Symbolic Algebra with sympy

from sympy import *

from sympy import init_session
init_session()

IPython console for SymPy 0.7.6.1 (Python 3.5.1-64-bit) (ground types: python)

These commands were executed:
>>> from __future__ import division
>>> from sympy import *
>>> x, y, z, t = symbols('x y z t')
>>> k, m, n = symbols('k m n', integer=True)
>>> f, g, h = symbols('f g h', cls=Function)
>>> init_printing()

Documentation can be found at http://www.sympy.org

Basics

from sympy.stats import *
E(Die('X', 6))

$$\frac{7}{2}$$

sqrt(8)

$$2 \sqrt{2}$$

expr = x + 2*y

expr2 = x*expr

expr2

$$x \left(x + 2 y\right)$$

expand(expr2)

$$x^{2} + 2 x y$$

factor(expand(expr2))

$$x \left(x + 2 y\right)$$

diff(sin(x) * exp(x), x)

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

integrate(exp(x)*sin(x) + exp(x)*cos(x), x)

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

integrate(sin(x**2), (x, -oo, oo))

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

dsolve(Eq(f(t).diff(t, t) - f(t), exp(t)), f(t))

$$f{\left (t \right )} = C_{2} e^{- t} + \left(C_{1} + \frac{t}{2}\right) e^{t}$$

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

$$\left \{ \frac{3}{2} + \frac{\sqrt{17}}{2} : 1, \quad - \frac{\sqrt{17}}{2} + \frac{3}{2} : 1\right \}$$

nu = symbols('nu')
besselj(nu, z).rewrite(jn)

$$\frac{\sqrt{2} \sqrt{z}}{\sqrt{\pi}} j_{\nu - \frac{1}{2}}\left(z\right)$$

latex(Integral(cos(x)**2, (x, 0, pi)))

'\\int_{0}^{\\pi} \\cos^{2}{\\left (x \\right )}\\, dx'

expr = cos(x) + 1

expr.subs(x, y)

$$\cos{\left (y \right )} + 1$$

expr = x**y
expr = expr.subs(y, x**y)
expr = expr.subs(y, x**y)
expr = expr.subs(x, x**x)

expr

$$\left(x^{x}\right)^{\left(x^{x}\right)^{\left(x^{x}\right)^{y}}}$$

expr = sin(2*x) + cos(2*x)
expand_trig(expr)

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

expr.subs(sin(2*x), 2*sin(x)*cos(x))

$$2 \sin{\left (x \right )} \cos{\left (x \right )} + \cos{\left (2 x \right )}$$

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)

$$- 4 x^{3} - 2 x + y^{4} + 4 y^{2} + 3$$

sexpr = "x**4 - 4*x**3 + 4*x**2 - 2*x +3"
sympify(sexpr)

$$x^{4} - 4 x^{3} + 4 x^{2} - 2 x + 3$$

sqrt(8).evalf()

$$2.82842712474619$$

pi.evalf(100)

$$3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068$$

expr = cos(2*x)
expr.evalf(subs = {x: 2.4})

$$0.0874989834394464$$

expr = cos(x)**2 + sin(x)**2
expr.evalf(subs = {x: 1}, chop=True)

$$1.0$$

a = range(10)
expr = sin(x)
f = lambdify(x, expr, "numpy")
f(a)

array([ 0.        ,  0.84147098,  0.90929743,  0.14112001, -0.7568025 ,
       -0.95892427, -0.2794155 ,  0.6569866 ,  0.98935825,  0.41211849])

Simplify

simplify(sin(x)**2 + cos(x)**2)

$$1$$

simplify(gamma(x) / gamma(x-2))

$$\left(x - 2\right) \left(x - 1\right)$$

expand((x + y)**3)

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

factor(x**3 - x**2 + x - 1)

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

factor_list(x**3 - x**2 + x - 1)

$$\left ( 1, \quad \left [ \left ( x - 1, \quad 1\right ), \quad \left ( x^{2} + 1, \quad 1\right )\right ]\right )$$

expand((cos(x) + sin(x))**2)

$$\sin^{2}{\left (x \right )} + 2 \sin{\left (x \right )} \cos{\left (x \right )} + \cos^{2}{\left (x \right )}$$

expr = x*y + x - 3 + 2*x**2 - z*x**2 + x**3
cexpr = collect(expr, x)
cexpr

$$x^{3} + x^{2} \left(- z + 2\right) + x \left(y + 1\right) - 3$$

cexpr.coeff(x**2)

$$- z + 2$$

cancel((x**2 + 2*x + 1) / (x**2 + x))

