Intro to SymPy

Here we give a (quick) introduction to SymPy. This tutorial assumes you are already familiar with SymPy expressions, so this notebook should serve as a refresher.

Expressions

The function init_printing() will enable LaTeX pretty printing in the notebook for SymPy expressions. We recommend calling it at the top of any notebook that uses SymPy.

SymPy symbols are created with the symbols() function. SymPy expressions are built up from symbols, numbers, and SymPy functions,

SymPy automatically pretty prints symbols with greek letters and subscripts.

Exercise

Write a symbolic expression for $$\frac{1}{\sqrt{2\pi\sigma^2} } \; e^{ -\frac{(x-\mu)^2}{2\sigma^2} }.$$ Remember that the function for $e^x$ is exp(x). You will need to create symbols for sigma and mu. Square root is sqrt.

Some reminders on common gotchas:

Dividing two integers in Python creates a float, like 1/2 -> 0.5. If you want a rational number, use Rational(1, 2) or S(1)/2.

^ is the XOR operator. Use ** for powers.

All SymPy expressions are immutable. Functions that operate on an expression return a new expression.

Floating point representation

Since most languages targeted will not support symbolic representation it is useful to let SymPy evaluate a floating point approximation (up to a user specified number of digits).

Exercise

Compute 100 digits of $\pi$.

Undefined functions

Undefined functions are created with Function(). Undefined are useful to state that one variable depends on another (for the purposes of differentiation).

Derivatives

Derivatives are computed with the diff() function, using the syntax diff(expr, var1, var2, ...).

Exercise

Write an expression representing the wave equation in one dimension: $${\partial^2 u\over \partial t^2 } = c^2 { \partial^2 u\over \partial x^2}.$$ Remember that $u$ is a function in two variables. You can represent an equation using Eq, like

Matrices

Matrices are created with Matrix. A matrix can contain any symbolic expression. See Matrix? for different ways to create a Matrix.

Matrices support all common operations, and have many methods for performing operations.

for example, calculating the Jacobian matrix is as easy as:

and for those of you who don't remember, the Jacobian is defined as:

$$ J = \begin{bmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & ~ \\ \vdots & ~ & \ddots \\ \end{bmatrix} $$

you will come across this mathematical entity in later notebooks in this tutorial.

Exercise

Create the following matrix $$\left[\begin{matrix}1 & 0 & 1\\-1 & 2 & 3\\1 & 2 & 3\end{matrix}\right]$$

Now create a matrix representing $$\left[\begin{matrix}x\\y\\z\end{matrix}\right]$$ and multiply it with the previous matrix to get $$\left[\begin{matrix}x + z\\- x + 2 y + 3 z\\x + 2 y + 3 z\end{matrix}\right].$$

Now take the Jacobian of that matrix with respect to your column vector, to get the original matrix back.

Matrix symbols

SymPy can also operate on matrices of symbolic dimension ($n \times m$). MatrixSymbol("M", n, m) creates a matrix $M$ of shape $n \times m$.

Some matrix expression functions do not evaluate unless you call doit.

Exercise

Write a matrix expression representing $$Au + Bv,$$ where $A$ and $B$ are $100\times 100$ and $u$ and $v$ are $100 \times 1$.

Indexed

Indexed symbols can be created with IndexedBase and Idx. IndexedBase("A") represents an array A and Idx('i') represents an index i. You can also use symbols('i') instead of Idx('i').

Exercise

Write an Indexed expression for $$A[i, j, k]$$

Solving systems of equations

A useful tool in your toolbelt when manipulating expressions is the solve function. solve solves equations symbolically (not numerically). The return value is a list of solutions.

You can give solve an Eq, or if you give it an expression, it automatically assumes that it is equal to 0.

It can also handle systems of equations. The return is a list of dictionaries, mapping symbols to solutions.

Exercises

Solve the following system of equations: $$\begin{align}z &= x^2 - y^2\\z^2 &= x^2 + y^2 + 4\\z &= x + y\end{align}$$

Solving differential equations

If you are dealing with a differential equation, say:

$$ f''(x) = -f(x) $$

SymPy's dsolve can (sometimes) produce an exact symbolic solution. Like solve, dsolve assumes that expressions are equal to 0.

Exercise

Solve the following ODE: $$f''(x) + 2f'(x) + f(x) = \sin(x)$$