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

In summary, the conversation discusses three non-linear equations and the use of the Newton-Raphson Method to solve for the variables $c$, $s$, and $q$. The method involves finding the derivatives of the equations and using an iteration formula to solve for the values of the variables.
  • #1
ra_forever8
129
0
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 $c,s,q$.
 
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  • #2
grandy said:
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}$
 

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

1. How does the Newton-Raphson Method work?

The Newton-Raphson Method is an iterative process used to solve non-linear equations. It starts with an initial guess for the solution and uses the derivative of the equation to find the tangent line at that point. The root of the tangent line is then used as the next guess, and this process is repeated until the solution is found.

2. What is the role of the derivative in the Newton-Raphson Method?

The derivative of the equation is used to find the slope of the tangent line at the current guess. This slope helps to determine the next guess for the solution, making the method more efficient and accurate.

3. How do I choose an initial guess for the Newton-Raphson Method?

It is important to choose an initial guess that is close to the actual solution in order for the method to converge quickly. A good way to do this is by graphing the equation and estimating the solution.

4. Can the Newton-Raphson Method be used for any type of non-linear equation?

The Newton-Raphson Method is most commonly used for equations with one variable. However, it can also be extended to solve systems of non-linear equations with multiple variables, such as in the case of solving for c, s, and a in the given equation.

5. What are the advantages and disadvantages of using the Newton-Raphson Method?

The main advantage of the Newton-Raphson Method is its speed and efficiency in finding solutions to non-linear equations. However, it may not always converge to the correct solution and can be affected by the choice of initial guess. Additionally, it requires knowledge of the derivative of the equation, which may not always be available or easy to calculate.

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