bf{f}}_i}{\partial{\bf{x}}_j}$