edited paper

README.md

@@ -42,4 +42,6 @@ $$[C]_{tot} = [C] \+ [AB_2C] \label{Eq. 5}$$
 
 We can define a system comprising these equations to be simultaneously solved.
 In this system, we can see that chemical equilibria consist of nonlinear functions, meaning that they cannot be expressed as a sum of their variables each raised to the power of one.
-`equpy` solves this problem by linearizing these equations to make them suitable to be solved employing linear algebra and an iterative numerical method equivalent to the Newton search of the logarithmic equations over the logarithm of the variables.
+`equpy` solves this problem by linearizing these equations to make them suitable to be solved employing linear algebra and an iterative numerical method equivalent to the Newton search of the logarithmic equations over the logarithm of the variables.
+
+# Example

paper.md

@@ -153,7 +153,7 @@ W_4\ln[B] + W_5\ln[AB_2] + W_6\ln[AB_2C] = \ln \Bigl\{[B]_{tot}* \Bigl( \frac{W_
 W_7\ln[C] + W_8\ln[AB_2C] = \ln \Bigl\{[C]_{tot}* \Bigl( \frac{W_7}{1} \Bigr)^{W_7}*\Bigl(\frac{W_8}{1}\Bigr)^{W_8}\Bigr\}
 \end{equation}
 
-With $W_1 ... W_8$ equal to:
+With $W_{1...8}$ equal to:
 \begin{equation} \nonumber
 W_1 = \frac{[A]}{[A]+[AB_2]+[AB_2C]}
 \end{equation}
@@ -193,6 +193,8 @@ K_5 = [C]_{tot}* \Bigl( \frac{W_7}{1} \Bigr)^{W_7}*\Bigl(\frac{W_8}{1}\Bigr)^{W_
 And finally put all linearized equations in a matrix form suitable for our implementation, so that the new set of equations in linearized form can be expressed as follows:
 ![Fig.1 Linear problem expressed in matrix form. \label{fig:1}](Fig1.png)
 
+With **r** being the vector of the residuals, calculated as **r** = **Mx** – **y**, whose norm can be used to probe the error in the approximate solution. Since $W_{1…8}$, $K_3$, $K_4$ and $K_5$ are functions of our unknown solutions, **M** and **y** must be calculated using an initial guess of **x**, and the system solved for a new vector of solutions that can used for the next iteration until a satisfying result is reached.
+
 # Acknowledgements
 
 M.T. acknowledges the support of HHMI and Jack W. Szostak, the authors would like to thank Aleksandar Radakovic for introducing them to the problem of multiple equilibria at the genesis of this project.