Lecture 10¶

Symbolic Computing¶

Basics¶

In [1]:

import sympy
# from sympy import *
sympy.init_printing()
# nicely formatted

In [2]:

from sympy import I, pi, oo
# different from np.pi

Symbols¶

In [3]:

x = sympy.Symbol("x")
y = sympy.Symbol("y", real=True)
y.is_real

Out[3]:

True

In [4]:

x.is_real is None

Out[4]:

True

Assumption Keyword Arguments	Attributes	Description
real, imaginary	is_real, is_imaginary	Specify that a symbol represents a real or imaginary number.
positive, negative	is_positive, is_negative	Specify that a symbol is positive or negative.
integer	is_integer	The symbol represents an integer.
odd, even	is_odd, is_even	The symbol represents an odd or even integer.
prime	is_prime	The symbol is a prime number and therefore also an integer.
finite, infinite	is_finite, is_infinite	The symbol represents a quantity that is finite or infinite.

In [5]:

x = sympy.Symbol("x")
y = sympy.Symbol("y", positive=True)
sympy.sqrt(x ** 2)

Out[5]:

$\displaystyle \sqrt{x^{2}}$

In [6]:

sympy.sqrt(y ** 2)
# Simplify the expression

Out[6]:

$\displaystyle y$

In [7]:

n1 = sympy.Symbol("n")
n2 = sympy.Symbol("n", integer=True)
n3 = sympy.Symbol("n", odd=True)

In [8]:

sympy.cos(n1 * pi)

Out[8]:

$\displaystyle \cos{\left(\pi n \right)}$

In [9]:

sympy.cos(n2 * pi)

Out[9]:

$\displaystyle \left(-1\right)^{n}$

In [10]:

sympy.cos(n3 * pi)

Out[10]:

$\displaystyle -1$

In [11]:

a, b, c = sympy.symbols("a, b, c", negative=True)
d, e, f = sympy.symbols("d, e, f", positive=True)

Numbers¶

we cannot directly use the built-in Python objects for integers, int, and floating-point numbers, float, and so on.

In [12]:

i = sympy.Integer(19)
type(i)

Out[12]:

sympy.core.numbers.Integer

In [13]:

i.is_Integer, i.is_real, i.is_odd

Out[13]:

(True, True, True)

In [14]:

f = sympy.Float(2.3)
type(f)

Out[14]:

sympy.core.numbers.Float

In [15]:

i, f = sympy.sympify(19), sympy.sympify(2.3)

In [16]:

# integer and Integer are different
n = sympy.Symbol("n", integer=True)
n.is_integer, n.is_Integer, n.is_positive, n.is_Symbol

Out[16]:

(True, False, None, True)

In [17]:

i = sympy.Integer(19)
i.is_integer, i.is_Integer, i.is_positive, i.is_Symbol

Out[17]:

(True, True, True, False)

In [18]:

sympy.Float(0.3, 25)

Out[18]:

$\displaystyle 0.2999999999999999888977698$

In [19]:

sympy.Float('0.3', 25)

Out[19]:

$\displaystyle 0.3$

In [20]:

sympy.Rational(11, 13)

Out[20]:

$\displaystyle \frac{11}{13}$

In [21]:

r1 = sympy.Rational(2, 3)
r2 = sympy.Rational(4, 5)
r1 * r2

Out[21]:

$\displaystyle \frac{8}{15}$

Mathematical Symbol	SymPy Symbol	Description
π	sympy.pi	Ratio of the circumference to the diameter of a circle.
e	sympy.E	The base of the natural logarithm, e = exp (1).
γ	sympy.EulerGamma	Euler’s constant.
i	sympy.I	The imaginary unit.
∞	sympy.oo	Infinity.