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

expr = (x*y**2 - 2*x*y*z + x*z**2 + y**2 - 2*y*z + z**2)/(x**2 - 1)

cancel(expr)

$$\frac{1}{x - 1} \left(y^{2} - 2 y z + z^{2}\right)$$

factor(expr)

$$\frac{\left(y - z\right)^{2}}{x - 1}$$

expr = (4*x**3 + 21*x**2 + 10*x + 12)/(x**4 + 5*x**3 + 5*x**2 + 4*x)

apart(expr)

$$\frac{2 x - 1}{x^{2} + x + 1} - \frac{1}{x + 4} + \frac{3}{x}$$

trigsimp(sin(x)*tan(x)/sec(x))

$$\sin^{2}{\left (x \right )}$$

trigsimp(cosh(x)**2 + sinh(x)**2)

$$\cosh{\left (2 x \right )}$$

expand_trig(sin(x + y))

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

x, y = symbols('x y', positive=True)
a, b = symbols('a b', real=True)
z, t, c = symbols('z t c')

powsimp(x**a*x**b)

$$x^{a + b}$$

powsimp(x**a*y**a)

$$\left(x y\right)^{a}$$

powsimp(t**c*z**c)

$$t^{c} z^{c}$$

powsimp(t**c*z**c, force=True)

$$\left(t z\right)^{c}$$

expand_power_exp(x**(a + b))

$$x^{a} x^{b}$$

expand_power_base((x*y)**a)

$$x^{a} y^{a}$$

powdenest((x**a)**b)

$$x^{a b}$$

n = symbols('n', real=True)

expand_log(log(x*y))

$$\log{\left (x \right )} + \log{\left (y \right )}$$

expand_log(log(x**n))

$$n \log{\left (x \right )}$$

logcombine(n*log(x))

$$\log{\left (x^{n} \right )}$$

x, y, z = symbols('x y z')
k, m, n = symbols('k m n')

factorial(n)

$$n!$$

binomial(n, k)

$${\binom{n}{k}}$$

hyper([1,2], [3], z)

$${{}_{2}F_{1}\left(\begin{matrix} 1, 2 \\ 3 \end{matrix}\middle| {z} \right)}$$

factorial(x).rewrite(gamma)

$$\Gamma{\left(x + 1 \right)}$$

tan(x).rewrite(sin)

$$\frac{2 \sin^{2}{\left (x \right )}}{\sin{\left (2 x \right )}}$$

expand_func(gamma(x + 3))

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

hyperexpand(hyper([1, 1], [2], z))

$$- \frac{1}{z} \log{\left (- z + 1 \right )}$$

expr =  meijerg([[1],[1]], [[1],[]], -z)

expr

$${G_{2, 1}^{1, 1}\left(\begin{matrix} 1 & 1 \\1 & \end{matrix} \middle| {- z} \right)}$$

hyperexpand(expr)

$$e^{\frac{1}{z}}$$

combsimp(binomial(n+1, k+1)/binomial(n, k))

$$\frac{n + 1}{k + 1}$$

def list_to_frac(l):
    expr = Integer(0)
    for i in reversed(l[1:]):
        expr += i
        expr = 1/expr
    return l[0] + expr

list_to_frac([1,2,3,4])

$$\frac{43}{30}$$

syms = symbols('a0:5')

syms

$$\left ( a_{0}, \quad a_{1}, \quad a_{2}, \quad a_{3}, \quad a_{4}\right )$$

frac = list_to_frac(syms)

frac

$$a_{0} + \frac{1}{a_{1} + \frac{1}{a_{2} + \frac{1}{a_{3} + \frac{1}{a_{4}}}}}$$

frac = cancel(frac)

frac

$$\frac{a_{0} a_{1} a_{2} a_{3} a_{4} + a_{0} a_{1} a_{2} + a_{0} a_{1} a_{4} + a_{0} a_{3} a_{4} + a_{0} + a_{2} a_{3} a_{4} + a_{2} + a_{4}}{a_{1} a_{2} a_{3} a_{4} + a_{1} a_{2} + a_{1} a_{4} + a_{3} a_{4} + 1}$$

from sympy.printing import print_ccode
print_ccode(frac)

(a0*a1*a2*a3*a4 + a0*a1*a2 + a0*a1*a4 + a0*a3*a4 + a0 + a2*a3*a4 + a2 + a4)/(a1*a2*a3*a4 + a1*a2 + a1*a4 + a3*a4 + 1)

Calculus

diff(cos(x), x)

