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Expressions and Substitutions

Expressions and Substitutions

Symbolic math variables can be combined into symbolic math expressions. Once in an expression, symbolic math variables can be exchanged with substituion.

Expressions

A symbolic math expression is a combination of symbolic math variables with numbers and mathematical operators such as +, -, / and *. The standard Python rules for calculating numbers apply in SymPy symbolic math expressions.

After the symbols x and y are created, a symbolic math expression using x and y can be defined.

In [1]:

from sympy import symbols

x, y = symbols('x y')
expr = 2*x + y

Substitution

Use SymPy's .subs() method to insert a numerical value into a symbolic math expression. The first argument of the .subs() method is the mathematical symbol and the second argument is the numerical value. In the expression below:

$$2x + y$$

If we substitute:

$$x = 2$$

The resulting expression is:

$$2(2) + y $$ $$ 4 + y$$

We can code the substitution above using SymPy's .subs() method as shown below.

In [2]:

expr.subs(x, 2)

Out[2]:

$\displaystyle y + 4$

The .subs() method does not replace variables in place, .subs() only completes a one-time substitution. If expr is called after the .subs() method is applied, the original expr expression is returned.

In [3]:

expr

Out[3]:

$\displaystyle 2 x + y$

To make the substitution permanent, a new expression object needs to be assigned to the output of the .subs() method.

In [4]:

expr = 2*x + y
expr2 = expr.subs(x, 2)
expr2

Out[4]:

$\displaystyle y + 4$

SymPy variables can also be substituted into SymPy expressions. In the code section below, the symbol $z$ is substituted for the symbol $x$ ($z$ replaces $x$).

In [5]:

x, y, z = symbols('x y z')
expr = 2*x + y
expr2 = expr.subs(x, z)
expr2

Out[5]:

$\displaystyle y + 2 z$

Expressions can also be substituted into other expressions. Consider the following:

$$y + 2x^2 + z^{-3}$$

substitute in

$$y = 2x$$

results in

$$2x + 2x^2 + z^{-3}$$

In [6]:

x, y, z = symbols('x y z')
expr = y + 2*x**2 + z**(-3)
expr2 = expr.subs(y, 2*x)
expr2

Out[6]:

$\displaystyle 2 x^{2} + 2 x + \frac{1}{z^{3}}$

A practical example involving symbolic math variables, expressions and substitutions can include a complex expression and several replacements.

$$n_0e^{-Q_v/RT}$$

$$n_0 = 3.48 \times 10^{-6}$$

$$Q_v = 12,700$$

$$R = 8.31$$

$$T = 1000 + 273$$

We can create four symbolic math variables and combine the variables into an expression with the code below.

In [7]:

from sympy import symbols, exp
n0, Qv, R, T = symbols('n0 Qv R T')
expr = n0exp(-Qv/(RT))

Multiple SymPy subs() methods can be chained together to substitute multiple variables in one line of code.

In [8]:

expr.subs(n0, 3.48e-6).subs(Qv,12700).subs(R, 8031).subs(T, 1000+273)

Out[8]:

$\displaystyle \frac{3.48 \cdot 10^{-6}}{e^{\frac{12700}{10223463}}}$

To evaluate an expression as a floating point number, use SymPy's .evalf() method.

In [9]:

expr2 = expr.subs(n0, 3.48e-6).subs(Qv,12700).subs(R, 8031).subs(T, 1000+273)

In [10]:

expr2.evalf()

Out[10]:

$\displaystyle 3.47567968697765 \cdot 10^{-6}$

You can control the number of digits the .evalf() method outputs by passing a number as an argument.

In [11]:

expr2.evalf(4)

Out[11]:

$\displaystyle 3.476 \cdot 10^{-6}$

Summary

The SymPy functions and methods used in this section are summarized in the table below.

SymPy function or method	Description	Example
symbols()	create symbolic math variables	x, y = symbols('x y')
.subs()	substitute a value into a symbolic math expression	expr.subs(x,2)
.evalf()	evaluate a symbolic math expression as a floating point number	expr.evalf()