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Solve Equations in Python

Main.PythonSolveEquations History

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Changed line 72 from:
h = sym.Eq(-2*x+y,-z)sourceend:)
to:
h = sym.Eq(-2*x+y,-z)
Changed lines 81-82 from:
{$z=c_1 - \frac{5}{2} +\frac{5 \sqrt{3}}{2}$}
to:
{$z=-c_1 - \frac{5}{2} +\frac{5 \sqrt{3}}{2}$}
Changed lines 87-89 from:
{$z=c_1 - \frac{5}{2} -\frac{5 \sqrt{3}}{2}$}
to:
{$z=-c_1 - \frac{5}{2} -\frac{5 \sqrt{3}}{2}$}

The same approach applies to linear or nonlinear equations.
Changed lines 79-87 from:
{$-\frac{1}{2}+\frac{\sqrt{3}}{2}$}
to:
{$x=-\frac{1}{2}+\frac{\sqrt{3}}{2}$}
{$y=c_1 - \frac{3 \sqrt{3}}{2} +\frac{3}{2}$}
{$z=c_1 - \frac{5}{2} +\frac{5 \sqrt{3}}{2}$}

and a second solution:

{$x=-\frac{1}{2}-\frac{\sqrt{3}}{2}$}
{$y=c_1 + \frac{3 \sqrt{3}}{2} +\frac{3}{2}$}
{$z=c_1 - \frac{5}{2} -\frac{5
\sqrt{3}}{2}$}
Changed line 79 from:
{$-\frac{1}{2}+\frac{\sqrt(3)}{2}$}
to:
{$-\frac{1}{2}+\frac{\sqrt{3}}{2}$}
Added lines 56-79:

!!!! Symbolic Solutions

Sympy is a package for symbolic solutions in Python that can be used to solve systems of equations.

{$2x^2+y+z=1$}
{$x+2y+z=c_1$}
{$-2*x+y=-z$}

(:source lang=python:)
import sympy as sym
sym.init_printing()
x,y,z = sym.symbols('x,y,z')
c1 = sym.Symbol('c1')
f = sym.Eq(2*x**2+y+z,1)
g = sym.Eq(x+2*y+z,c1)
h = sym.Eq(-2*x+y,-z)sourceend:)

sym.solve([f,g,h],(x,y,z))
(:sourceend:)

When solved in an IPython environment such as a Jupyter notebook, the result is displayed as:

{$-\frac{1}{2}+\frac{\sqrt(3)}{2}$}
Changed line 61 from:
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]].
to:
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]].
Added lines 60-61:

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]].
March 07, 2017, at 01:05 PM by 45.56.3.173 -
Changed lines 15-26 from:
 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)
to:
(:source lang=python:)
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)
(:sourceend:
)
Changed lines 37-43 from:
 from numpy import *
 from scipy.optimize import *

 def myFunction(z):
   x = z[0]
   y = z[1]
   w = z[2]
to:
(:source lang=python:)
from numpy import *
from scipy.optimize import *

def myFunction(z):
  x = z[0]
  y = z[1]
  w = z[2]
Changed lines 46-54 from:
    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)
to:
   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)
(:sourceend:
)
Changed line 55 from:
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.
to:
Linear and nonlinear equations can also be solved with [[Main/ExcelSolveEquations|Excel]] and [[Main/MatlabSolveEquations|MATLAB]]. Click on the appropriate link for additional information and source code.
Changed line 17 from:
 A = np.array([[3,-9],[2,4]])
to:
 A = np.array([ [3,-9], [2,4] ])
Changed line 22 from:
 M = np.array([[1,-2,-1],[2,2,-1],[-1,-1,2]])
to:
 M = np.array([ [1,-2,-1], [2,2,-1], [-1,-1,2] ])
Added lines 13-26:
!!!!Source Code for Linear Solutions

 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)

Changed line 33 from:
!!!!Source Code
to:
!!!!Source Code for Nonlinear Solution
Added lines 1-60:
(:title Solve Equations in Python:)
(:keywords Python, solve equations, linear, nonlinear:)
(:description Python tutorial on solving linear and nonlinear equations with matrix operations (linear) or fsolve NumPy(nonlinear):)

The following tutorials are an introduction to solving linear and nonlinear equations with Python. The solution to linear equations is through matrix operations while sets of nonlinear equations require a solver to numerically find a solution.

!!!!Solve Linear Equations with Python

(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/44pAWI7v5Zk" frameborder="0" allowfullscreen></iframe>
(:htmlend:)

!!!!Solve Nonlinear Equations with Python

(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/S4Qg2CsiIj8" frameborder="0" allowfullscreen></iframe>
(:htmlend:)

!!!!Source Code

 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)

!!!!Additional Tutorials

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.

----

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