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

diff(x**4, x, 3)

$$24 x$$

expr = exp(x*y*z)
diff(expr, x, y, 2, z, 4)

$$x^{3} y^{2} \left(x^{3} y^{3} z^{3} + 14 x^{2} y^{2} z^{2} + 52 x y z + 48\right) e^{x y z}$$

deriv = Derivative(expr, x, y, 2, z, 4)
deriv

$$\frac{\partial^{7}}{\partial x\partial y^{2}\partial z^{4}} e^{x y z}$$

deriv.doit()

$$x^{3} y^{2} \left(x^{3} y^{3} z^{3} + 14 x^{2} y^{2} z^{2} + 52 x y z + 48\right) e^{x y z}$$

integrate(cos(x), x)

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

integrate(exp(-x), (x, 0, oo))

$$1$$

integrate(exp(-x**2 - y**2), (x, -oo, oo), (y, -oo, oo))

$$\pi$$

integrate(x**x, x)

$$\int x^{x}\, dx$$

expr = Integral(log(x)**2, x)
expr

$$\int \log^{2}{\left (x \right )}\, dx$$

expr.doit()

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

integrate(sin(x**2), x)

$$\frac{3 \sqrt{2} \sqrt{\pi} S\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{8 \Gamma{\left(\frac{7}{4} \right)}} \Gamma{\left(\frac{3}{4} \right)}$$

integrate(x**y*exp(-x), (x, 0, oo))

$$\begin{cases} \Gamma{\left(y + 1 \right)} & \text{for}\: - \Re{y} < 1 \\\int_{0}^{\infty} x^{y} e^{- x}\, dx & \text{otherwise} \end{cases}$$

limit(sin(x)/x, x, 0)

$$1$$

expr = Limit((cos(x) - 1)/x, x, 0)
expr

$$\lim_{x \to 0^+}\left(\frac{1}{x} \left(\cos{\left (x \right )} - 1\right)\right)$$

expr.doit()

$$0$$

limit(1/x, x, 0, '-')

$$-\infty$$

expr = exp(sin(x))
expr.series(x, 0, 4)

$$1 + x + \frac{x^{2}}{2} + \mathcal{O}\left(x^{4}\right)$$

x + x**3 + x**6 + O(x**4)

$$x + x^{3} + \mathcal{O}\left(x^{4}\right)$$

x * O(1)

$$\mathcal{O}\left(x\right)$$

expr.series(x, 0, 4).removeO()

$$\frac{x^{2}}{2} + x + 1$$

exp(x - 6).series(x, 6)

$$-5 + \frac{1}{2} \left(x - 6\right)^{2} + \frac{1}{6} \left(x - 6\right)^{3} + \frac{1}{24} \left(x - 6\right)^{4} + \frac{1}{120} \left(x - 6\right)^{5} + x + \mathcal{O}\left(\left(x - 6\right)^{6}; x\rightarrow6\right)$$

exp(x - 6).series(x, 6).removeO().subs(x, x - 6)

$$x + \frac{1}{120} \left(x - 12\right)^{5} + \frac{1}{24} \left(x - 12\right)^{4} + \frac{1}{6} \left(x - 12\right)^{3} + \frac{1}{2} \left(x - 12\right)^{2} - 11$$

Working with matrices

M = Matrix([[1,2,3],[3,2,1]])
P = Matrix([0,1,1])
M*P

$$\left[\begin{matrix}5\\3\end{matrix}\right]$$

M

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

M.shape

$$\left ( 2, \quad 3\right )$$

M.col(-1)

$$\left[\begin{matrix}3\\1\end{matrix}\right]$$

M = Matrix([[1,3], [-2,3]])
M**-1

$$\left[\begin{matrix}\frac{1}{3} & - \frac{1}{3}\\\frac{2}{9} & \frac{1}{9}\end{matrix}\right]$$

M.T

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

eye(3)

$$\left[\begin{matrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{matrix}\right]$$

diag(1,2,3)

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

M.det()

$$9$$

M= Matrix([[1,0,1,3],[2,3,4,7],[-1,-3,-3,-4]])
M

$$\left[\begin{matrix}1 & 0 & 1 & 3\\2 & 3 & 4 & 7\\-1 & -3 & -3 & -4\end{matrix}\right]$$

M.rref()

$$\left ( \left[\begin{matrix}1 & 0 & 1 & 3\\0 & 1 & \frac{2}{3} & \frac{1}{3}\\0 & 0 & 0 & 0\end{matrix}\right], \quad \left [ 0, \quad 1\right ]\right )$$

