Solving set of non-linear equations with two unknowns in MATLAB

[bilalcisco]

Hi,
I have two non-linear equations with two unknowns, i.e., tau and p. both equations are:
p=1-(1-tau).^(n-1)
and
tau - 2*(1-2*p) ./ ( (1-2*p)*(W+1)+(p*W).*(1-(2*p).^m)).

I am interested to find the value of tau.
After doing some research on internet I came to know that these equations can be solved by finding roots and finding fixed points. However, the problem is not that straight as it involves two non-linear equations, as opposed to various examples I found on internet which involves only one non-linear equation.
Additionally, I have MATLAB code for solving this problem, but still spending few days to understand and searching internet relentlessly, I couldn't understand how this solution actually works. Below I am giving that MATLAB code and need your helping hand to explain it to me the actual logic behind solving 'set of non-linear equations.

Matlab M-file is:

function result=tau_eq(tau)

n=6;
W=32;
m=5;

p=1-(1-tau).^(n-1);
result=tau - 2*(1-2*p) ./ ( (1-2*p)*(W+1)+(p*W).*(1-(2*p).^m));

Command at the command window:
result=fzero(@tau_eq,[0,1],[])
output is:
result =

0.0448

The given result is satisfactory, however I do not understand the logic behind it. Any explanation or referring to useful resources will be highly appreciated.

 

[lippyka]

Thanks in advance. The given Matlab code is an example of using a numerical root-finding algorithm to solve a system of non-linear equations. The function "fzero" is used to find a root (in this case, the root is tau) of a function (in this case, the function is given by the equations you provided). The input of the fzero function is the equation (in this case, it is the @tau_eq function) and the interval ([0,1]) in which the root is expected to lie. The output is the value of tau that satisfies the system of equations. Root finding algorithms are iterative methods which use an initial guess for the root and iteratively refine it until the desired accuracy is achieved. The process of refinement is based on the derivatives of the equations (in this case, the Jacobian matrix of the system of equations) and can be thought of as a method for minimizing the error between the equations and their solutions. This process is repeated until the error is sufficiently small and the solution is accepted as the root. There are several root-finding algorithms available and they differ in terms of their efficiency, accuracy, and robustness. In this case, the fzero function is using the Brent's method, which is a combination of the bisection and secant methods. For more information, you can refer to the Mathworks website: https://www.mathworks.com/help/matlab/ref/fzero.html