In [22]:

x, y, z = sympy.symbols("x, y, z")
f = sympy.Function("f")
f(x)

Out[22]:

$\displaystyle f{\left(x \right)}$

In [23]:

g = sympy.Function("g")(x, y, z)
g

Out[23]:

$\displaystyle g{\left(x,y,z \right)}$

In [24]:

g.free_symbols

Out[24]:

$\displaystyle \left\{x, y, z\right\}$

Naturally, SymPy has built-in functions for many standard mathematical functions that are available in the global SymPy namespace

In [25]:

sympy.sin(x)

Out[25]:

$\displaystyle \sin{\left(x \right)}$

In [26]:

h = sympy.Lambda(x, x**2)
h

Out[26]:

$\displaystyle \left( x \mapsto x^{2} \right)$

In [27]:

h(5)

Out[27]:

$\displaystyle 25$

In [28]:

h(1 + x)

Out[28]:

$\displaystyle \left(x + 1\right)^{2}$

Expressions¶

In [29]:

x = sympy.Symbol("x")
expr = 1 + 2 * x**2 + 3 * x**3
expr

Out[29]:

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

For an operator, the args attribute is a tuple of subexpressions.

In [30]:

expr.args

Out[30]:

$\displaystyle \left( 1, \ 2 x^{2}, \ 3 x^{3}\right)$

In [31]:

expr.args[1]

Out[31]:

$\displaystyle 2 x^{2}$

In [32]:

expr.args[1].args[1]

Out[32]:

$\displaystyle x^{2}$

In [33]:

expr.args[1].args[1].args[0]

Out[33]:

$\displaystyle x$

Manipulating Expressions¶

In [34]:

expr = 2 * (x**2 - x) - x * (x + 1)
expr

Out[34]:

$\displaystyle 2 x^{2} - x \left(x + 1\right) - 2 x$

In [35]:

sympy.simplify(expr)

Out[35]:

$\displaystyle x \left(x - 3\right)$

In [36]:

expr = sympy.exp(x) * sympy.exp(y)
expr

Out[36]:

$\displaystyle e^{x} e^{y}$

In [37]:

sympy.simplify(expr)

Out[37]:

$\displaystyle e^{x + y}$

Function	Description
sympy.simplify	Attempt various methods and approaches to obtain a simpler form of a given expression.
sympy.trigsimp	Attempt to simplify an expression using trigonometric identities.
sympy.powsimp	Attempt to simplify an expression using laws of powers.
sympy.compsimp	Simplify combinatorial expressions.
sympy.ratsimp	Simplify an expression by writing on a common denominator.

Expand¶

In [38]:

expr = (x + 1) * (x + 2)
sympy.expand(expr)

Out[38]:

$\displaystyle x^{2} + 3 x + 2$

In [39]:

sympy.sin(x + y).expand(trig=True)

Out[39]:

$\displaystyle \sin{\left(x \right)} \cos{\left(y \right)} + \sin{\left(y \right)} \cos{\left(x \right)}$

In [40]:

a, b = sympy.symbols("a, b", positive=True)
sympy.log(a * b).expand(log=True)

Out[40]:

$\displaystyle \log{\left(a \right)} + \log{\left(b \right)}$

In [41]:

sympy.expand((a * b)**x, power_base=True)

Out[41]:

$\displaystyle a^{x} b^{x}$

Factor, Collect, and Combine¶

In [42]:

sympy.factor(x**2 - 1)
# the opposite to sympy.expand with mul=True.

Out[42]:

$\displaystyle \left(x - 1\right) \left(x + 1\right)$

In [43]:

sympy.factor(x * sympy.cos(y) + sympy.sin(z) * x)

Out[43]:

$\displaystyle x \left(\sin{\left(z \right)} + \cos{\left(y \right)}\right)$

In [44]:

expr = x + y + x * y * z
expr.collect(x)

Out[44]:

$\displaystyle x \left(y z + 1\right) + y$

In [45]:

expr.collect(y)

Out[45]:

$\displaystyle x + y \left(x z + 1\right)$

Apart, Together, and Cancel¶

In [46]:

sympy.apart(1/(x**2 + 3*x + 2), x)

Out[46]:

$\displaystyle - \frac{1}{x + 2} + \frac{1}{x + 1}$

In [47]:

sympy.together(1 / (y * x + y) + 1 / (1+x))

Out[47]:

$\displaystyle \frac{y + 1}{y \left(x + 1\right)}$

In [48]:

sympy.cancel(y / (y * x + y))

Out[48]:

$\displaystyle \frac{1}{x + 1}$

Substitutions¶

In [49]:

(x + y).subs(x, y)

Out[49]:

$\displaystyle 2 y$

In [50]:

sympy.sin(x * sympy.exp(x)).subs(x, y)

Out[50]:

$\displaystyle \sin{\left(y e^{y} \right)}$

In [51]:

sympy.sin(x * z).subs({z: sympy.exp(y), x: y, sympy.sin: sympy.cos})

Out[51]:

$\displaystyle \cos{\left(y e^{y} \right)}$

In [52]:

expr = x * y + z**2 *x
values = {x: 1.25, y: 0.4, z: 3.2}
expr.subs(values)

Out[52]:

$\displaystyle 13.3$

Numerical Evaluation¶

In [53]:

sympy.N(1 + pi)

Out[53]:

$\displaystyle 4.14159265358979$

In [54]:

(x + 1/pi).evalf(10)

Out[54]:

$\displaystyle x + 0.3183098862$

In [55]:

expr = sympy.sin(pi * x * sympy.exp(x))
expr_func = sympy.lambdify(x, expr)
expr_func(1.0)

Out[55]:

$\displaystyle 0.773942685266709$

In [56]:

expr_func = sympy.lambdify(x, expr, 'numpy')
import numpy as np
xvalues = np.arange(0, 10)
expr_func(xvalues)

Out[56]:

array([ 0.        ,  0.77394269,  0.64198244,  0.72163867,  0.94361635,
        0.20523391,  0.97398794,  0.97734066, -0.87034418, -0.69512687])

Calculus¶

In [57]:

f = sympy.Function('f')(x)
sympy.diff(f, x)
# equivalent to f.diff(x)

Out[57]:

$\displaystyle \frac{d}{d x} f{\left(x \right)}$

In [58]:

sympy.diff(f, x, x)

Out[58]:

$\displaystyle \frac{d^{2}}{d x^{2}} f{\left(x \right)}$

In [59]:

sympy.diff(f, x, 3)

Out[59]:

$\displaystyle \frac{d^{3}}{d x^{3}} f{\left(x \right)}$

In [60]:

g = sympy.Function('g')(x, y)
g.diff(x, y)

Out[60]:

$\displaystyle \frac{\partial^{2}}{\partial y\partial x} g{\left(x,y \right)}$

In [61]:

g.diff(x, 3, y, 2)

Out[61]:

$\displaystyle \frac{\partial^{5}}{\partial y^{2}\partial x^{3}} g{\left(x,y \right)}$

In [62]:

expr = x**4 + x**3 + x**2 + x + 1
expr.diff(x)

Out[62]:

$\displaystyle 4 x^{3} + 3 x^{2} + 2 x + 1$

In [63]:

expr.diff(x, x)

Out[63]:

$\displaystyle 2 \cdot \left(6 x^{2} + 3 x + 1\right)$

In [64]:

expr = (x + 1)**3 * y ** 2 * (z - 1)
expr.diff(x, y, z)

Out[64]:

$\displaystyle 6 y \left(x + 1\right)^{2}$

In [65]:

d = sympy.Derivative(sympy.exp(sympy.cos(x)), x)
d

Out[65]:

$\displaystyle \frac{d}{d x} e^{\cos{\left(x \right)}}$

In [66]:

d.doit()

Out[66]:

$\displaystyle - e^{\cos{\left(x \right)}} \sin{\left(x \right)}$

Integrals¶

In [67]:

a, b, x, y = sympy.symbols("a, b, x, y")
f = sympy.Function("f")(x)
sympy.integrate(f)

Out[67]:

$\displaystyle \int f{\left(x \right)}\, dx$

In [68]:

sympy.integrate(f, (x, a, b))

Out[68]:

$\displaystyle \int\limits_{a}^{b} f{\left(x \right)}\, dx$

In [69]:

sympy.integrate(sympy.sin(x))

