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The [[https://apmonitor.com|APMonitor Modeling Language]] with a Python interface is optimization software for mixed-integer and differential algebraic equations. It is coupled with large-scale solvers for linear, quadratic, nonlinear, and mixed integer programming (LP, QP, NLP, MILP, MINLP). Modes of operation include data reconciliation, real-time optimization, dynamic simulation, and nonlinear predictive control. It is freely available through MATLAB, Python, Julia, or from a [[https://apmonitor.com/online/view_pass.php|web browser interface]].

where ''T'' is the temperature, ''V'' is the molar volume, {`R_g`} is the universal gas constant, and ''a'' and ''b'' are compound-specific constants.  The solution for the molar volume of ethane for each phase at ''T = 77°C'' and ''P = 1 bar'' is shown below with [[https://youtu.be/EIgP6Q20xm8|Scipy fsolve]] and [[https://apmonitor.com/wiki/index.php/Main/GekkoPythonOptimization|Gekko]].  For ethane, ''a = 2.877e8 cm^6 K^0.5 bar / mol^2 and ''b = 60.211 cm^3 / mol''. The ideal gas law is a guess of the vapor volume and ''1.1*b'' is a guess of the liquid volume.

where ''T'' is the temperature, ''V'' is the molar volume, {`R_g`} is the universal gas constant, and ''a'' and ''b'' are compound-specific constants.  The solution for the molar volume of ethane for each phase at ''T = 77°C'' and ''P = 1 bar'' is shown below with [[https://youtu.be/EIgP6Q20xm8|Scipy fsolve]] and [[https://apmonitor.com/wiki/index.php/Main/GekkoPythonOptimization|Gekko]].  For ethane, ''a = 2.877e8'' {`\frac{cm^6 K^{0.5} bar}{mol^2}`} and ''b = 60.211'' {`\frac{cm^3}{mol}`}. The ideal gas law is a guess of the vapor volume and ''1.1*b'' is a guess of the liquid volume.

where ''T'' is the temperature, ''V'' is the molar volume, {`R_g`} is the universal gas constant, and ''a'' and ''b'' are compound-specific constants.  The solution for the molar volume of ethane for each phase at ''T = 77°C'' and ''P = 1 bar'' is shown below with [[https://youtu.be/EIgP6Q20xm8|Scipy fsolve]] and [[https://apmonitor.com/wiki/index.php/Main/GekkoPythonOptimization|Gekko]].  For ethane, ''a = 2.877e8'' {`\frac{cm^6-bar-K^{0.5}}{mol^2}`} and ''b = 60.211'' {`\frac{cm^3}{mol}`}. The ideal gas law is a guess of the vapor volume and ''1.1*b'' is a guess of the liquid volume.

where ''T'' is the temperature, ''V'' is the molar volume, {`R_g`} is the universal gas constant, and ''a'' and ''b'' are compound-specific constants.  The solution for the molar volume of ethane for each phase at ''T = 77°C'' and ''P = 1 bar'' is shown below with Scipy fsolve and [[https://apmonitor.com/wiki/index.php/Main/GekkoPythonOptimization|Gekko]].  For ethane, ''a = 2.877×10^8'' {`\frac{cm^6 bar K^{0.5}}{mol^2}`} and ''b = 60.211'' {`\frac{cm^3}{mol}`}. The ideal gas law is used to obtain the guess of the vapor volume and ''1.1*b'' is used to obtain the guess for the liquid volume.

{$-2*x+y=-z$}

(:source lang=python:)

!!!!Source Code for Nonlinear Solution

from numpy import *
from scipy.optimize import *

  
   F = empty((3))
  F[0] = pow(x,2)+pow(y,2)-20
  F[1] = y - pow(x,2)

zGuess = array([1,1,1])

!!!! Symbolic Solutions

h = sym.Eq(-2*x+y,-z)sourceend:)

{$z=c_1 - \frac{5}{2} +\frac{5 \sqrt{3}}{2}$}

{$z=c_1 - \frac{5}{2} -\frac{5 \sqrt{3}}{2}$}

{$-\frac{1}{2}+\frac{\sqrt{3}}{2}$}

{$-\frac{1}{2}+\frac{\sqrt(3)}{2}$}

The [[https://apmonitor.com|APMonitor Modeling Language]] is optimization software for mixed-integer and differential algebraic equations. It is coupled with large-scale solvers for linear, quadratic, nonlinear, and mixed integer programming (LP, QP, NLP, MILP, MINLP). Modes of operation include data reconciliation, real-time optimization, dynamic simulation, and nonlinear predictive control. It is freely available through MATLAB, Python, Julia, or from a [[https://apmonitor.com/online/view_pass.php|web browser interface]].

 import numpy as np

 A = np.array([ [3,-9], [2,4] ])
 b = np.array([-42,2])
 z = np.linalg.solve(A,b)
 print(z)

 M = np.array([ [1,-2,-1], [2,2,-1], [-1,-1,2] ])
 c = np.array([6,1,1])
 y = np.linalg.solve(M,c)
 print(y)

 from numpy import *
 from scipy.optimize import *

 def myFunction(z):
   x = z[0]
   y = z[1]
   w = z[2]

    F = empty((3))
   F[0] = pow(x,2)+pow(y,2)-20
   F[1] = y - pow(x,2)
   F[2] = w + 5 - x*y
   return F

 zGuess = array([1,1,1])
 z = fsolve(myFunction,zGuess)
 print(z)

Linear and nonlinear equations can also be solved with [[Main/ExcelSolveEquations|Excel]] and [[Main/PythonSolveEquations|Python]]. Click on the appropriate link for additional information and source code.

 A = np.array([[3,-9],[2,4]])

 M = np.array([[1,-2,-1],[2,2,-1],[-1,-1,2]])

!!!!Source Code