Best way to numericallty solve this system of equations

[Ax_xiom]

TL;DR

I have a system of equations that I need solving numerically

So I'm working on a project where I am trying to work out what the ideal sizes of rocket stages are and I am using Excel to allow the user to interact with this quickly. The method I am using is derived from this video, and I end up with this system of equations: \begin{align}
1 - c_{1}\lambda + \frac{sN_{1}}{1 - sN_{1}} = 0 \\
1 - c_{2}\lambda + \frac{sN_{2}}{1 - sN_{2}} = 0 \\
c_{1}\ln(N_{1}) + c_{2}\ln(N_{2}) - V_{f} = 0 \end{align}
Where I am trying to solve for $N_{1}$ and $N_{2}$ The number of equations like (1) and (2) can be as many as 5, and the number of $c_{n}\ln(N_{n})$ terms in (3) is also variable. As far as I know, these equations cannot be solved analytically, so they must be solved numerically. Here are the options that I've considered:

1. Newton's method with the entire system
   • Pros:
     • Easiest to think about conceptually
     • Converges quickly from a reasonable starting point
   • Cons:
     • Hard to implement a variable amount of stages
     • More work to set up
     • Have to consider the order of the columns and rows of the Jacobian
2. Solve for $\lambda$ and $N_{n}$ after
   • It is possible to show that $N_{n} = \frac{1-\frac{1}{c_{n}\lambda}}{s}$. Using this, it is possible to substitute these into equation (3) and numerically solve for $\lambda$ before solving for $N_{n}$ using the aforementioned expression.
   • Pros:
     • Less computation
     • Easier to adjust for different amounts of stages
   • Cons:
     • The resulting expression is extremely badly behaved, diverging to $-\infty$ when using Newton-Raphson unless given an extremely close guess

So what method should I use? Have I overlooked any methods that I may be better off using?

 

[anuttarasammyak]

I would try let computer draw three graphs (1)(2)(3) and observe crossing points, if not ask AI the solutions direct.

 

[Ax_xiom]

I would try let computer draw three graphs (1)(2)(3) and observe crossing points.

A few problems:

1. I want to implement up to 5 stages, which would result in 6 equations and 6 variables that would need to be graphed which would require a 6D space to visualise
2. I'm not sure how easy that would be to use or how accurate it would be
3. Can Excel even draw 3D graphs?

 

[anuttarasammyak]

There are six kind of alphabets in your equations. 3 equations for 6 unknowns seems desperate. Which are numbers given for numerical calculation and which are parameters for which we would like to get solutions ?

 

[renormalize]

Which are numbers given for numerical calculation and which are parameters for which we would like to get solutions ?

From skimming through the video referenced by the OP, the known rocket parameters are:

• $c_n$ is the exhaust velocity of the $n^\text{th}-$stage rocket motor.
• $s_n$ is the $n^\text{th}-$stage structural factor (fraction of the total stage-mass that is not fuel).
• $V_f$ is the desired final velocity of the last stage.

Unknowns to be solved for:

• $N_n$ are mass ratios of the various stages. These values ultimately determine the mass of fuel in each stage that's required to achieve the final velocity.
• $\lambda$ is a Lagrange-multiplier parameter.

But the OP is essentially reinventing the wheel: there's already a nice online calculator that already does what they want: https://space.geometrian.com/calcs/opt-multi-stage.php.