Out[69]:

$\displaystyle - \cos{\left(x \right)}$

In [70]:

sympy.integrate(sympy.sin(x), (x, a, b))

Out[70]:

$\displaystyle \cos{\left(a \right)} - \cos{\left(b \right)}$

In [71]:

sympy.integrate(sympy.exp(-x**2), (x, 0, oo))

Out[71]:

$\displaystyle \frac{\sqrt{\pi}}{2}$

In [72]:

a, b, c = sympy.symbols("a, b, c", positive=True)
sympy.integrate(a * sympy.exp(-((x-b)/c)**2), (x, -oo, oo))

Out[72]:

$\displaystyle \sqrt{\pi} a c$

In [73]:

sympy.integrate(sympy.sin(x * sympy.cos(x)))

Out[73]:

$\displaystyle \int \sin{\left(x \cos{\left(x \right)} \right)}\, dx$

In [74]:

expr = sympy.sin(x*sympy.exp(y))
sympy.integrate(expr, x)

Out[74]:

$\displaystyle - e^{- y} \cos{\left(x e^{y} \right)}$

In [75]:

expr = (x + y)**2
sympy.integrate(expr, x)

Out[75]:

$\displaystyle \frac{x^{3}}{3} + x^{2} y + x y^{2}$

In [76]:

sympy.integrate(expr, x, y)

Out[76]:

$\displaystyle \frac{x^{3} y}{3} + \frac{x^{2} y^{2}}{2} + \frac{x y^{3}}{3}$

In [77]:

sympy.integrate(expr, (x, 0, 1), (y, 0, 1))

Out[77]:

$\displaystyle \frac{7}{6}$

Sums and Products¶

In [78]:

# We omit Series and Limits!
n = sympy.symbols("n", integer=True)
x = sympy.Sum(1/(n**2), (n, 1, oo))
x

Out[78]:

$\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{2}}$

In [79]:

x.doit()

Out[79]:

$\displaystyle \frac{\pi^{2}}{6}$

In [80]:

x = sympy.Product(n, (n, 1, 7))
x

Out[80]:

$\displaystyle \prod_{n=1}^{7} n$

In [81]:

x.doit()

Out[81]:

$\displaystyle 5040$

Equations¶

In [82]:

x = sympy.Symbol("x")
sympy.solve(x**2 + 2*x - 3)

Out[82]:

$\displaystyle \left[ -3, \ 1\right]$

In [83]:

a, b, c = sympy.symbols("a, b, c")
sympy.solve(a * x**2 + b * x + c, x)

Out[83]:

$\displaystyle \left[ \frac{- b - \sqrt{- 4 a c + b^{2}}}{2 a}, \ \frac{- b + \sqrt{- 4 a c + b^{2}}}{2 a}\right]$

In [84]:

sympy.solve(sympy.sin(x) - sympy.cos(x), x)

Out[84]:

$\displaystyle \left[ \frac{\pi}{4}\right]$

In [85]:

sympy.solve(sympy.exp(x) + 2 * x, x)
# LambertW function

Out[85]:

$\displaystyle \left[ - W\left(\frac{1}{2}\right)\right]$

It is not uncommon to encounter equations that are not solvable algebraically or which SymPy is unable to solve.

In [86]:

sympy.solve(x**5 - x**2 + 1, x)

Out[86]:

$\displaystyle \left[ \operatorname{CRootOf} {\left(x^{5} - x^{2} + 1, 0\right)}, \ \operatorname{CRootOf} {\left(x^{5} - x^{2} + 1, 1\right)}, \ \operatorname{CRootOf} {\left(x^{5} - x^{2} + 1, 2\right)}, \ \operatorname{CRootOf} {\left(x^{5} - x^{2} + 1, 3\right)}, \ \operatorname{CRootOf} {\left(x^{5} - x^{2} + 1, 4\right)}\right]$

In [87]:

eq1 = x + 2 * y - 1
eq2 = x - y + 1
sympy.solve([eq1, eq2], [x, y], dict=True)