M = Matrix([[1,2,3,0,0],[4,10,0,0,1]])
M

$$\left[\begin{matrix}1 & 2 & 3 & 0 & 0\\4 & 10 & 0 & 0 & 1\end{matrix}\right]$$

M.nullspace()

$$\left [ \left[\begin{matrix}-15\\6\\1\\0\\0\end{matrix}\right], \quad \left[\begin{matrix}0\\0\\0\\1\\0\end{matrix}\right], \quad \left[\begin{matrix}1\\- \frac{1}{2}\\0\\0\\1\end{matrix}\right]\right ]$$

M = Matrix([[3, -2, 4, -2], [5,3,-3,-2], [5,-2,2,-2], [5,-2,-3,3]])
M

$$\left[\begin{matrix}3 & -2 & 4 & -2\\5 & 3 & -3 & -2\\5 & -2 & 2 & -2\\5 & -2 & -3 & 3\end{matrix}\right]$$

M.eigenvals()

$$\left \{ -2 : 1, \quad 3 : 1, \quad 5 : 2\right \}$$

M.eigenvects()

$$\left [ \left ( -2, \quad 1, \quad \left [ \left[\begin{matrix}0\\1\\1\\1\end{matrix}\right]\right ]\right ), \quad \left ( 3, \quad 1, \quad \left [ \left[\begin{matrix}1\\1\\1\\1\end{matrix}\right]\right ]\right ), \quad \left ( 5, \quad 2, \quad \left [ \left[\begin{matrix}1\\1\\1\\0\end{matrix}\right], \quad \left[\begin{matrix}0\\-1\\0\\1\end{matrix}\right]\right ]\right )\right ]$$

P, D = M.diagonalize()

P

$$\left[\begin{matrix}0 & 1 & 1 & 0\\1 & 1 & 1 & -1\\1 & 1 & 1 & 0\\1 & 1 & 0 & 1\end{matrix}\right]$$

D

$$\left[\begin{matrix}-2 & 0 & 0 & 0\\0 & 3 & 0 & 0\\0 & 0 & 5 & 0\\0 & 0 & 0 & 5\end{matrix}\right]$$

P*D*P**-1

$$\left[\begin{matrix}3 & -2 & 4 & -2\\5 & 3 & -3 & -2\\5 & -2 & 2 & -2\\5 & -2 & -3 & 3\end{matrix}\right]$$

lamda = symbols('lamda')
p = M.charpoly(lamda)
p

$$\operatorname{PurePoly}{\left( \lambda^{4} - 11 \lambda^{3} + 29 \lambda^{2} + 35 \lambda - 150, \lambda, domain=\mathbb{Z} \right)}$$

factor(p)

$$\left(\lambda - 5\right)^{2} \left(\lambda - 3\right) \left(\lambda + 2\right)$$

Solving Algebraic and Differential Equations

solve(x**2 - 1, x)

$$\left [ -1, \quad 1\right ]$$

solve((x - y + 2, x + y -3), (x, y))

$$\left \{ x : \frac{1}{2}, \quad y : \frac{5}{2}\right \}$$

solve(x**3 - 6*x**2 + 9*x, x)

$$\left [ 0, \quad 3\right ]$$

roots(x**3 - 6*x**2 + 9*x, x)

$$\left \{ 0 : 1, \quad 3 : 2\right \}$$

f, g = symbols('f g', cls=Function)

f(x).diff(x)

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

diffeq = Eq(f(x).diff(x, 2) - 2*f(x).diff(x) + f(x), sin(x))

diffeq

$$f{\left (x \right )} - 2 \frac{d}{d x} f{\left (x \right )} + \frac{d^{2}}{d x^{2}} f{\left (x \right )} = \sin{\left (x \right )}$$

dsolve(diffeq, f(x))

$$f{\left (x \right )} = \left(C_{1} + C_{2} x\right) e^{x} + \frac{1}{2} \cos{\left (x \right )}$$

dsolve(f(x).diff(x)*(1 - sin(f(x))), f(x))

$$f{\left (x \right )} + \cos{\left (f{\left (x \right )} \right )} = C_{1}$$

a, t = symbols('a t')
f(t).diff(t)
diffeq = Eq(f(t).diff(t), a*t)

diffeq

$$\frac{d}{d t} f{\left (t \right )} = a t$$

dsolve(diffeq, f(t))

$$f{\left (t \right )} = C_{1} + \frac{a t^{2}}{2}$$

x = symbols('x', cls=Function)
diffeq = Eq(x(t).diff(t), a*x(t))
diffeq

$$\frac{d}{d t} x{\left (t \right )} = a x{\left (t \right )}$$

dsolve(diffeq, x(t))

$$x{\left (t \right )} = C_{1} e^{a t}$$

Numerics

N(pi, 10)

$$3.141592654$$

x = symbols('x')

