from sympy import *
init_printing()
x, y, z = symbols('x,y,z')
Equation solving is both a common need also a common building block for more complicated symbolic algorithms.
Here we introduce the solveset function.
solveset(x**2 - 4, x)
Solveset takes two arguments and one optional argument specifying the domain, an equation like $x^2 - 4$ and a variable on which we want to solve, like $x$ and an optional argument domain specifying the region in which we want to solve.
Solveset returns the values of the variable, $x$, for which the equation, $x^2 - 4$ equals 0.
What would the following code produce? Are you sure?
solveset(x**2 - 4 == 0, x)
One of the major improvements of solveset over solve is that it also supports infinite solution.
solveset(sin(x), x)
solveset(exp(x) - 1, x)
solveset by default solves everything in the complex domain. In complex domain $exp(x) == cos(x) + i\ sin(x)$ and solution is basically equal to solution to $cos(x) == 1$. If you want only real solution, you can specify the domain as S.Reals.
solveset(exp(x) - 1, x, domain=S.Reals)
solveset isn't always able to solve a given equation, such cases it returns a ConditionSet object. ConditionSet represents a set satisfying a given condition.
solveset(exp(x) + cos(x) + 1, x, domain=S.Reals)
solveset aims to return all the solutions of the equation. In cases where it able to find some solution but not all it returns a union of the known solutions and ConditionSet.
solveset((x - 1)*(exp(x) + cos(x) + 1), x, domain=S.Reals)
solveset¶Results of solveset don't need to be numeric, like {-2, 2}. We can use solveset to perform algebraic manipulations. For example if we know a simple equation for the area of a square
area = height * width
we can solve this equation for any of the variables. For example how would we solve this system for the height, given the area and width?
height, width, area = symbols('height, width, area')
solveset(area - height*width, height)
Note that we would have liked to have written
solveset(area == height * width, height)
But the == gotcha bites us. Instead we remember that solveset expects an expression that is equal to zero, so we rewrite the equation
area = height * width
into the equation
0 = height * width - area
and that is what we give to solveset.
Compute the radius of a sphere, given the volume. Reminder, the volume of a sphere of radius r is given by
Assume r, V to be positive and domain to be Real.
%load_ext exercise
Use either the * %exercise * or * %load * magic to get the exercise / solution respecitvely (i.e. delete the whole contents of the cell except for the uncommented magic command). Replace ??? with the correct expression.
# %load exercise_volume.py
r, V = symbols(???)
solveset(???)
We often want to substitute in one expression for another. For this we use the subs method
x**2
# Replace x with y
(x**2).subs({x: y})
Subsitute $x$ for $sin(x)$ in the equation $x^2 + 2\cdot x + 1$
# Replace x with sin(x)
We can use subs and solveset together to plug the solution of one equation into another
# Solve for the height of a rectangle given area and width
soln = list(solveset(area - height*width, height))[0]
soln
# Define perimeter of rectangle in terms of height and width
perimeter = 2*(height + width)
perimeter
# Substitute the solution for height into the expression for perimeter
perimeter.subs({height: soln})
In the last section you solved for the radius of a sphere given its volume
V, r = symbols('V, r', real=True)
4*pi/3 * r**3
r_v = list(solveset(V - 4*pi/3 * r**3, r))[0]
r_v
Now lets compute the surface area of a sphere in terms of the volume. Recall that the surface area of a sphere is given by
$$ 4 \pi r^2 $$# %load exercise_surface.py
(4*pi*r**2).subs(???)
Does the expression look right? How would you expect the surface area to scale with respect to the volume? What is the exponent on $V$?
SymPy can plot expressions easily using the plot function. By default this links against matplotlib.
%matplotlib inline
plot(x**2, (x, -100, 100))
In the last exercise you derived a relationship between the volume of a sphere and the surface area. Plot this relationship using plot.
# %load exercise_plot_surface.py
plot(???)
You may know that SymPy tries to be a very low-dependency project. Our user base is very broad. Some entertaining aspects result. For example, textplot.
textplot(x**2, -3, 3)
Play with textplot and enjoy :)