Out[87]:

$\displaystyle \left[ \left\{ x : - \frac{1}{3}, \ y : \frac{2}{3}\right\}\right]$

In [88]:

eq1 = x**2 - y
eq2 = y**2 - x
sols = sympy.solve([eq1, eq2], [x, y], dict=True)
sols

Out[88]:

$\displaystyle \left[ \left\{ x : 0, \ y : 0\right\}, \ \left\{ x : 1, \ y : 1\right\}, \ \left\{ x : \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)^{2}, \ y : - \frac{1}{2} - \frac{\sqrt{3} i}{2}\right\}, \ \left\{ x : \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)^{2}, \ y : - \frac{1}{2} + \frac{\sqrt{3} i}{2}\right\}\right]$

Linear Algebra¶

In [89]:

sympy.Matrix([1, 2])

Out[89]:

$\displaystyle \left[\begin{matrix}1\\2\end{matrix}\right]$

In [90]:

sympy.Matrix([[1, 2]])

Out[90]:

$\displaystyle \left[\begin{matrix}1 & 2\end{matrix}\right]$

In [91]:

sympy.Matrix([[1, 2], [3, 4]])

Out[91]:

$\displaystyle \left[\begin{matrix}1 & 2\\3 & 4\end{matrix}\right]$

In [92]:

sympy.Matrix(3, 4, lambda m, n: 10 * m + n)

Out[92]:

$\displaystyle \left[\begin{matrix}0 & 1 & 2 & 3\\10 & 11 & 12 & 13\\20 & 21 & 22 & 23\end{matrix}\right]$

In [93]:

a, b, c, d = sympy.symbols("a, b, c, d")
M = sympy.Matrix([[a, b], [c, d]])
M

Out[93]:

$\displaystyle \left[\begin{matrix}a & b\\c & d\end{matrix}\right]$

In [94]:

M * M

Out[94]:

$\displaystyle \left[\begin{matrix}a^{2} + b c & a b + b d\\a c + c d & b c + d^{2}\end{matrix}\right]$

In [95]:

x = sympy.Matrix(sympy.symbols("x_1, x_2"))
M * x

Out[95]:

$\displaystyle \left[\begin{matrix}a x_{1} + b x_{2}\\c x_{1} + d x_{2}\end{matrix}\right]$

$$ x + py =b_1\\ qx + y =b_2 $$

In [96]:

p, q = sympy.symbols("p, q")
M = sympy.Matrix([[1, p], [q, 1]])
M

Out[96]:

$\displaystyle \left[\begin{matrix}1 & p\\q & 1\end{matrix}\right]$

In [97]:

b = sympy.Matrix(sympy.symbols("b_1, b_2"))
b

Out[97]:

$\displaystyle \left[\begin{matrix}b_{1}\\b_{2}\end{matrix}\right]$

In [98]:

# computing the inverse of a matrix is more difficult
x = M.LUsolve(b)
x

Out[98]:

$\displaystyle \left[\begin{matrix}b_{1} - \frac{p \left(- b_{1} q + b_{2}\right)}{- p q + 1}\\\frac{- b_{1} q + b_{2}}{- p q + 1}\end{matrix}\right]$

Latex¶

In [99]:

x = sympy.Symbol("x")
# Greek alphabet 
print(sympy.latex(sympy.Integral(sympy.sqrt(1/x), x)))
# use print() !!!

\int \sqrt{\frac{1}{x}}\, dx

$$ \int \sqrt{\frac{1}{x}}\, dx $$

In [100]:

print(sympy.latex(sympy.Integral(sympy.sqrt(1/x), x).doit()))

2 x \sqrt{\frac{1}{x}}

$$ 2 x \sqrt{\frac{1}{x}} $$

In [107]:

alpha, beta, gamma= sympy.symbols("alpha beta gamma")
alpha, beta, gamma

Out[107]:

$\displaystyle \left( \alpha, \ \beta, \ \gamma\right)$

In [108]:

print(sympy.latex(alpha+beta+gamma))

\alpha + \beta + \gamma