expr = Integral(sin(x)/(x**2), (x, 1, oo))

expr.evalf()

$$0.5$$

expr.evalf(maxn=20)

$$0.5$$

expr.evalf(quad='osc')

$$0.504067061906928$$

expr.evalf(20, quad='osc')

$$0.50406706190692837199$$

expr = Integral(sin(1/x), (x, 0, 1))
expr

$$\int_{0}^{1} \sin{\left (\frac{1}{x} \right )}\, dx$$

expr.evalf()

$$0.5$$

expr = expr.transform(x, 1/x)
expr

$$\int_{1}^{\infty} \frac{1}{x^{2}} \sin{\left (x \right )}\, dx$$

expr.evalf(quad='osc')

$$0.504067061906928$$

nsimplify(pi, tolerance=0.001)

$$\frac{355}{113}$$

expr = sin(x)/x

%timeit expr.evalf(subs={x: 3.14})

1000 loops, best of 3: 397 µs per loop

f1 = lambdify(x, expr)
%timeit f1(3.14)

The slowest run took 13.44 times longer than the fastest. This could mean that an intermediate result is being cached
1000000 loops, best of 3: 308 ns per loop

f2 = lambdify(x, expr, 'numpy')
%timeit f2(3.14)

The slowest run took 16.41 times longer than the fastest. This could mean that an intermediate result is being cached
100000 loops, best of 3: 2.2 µs per loop

%timeit f2(np.linspace(1, 10, 10000))

1000 loops, best of 3: 306 µs per loop

%timeit [f1(x) for x in np.linspace(1, 10, 10000)]

100 loops, best of 3: 5.19 ms per loop

from mpmath import *

f = odefun(lambda x, y: [-y[1], y[0]], 0, [1, 0])
for x in [0, 1, 2.5, 10]:
    nprint(f(x), 15)
    nprint([cos(x), sin(x)], 15)

[1.0, 0.0]
[1.0, 0.0]
[0.54030230586814, 0.841470984807897]
[0.54030230586814, 0.841470984807897]
[-0.801143615546934, 0.598472144103957]
[-0.801143615546934, 0.598472144103957]
[-0.839071529076452, -0.54402111088937]
[-0.839071529076452, -0.54402111088937]

from sympy.plotting import plot
%matplotlib inline

plot(x*y**3 - y*x**3)
pass

from sympy.plotting import plot3d_parametric_surface
from sympy import sin, cos
u, v = symbols('u v')
plot3d_parametric_surface(cos(u + v), sin(u - v), u-v, (u, -5, 5), (v, -5, 5))
pass

Statistics

from sympy.stats import *

k = Symbol("k", positive=True)
theta = Symbol("theta", positive=True)
z = Symbol("z")
X = Gamma("x", k, theta)

D = density(X)(z)
D

$$\frac{z^{k - 1} e^{- \frac{z}{\theta}}}{\theta^{k} \Gamma{\left(k \right)}}$$

C = cdf(X, meijerg=True)(z)
C

$$\begin{cases} - \frac{k \gamma\left(k, 0\right)}{\Gamma{\left(k + 1 \right)}} + \frac{k \gamma\left(k, \frac{z}{\theta}\right)}{\Gamma{\left(k + 1 \right)}} & \text{for}\: z \geq 0 \\0 & \text{otherwise} \end{cases}$$

E(X)

$$\frac{\theta}{\Gamma{\left(k \right)}} \Gamma{\left(k + 1 \right)}$$

V = variance(X)
V

$$\frac{\theta^{3} \theta^{k - 1}}{\theta^{k} \Gamma^{2}{\left(k \right)}} \Gamma^{2}{\left(k + 1 \right)} - \frac{2 \theta^{2}}{\Gamma^{2}{\left(k \right)}} \Gamma^{2}{\left(k + 1 \right)} + \frac{\theta \theta^{k + 1}}{\theta^{k} \Gamma{\left(k \right)}} \Gamma{\left(k + 2 \right)}$$

simplify(V)

$$k \theta^{2}$$

N = Normal('Gaussian', 10, 5)
density(N)(z)

$$\frac{\sqrt{2}}{10 \sqrt{\pi}} e^{- \frac{1}{50} \left(z - 10\right)^{2}}$$

density(N)(3).evalf()

$$0.029945493127149$$

simplify(cdf(N)(z))

$$\frac{1}{2} \operatorname{erf}{\left (\frac{\sqrt{2}}{10} \left(z - 10\right) \right )} + \frac{1}{2}$$

P(N > 10)

$$\frac{1}{2}$$

sample(N)

$$6.30254802079348$$