Tutorial 4: SymPy¶

from sympy import *

# numerical expression
simplify('5**3')
125

simplify('(5**2)/2 + 6')
37/2

# symbolic expressions
init_printing(use_unicode=True)
x = symbols('x')

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

simplify((x**2 + 2*x + 1)/(sqrt(x + 1)))
 2
x  + 2⋅x + 1
────────────
  _______
         ╲╱ x + 1

simplify(sin(x)**2 + cos(x)**2)
1

# polynomials, rational functions
expand((x+1)**2)
 2
x  + 2⋅x + 1

factor(x**2 + 5*x + 6)
(x + 2)⋅(x + 3)

apart((x**2 + x + 1)/(x**3 + 3*x**2 + 3*x + 1))
  1        1          1
───── - ──────── + ────────
x + 1          2          3
        (x + 1)    (x + 1)

# single equation
solveset(x**2 - 1 , x)
{-1, 1}

solveset(sin(x) - 1, x)
            ⎧        π           ⎫
            ⎨2⋅n⋅π + ─ | n ∊ ℤ⎬
            ⎩        2           ⎭

solveset(1/x, x)
            ∅

# linear equations
linsolve([x + y + z - 1, x + y + 2*z - 3, x - z + 4], (x, y, z))
{(-2, 1, 2)}

linsolve(Matrix(([1, 1, 1, 1], [1, 1, 2, 3])), (x, y, z))
{(-y - 1, y, 2)}

# differential equation
f = symbols('f', cls=Function)

dsolve(f(x).diff(x,x) + f(x), f(x))
f(x) = C₁⋅sin(x) + C₂⋅cos(x)

dsolve(f(x).diff(x,x) - 2*f(x)*diff(x) + f(x) - sin(x), f(x))
         -x    x   sin(x)
f(x) = C₁⋅ℯ   + C₂⋅ℯ  - ──────
                     2

# definition
Matrix([1,2,3])
          ⎡1⎤
          ⎢   ⎥
          ⎢2⎥
          ⎢   ⎥
          ⎣3⎦

Matrix(([1,2,3], [4,5,6], [7,8,9]))
          ⎡1  2  3⎤
          ⎢                     ⎥
          ⎢4  5  6⎥
          ⎢                     ⎥
          ⎣7  8  9⎦

# basic operations
Matrix(([1,2,3], [4,5,6], [7,8,9])) * Matrix([1,1,1])
          ⎡6 ⎤
          ⎢      ⎥
          ⎢15⎥
          ⎢      ⎥
          ⎣24⎦

Matrix(([1,2,3], [4,5,6])) + Matrix(([1,1,1], [0,0,1]))
          ⎡2  3  4⎤
          ⎢                     ⎥
          ⎣4  5  7⎦

Matrix([1,2,3]).T
[1  2  3]

# advanced operations
Matrix(([1,2], [1,0])).det()
-2

Matrix([[1, 0, 1, 3], [2, 3, 4, 7], [-1, -3, -3, -4]]).rref()
          ⎛⎡1  0   1    3 ⎤, [0, 1]⎞
          ⎜⎢                                          ⎥                        ⎟
          ⎜⎢0  1  2/3  1/3⎥                        ⎟
          ⎜⎢                                          ⎥                        ⎟
          ⎝⎣0  0   0    0 ⎦                        ⎠

Matrix([[1, 0, 1, 3], [2, 3, 4, 7], [-1, -3, -3, -4]]).columnspace()
         ⎡⎡1 ⎤, ⎡0 ⎤⎤
         ⎢⎢      ⎥      ⎢      ⎥⎥
         ⎢⎢2 ⎥      ⎢3 ⎥⎥
         ⎢⎢      ⎥      ⎢      ⎥⎥
         ⎣⎣-1⎦      ⎣-3⎦⎦

Matrix([[3, -2,  4, -2], [5,  3, -3, -2], [5, -2,  2, -2], [5, -2, -3,  3]]).eigenvects()
         ⎡⎛-2, 1, ⎡⎡0⎤⎤⎞, ⎛3, 1, ⎡⎡1⎤⎤⎞, ⎛5, 2, ⎡⎡1⎤, ⎡0 ⎤⎤⎞⎤
         ⎢⎜                     ⎢⎢   ⎥⎥⎟      ⎜                  ⎢⎢   ⎥⎥⎟      ⎜                  ⎢⎢   ⎥      ⎢      ⎥⎥⎟⎥
         ⎢⎜                     ⎢⎢1⎥⎥⎟      ⎜                  ⎢⎢1⎥⎥⎟      ⎜                  ⎢⎢1⎥      ⎢-1⎥⎥⎟⎥
         ⎢⎜                     ⎢⎢   ⎥⎥⎟      ⎜                  ⎢⎢   ⎥⎥⎟      ⎜                  ⎢⎢   ⎥      ⎢      ⎥⎥⎟⎥
         ⎢⎜                     ⎢⎢1⎥⎥⎟      ⎜                  ⎢⎢1⎥⎥⎟      ⎜                  ⎢⎢1⎥      ⎢0 ⎥⎥⎟⎥
         ⎢⎜                     ⎢⎢   ⎥⎥⎟      ⎜                  ⎢⎢   ⎥⎥⎟      ⎜                  ⎢⎢   ⎥      ⎢      ⎥⎥⎟⎥
         ⎣⎝                     ⎣⎣1⎦⎦⎠      ⎝                  ⎣⎣1⎦⎦⎠      ⎝                  ⎣⎣0⎦      ⎣1 ⎦⎦⎠⎦

# derivatives
diff(cos(x), x)
-sin(x)

diff(-1/(x**2), x)
2
──
 3
x

diff(-y*tan(y)/(x**2) + y, x, y)
              ⎛     ⎛   2         ⎞                        ⎞
2⋅⎝y⋅⎝tan (y) + 1⎠ + tan(y)⎠
────────────────────────────
            3
           x

# integrals
integrate(x**2, x)
 3
x
──
3

integrate(exp(-x**2), x)
√π⋅erf(x)
─────────
    2

integrate(exp(-x**2), (x, -oo, oo))
√π

# limits
limit((exp(x) - 1)/x, x, 0)
1

limit(1/x, x, 0, '-')
-∞