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Solving simultaneous equations with sympy
'''''''''''''''''''''''''''''''''''''''''

This document is a tutorial for how to use the Python module `sympy` to solve simultaneous equations. Since sympy does this so well, there is no need to implement it within `reliability`, but users may find this tutorial helpful as problems involving physics of failure will often require the solution of simultaneous equations. `sympy` is not installed by default when you install `reliability` so users following this tutorial will need to ensure `sympy` is installed on their machine. The following three examples should be sufficient to illustrate how to use `sympy` for solving simultaneous equations. Further examples are available in the `sympy documentation <https://docs.sympy.org/latest/modules/solvers/solvers.html>`_.

**Example 1**

:math:`\text{Eqn 1:} \hspace{11mm} x + y = 5` 

:math:`\text{Eqn 2:} \hspace{11mm} x^2 + y^2 = 17`

Solving with sympy:

.. code:: python

    import sympy as sym
    x,y = sym.symbols('x,y')
    eq1 = sym.Eq(x+y,5)
    eq2 = sym.Eq(x**2+y**2,17)
    result = sym.solve([eq1,eq2],(x,y))
    print(result)

    '''
    [(1, 4), (4, 1)] #these are the solutions for x,y. There are 2 solutions because the equations represent a line passing through a circle.
    '''

**Example 2**

:math:`\text{Eqn 1:} \hspace{11mm} a1000000^b = 119.54907` 

:math:`\text{Eqn 2:} \hspace{11mm} a1000^b = 405`

Solving with sympy:

.. code:: python

    import sympy as sym
    a,b = sym.symbols('a,b')
    eq1 = sym.Eq(a*1000000**b,119.54907)
    eq2 = sym.Eq(a*1000**b,405)
    result = sym.solve([eq1,eq2],(a,b))
    print(result)

    '''
    [(1372.03074854535, -0.176636273742481)] #these are the solutions for a,b
    '''

**Example 3**

:math:`\text{Eqn 1:} \hspace{11mm} 2x^2 +y + z = 1` 

:math:`\text{Eqn 2:} \hspace{11mm} x + 2y + z = c_1`

:math:`\text{Eqn 3:} \hspace{11mm} -2x + y = -z`

The actual solution to the above set of equations is:

:math:`\hspace{21mm} x = -\frac{1}{2}+\frac{\sqrt{3}}{2}` 

:math:`\hspace{21mm} y = c_1 - \frac{3\sqrt{3}}{2}+\frac{3}{2}` 

:math:`\hspace{21mm} z = -c_1 - \frac{5}{2}+\frac{5\sqrt{3}}{2}` 

and a second solution:

:math:`\hspace{21mm} x = -\frac{1}{2}-\frac{\sqrt{3}}{2}` 

:math:`\hspace{21mm} y = c_1 + \frac{3\sqrt{3}}{2}+\frac{3}{2}` 

:math:`\hspace{21mm} z = -c_1 - \frac{5}{2}-\frac{5\sqrt{3}}{2}` 

Solving with sympy:

.. code:: python

    import sympy as sym
    x,y,z = sym.symbols('x,y,z')
    c1 = sym.Symbol('c1')
    eq1 = sym.Eq(2*x**2+y+z,1)
    eq2 = sym.Eq(x+2*y+z,c1)
    eq3 = sym.Eq(-2*x+y,-z)
    result = sym.solve([eq1,eq2,eq3],(x,y,z))
    print(result)

    '''
    [(-1/2 + sqrt(3)/2, c1 - 3*sqrt(3)/2 + 3/2, -c1 - 5/2 + 5*sqrt(3)/2), (-sqrt(3)/2 - 1/2, c1 + 3/2 + 3*sqrt(3)/2, -c1 - 5*sqrt(3)/2 - 5/2)]
    '''

.. note:: If you are using an iPython notebook, the display abilities are much better than the command line interface, so you can simply add sym.init_printing() after the import line and your equations should be displayed nicely.

A special thanks to Brigham Young University for offering `this tutorial <https://apmonitor.com/che263/index.php/Main/PythonSolveEquations>`_.