Tutorial Release 10.4 The Sage Development Team https://doc.sagemath.org/pdf/en/tutorial/sage_tutorial.pdf

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Blaec (CSO) Tasks and Strategy Outline

tutorials/CoCalc's Official Sage Tutorial / Chapter Two / Chapter 2-4-1: Solving Equations (Continued).ipynb

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SageMath Example: Solving Non-Linear Systems

The following example of using Sage to solve a system of non-linear equations was provided by Jason Grout:

Solve the System Symbolically

First, we solve the system symbolically.

Declare the variables:

In [4]:

var('x y p q')

Out[4]:

(x, y, p, q)

Define the equations:

In [5]:

eq1 = p + q == 9
eq2 = q * y + p * x == -6
eq3 = q * y^2 + p * x^2 == 24

Solve the system of equations:

In [6]:

solve([eq1, eq2, eq3, p == 1], p, q, x, y)

Out[6]:

[[p == 1, q == 8, x == -4/3*sqrt(10) - 2/3, y == 1/6*sqrt(10) - 2/3], [p == 1, q == 8, x == 4/3*sqrt(10) - 2/3, y == -1/6*sqrt(10) - 2/3]]

Numerical Approximations of the Solutions

For numerical approximations of the solutions, you can use the following approach:

Solve the system of equations:

In [7]:

solns = solve([eq1, eq2, eq3, p == 1], p, q, x, y, solution_dict=True)

Print the numerical solutions:

In [8]:

[[s[p].n(30), s[q].n(30), s[x].n(30), s[y].n(30)] for s in solns]

Out[8]:

[[1.0000000, 8.0000000, -4.8830369, -0.13962039],
 [1.0000000, 8.0000000, 3.5497035, -1.1937129]]

Solving Equations Numerically

Often times, solve will not be able to find an exact solution to the equation or equations specified. When it fails, you can use find_root to find a numerical solution. For example, solve does not return anything interesting for the following equation:

Declare the variable $\theta$ :

In [9]:

theta = var('theta')

Solve the equation $\cos(\theta) = \sin(\theta)$ :

In [10]:

solve(cos(theta) == sin(theta), theta)

Out[10]:

[sin(theta) == cos(theta)]

On the other hand, we can use find_root to find a solution to the above equation in the range $0 < \phi < \frac{\pi}{2}$ :

Declare the variable $\phi$ :

In [11]:

phi = var('phi')

Find the numerical root:

In [12]:

find_root(cos(phi) == sin(phi), 0, pi/2)

Out[12]:

/ext/sage/10.4/local/var/lib/sage/venv-python3.12.4/lib/python3.12/site-packages/scikits/__init__.py:1: DeprecationWarning: pkg_resources is deprecated as an API. See https://setuptools.pypa.io/en/latest/pkg_resources.html
  __import__("pkg_resources").declare_namespace(__name__)
/ext/sage/10.4/local/var/lib/sage/venv-python3.12.4/lib/python3.12/site-packages/scikits/__init__.py:1: DeprecationWarning: Deprecated call to `pkg_resources.declare_namespace('scikits')`.
Implementing implicit namespace packages (as specified in PEP 420) is preferred to `pkg_resources.declare_namespace`. See https://setuptools.pypa.io/en/latest/references/keywords.html#keyword-namespace-packages
  __import__("pkg_resources").declare_namespace(__name__)

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SageMath Example: Solving Non-Linear Systems

Solve the System Symbolically

Numerical Approximations of the Solutions

Solving Equations Numerically

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