Basic Operations
Contents
Basic Operations¶
Here we discuss some of the most basic operations needed for expression manipulation in SymPy. Some more advanced operations will be discussed later in the advanced expression manipulation section.
>>> from sympy import *
>>> x, y, z = symbols("x y z")
Substitution¶
One of the most common things you might want to do with a mathematical
expression is substitution. Substitution replaces all instances of something
in an expression with something else. It is done using the subs method.
For example
>>> expr = cos(x) + 1
>>> expr.subs(x, y)
cos(y) + 1
Substitution is usually done for one of two reasons:
Evaluating an expression at a point. For example, if our expression is
cos(x) + 1and we want to evaluate it at the pointx = 0, so that we getcos(0) + 1, which is 2.>>> expr.subs(x, 0) 2
Replacing a subexpression with another subexpression. There are two reasons we might want to do this. The first is if we are trying to build an expression that has some symmetry, such as \(x^{x^{x^x}}\). To build this, we might start with
x**y, and replaceywithx**y. We would then getx**(x**y). If we replacedyin this new expression withx**x, we would getx**(x**(x**x)), the desired expression.>>> expr = x**y >>> expr x**y >>> expr = expr.subs(y, x**y) >>> expr x**(x**y) >>> expr = expr.subs(y, x**x) >>> expr x**(x**(x**x))
The second is if we want to perform a very controlled simplification, or perhaps a simplification that SymPy is otherwise unable to do. For example, say we have \(\sin(2x) + \cos(2x)\), and we want to replace \(\sin(2x)\) with \(2\sin(x)\cos(x)\). As we will learn later, the function
expand_trigdoes this. However, this function will also expand \(\cos(2x)\), which we may not want. While there are ways to perform such precise simplification, and we will learn some of them in the advanced expression manipulation section, an easy way is to just replace \(\sin(2x)\) with \(2\sin(x)\cos(x)\).>>> expr = sin(2*x) + cos(2*x) >>> expand_trig(expr) 2*sin(x)*cos(x) + 2*cos(x)**2 - 1 >>> expr.subs(sin(2*x), 2*sin(x)*cos(x)) 2*sin(x)*cos(x) + cos(2*x)
There are two important things to note about subs. First, it returns a
new expression. SymPy objects are immutable. That means that subs does
not modify it in-place. For example
>>> expr = cos(x)
>>> expr.subs(x, 0)
1
>>> expr
cos(x)
>>> x
x
Here, we see that performing expr.subs(x, 0) leaves expr unchanged.
In fact, since SymPy expressions are immutable, no function will change them
in-place. All functions will return new expressions.
To perform multiple substitutions at once, pass a list of (old, new) pairs
to subs.
>>> expr = x**3 + 4*x*y - z
>>> expr.subs([(x, 2), (y, 4), (z, 0)])
40
It is often useful to combine this with a list comprehension to do a large set of similar replacements all at once. For example, say we had \(x^4 - 4x^3 + 4x^2 - 2x + 3\) and we wanted to replace all instances of \(x\) that have an even power with \(y\), to get \(y^4 - 4x^3 + 4y^2 - 2x + 3\).
>>> expr = x**4 - 4*x**3 + 4*x**2 - 2*x + 3
>>> replacements = [(x**i, y**i) for i in range(5) if i % 2 == 0]
>>> expr.subs(replacements)
-4*x**3 - 2*x + y**4 + 4*y**2 + 3
Converting Strings to SymPy Expressions¶
The sympify function (that’s sympify, not to be confused with
simplify) can be used to convert strings into SymPy expressions.
For example
>>> str_expr = "x**2 + 3*x - 1/2"
>>> expr = sympify(str_expr)
>>> expr
x**2 + 3*x - 1/2
>>> expr.subs(x, 2)
19/2
Warning
sympify uses eval. Don’t use it on unsanitized input.
evalf¶
To evaluate a numerical expression into a floating point number, use
evalf.
>>> expr = sqrt(8)
>>> expr.evalf()
2.82842712474619
SymPy can evaluate floating point expressions to arbitrary precision. By
default, 15 digits of precision are used, but you can pass any number as the
argument to evalf. Let’s compute the first 100 digits of \(\pi\).
>>> pi.evalf(100)
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068
To numerically evaluate an expression with a Symbol at a point, we might use
subs followed by evalf, but it is more efficient and numerically
stable to pass the substitution to evalf using the subs flag, which
takes a dictionary of Symbol: point pairs.
>>> expr = cos(2*x)
>>> expr.evalf(subs={x: 2.4})
0.0874989834394464
Sometimes there are roundoff errors smaller than the desired precision that
remain after an expression is evaluated. Such numbers can be removed at the
user’s discretion by setting the chop flag to True.
>>> one = cos(1)**2 + sin(1)**2
>>> (one - 1).evalf()
-0.e-124
>>> (one - 1).evalf(chop=True)
0
lambdify¶
subs and evalf are good if you want to do simple evaluation, but if
you intend to evaluate an expression at many points, there are more efficient
ways. For example, if you wanted to evaluate an expression at a thousand
points, using SymPy would be far slower than it needs to be, especially if you
only care about machine precision. Instead, you should use libraries like
NumPy and SciPy.
The easiest way to convert a SymPy expression to an expression that can be
numerically evaluated is to use the lambdify function. lambdify acts
like a lambda function, except it converts the SymPy names to the names of
the given numerical library, usually NumPy. For example
>>> import numpy
>>> a = numpy.arange(10)
>>> expr = sin(x)
>>> f = lambdify(x, expr, "numpy")
>>> f(a)
[ 0. 0.84147098 0.90929743 0.14112001 -0.7568025 -0.95892427
-0.2794155 0.6569866 0.98935825 0.41211849]
Warning
lambdify uses eval. Don’t use it on unsanitized input.
You can use other libraries than NumPy. For example, to use the standard
library math module, use "math".
>>> f = lambdify(x, expr, "math")
>>> f(0.1)
0.0998334166468
To use lambdify with numerical libraries that it does not know about, pass a
dictionary of sympy_name:numerical_function pairs. For example
>>> def mysin(x):
... """
... My sine. Note that this is only accurate for small x.
... """
... return x
>>> f = lambdify(x, expr, {"sin":mysin})
>>> f(0.1)
0.1