chapter 4: Algebra And Symbolic Math with Sympy

이 장에서는 SymPy 라이브러리를 집중 분석해 보도록 하자.

4.1 Defining Symbols and Symbolic Operations

>>> from sympy import Symbol
>>> x = Symbol('x')

>>> from sympy import Symbol
>>> x = Symbol('x')
>>> x + x + 1
2*x + 1


>>> x = Symbol('x')
>>> y = Symbol('y')
>>> z = Symbol('z')


>>> from sympy import symbols
>>> x,y,z = symbols('x,y,z')

>>> from sympy import Symbol
>>> x = Symbol('x')
>>> y = Symbol('y')
96 Chapter 4
>>> s = x*y + x*y
>>> s
2*x*y

>>> p = x*(x + x)
>>> p
2*x**2

>>> p = (x + 2)*(x + 3)
>>> p
(x + 2)*(x + 3)

4.2 Working with Expressions

Factorizing and Expanding Expressions

x2 − y2 = (x + y)(x − y).

from sympy import Symbol
x = Symbol('x')
y = Symbol('y')

from sympy import factor,expand
expr = x**2 - y**2
print(factor(expr))
#(x - y)*(x + y)

factors = factor(expr)
print(expand(factors))
#x**2 - y**2

x3 + 3x2y + 3xy2 + y3 = (x + y)3

from sympy import factor,expand
expr = x**3 + 3*x**2*y + 3*x*y**2 + y**3
factors = factor(expr)
print(factors)
#(x + y)**3
print(expand(factors))
#x**3 + 3*x**2*y + 3*x*y**2 + y**3

Pretty Printing

from sympy import Symbol,factor,expand
x=Symbol('x')
y=Symbol('y')
expr = x*x + 2*x*y + y*y
print(expr)


from sympy import Symbol,factor,expand
x=Symbol('x')
y=Symbol('y')
expr = x*x + 2*x*y + y*y
print(expr)

from sympy import pprint

pprint(expr)

expr = 1 + 2*x + 2*x**2

pprint(expr)

#-*-coding:utf-8
from sympy import Symbol,factor,expand
x=Symbol('x')
y=Symbol('y')
expr = x*x + 2*x*y + y*y
print(expr)

from sympy import pprint

pprint(expr)

expr = 1 + 2*x + 2*x**2

pprint(expr)


from sympy import init_printing
init_printing(order='rev-lex')
pprint(expr)
#1 + 2·x + 2·x 2

Printing a Series

_images/chapter4-1.png 
'''
Print the series:
x + x**2 + x**3 + ... + x**n
____ _____ _____
2 3 n
'''
from sympy import Symbol, pprint, init_printing
def print_series(n):
    # Initialize printing system with reverse order
    init_printing(order='rev-lex')
    x = Symbol('x')
    series = x
    for i in range(2, n+1):
        series = series + (x**i)/i
    pprint(series)

if __name__ == '__main__':
        n = input('Enter the number of terms you want in the series: ')
        print_series(int(n))

Substituting in Values

x2 + 2xy + y2

from sympy import Symbol

x=Symbol('x')
y=Symbol('y')

print(x*x + x*y + x*y + y*y)

expr = x*x + x*y + x*y + y*y

res = expr.subs({x:1, y:2})

print(res)

print(expr.subs({x:1-y}))


expr_subs = expr.subs({x:1-y})

from sympy import simplify

print(simplify(expr_subs))

Calculating the Value of a Series

'''
Print the series:
x + x**2 + x**3 + ... + x**n
____ _____ _____
2 3 n
'''
from sympy import Symbol, pprint, init_printing
def print_series(n, x_value):
    # Initialize printing system with reverse order
    init_printing(order='rev-lex')
    x = Symbol('x')
    series = x
    for i in range(2, n+1):
        series = series + (x**i)/i
    pprint(series)
    # Evaluate the series at x_value
    series_value = series.subs({x:x_value})
    print('Value of the series at {0}: {1}'.format(x_value, series_value))
if __name__ == '__main__':
    n = input('Enter the number of terms you want in the series: ')
    x_value = input('Enter the value of x at which you want to evaluate the series: ')
    print_series(int(n), float(x_value))

Converting Strings to Mathematical Expressions

from sympy import sympify
expr = input('Enter a mathematical expression: ')
#Enter a mathematical expression: x**2 + 3*x + x**3 + 2*x
expr = sympify(expr)

print(2*expr)

#expr = input('Enter a mathematical expression: ')
#Enter a mathematical expression: x**2 + 3*x + x**3 + 2x
#expr = sympify(expr)
#print(expr)


from sympy import sympify
from sympy.core.sympify import SympifyError
expr = input('Enter a mathematical expression: ')
#Enter a mathematical expression: x**2 + 3*x + x**3 + 2x
try:
    expr = sympify(expr)
except SympifyError:
    print('Invalid input')

Expression Multiplier

'''
Product of two expressions
'''
from sympy import expand, sympify
from sympy.core.sympify import SympifyError
def product(expr1, expr2):
    prod = expand(expr1*expr2)
    print(prod)
if __name__=='__main__':
    expr1 = input('Enter the first expression: ')
    #Enter the first expression: x**2 + x*2 + x
    expr2 = input('Enter the second expression: ')
    #Enter the second expression: x**3 + x*3 + x
    try:
        expr1 = sympify(expr1)
        expr2 = sympify(expr2)
    except SympifyError:
        print('Invalid input')
    else:
        product(expr1, expr2)

#x**5 + 3*x**4 + 4*x**3 + 12*x**2


#===============================================================================
# Enter the first expression: x*y+x
# Enter the second expression: x*x+y
# x**3*y + x**3 + x*y**2 + x*y
#===============================================================================

Solving Equations

from sympy import Symbol, solve
x = Symbol('x')
expr = x - 5 - 7
print(solve(expr))

Solving Quadratic Equations

from sympy import solve,Symbol
x = Symbol('x')
expr = x**2 + 5*x + 4
print(solve(expr, dict=True))
#x [{x: -4}, {x: -1}]

x2 + x + 1 = 0

   from sympy import Symbol,solve
   x=Symbol('x')
   expr = x**2 + x + 1
   print(solve(expr, dict=True))
   #[{x: -1/2 - sqrt(3)*I/2}, {x: -1/2 + sqrt(3)*I/2}]


Solving for One Variable in Terms of Others

ax2 + bx + c = 0

from sympy import symbol

x = Symbol('x')
a = Symbol('a')
b = Symbol('b')
c = Symbol('c')

expr = a*x*x + b*x + c
print(solve(expr, x, dict=True))
#[{x: (-b + sqrt(-4*a*c + b**2))/(2*a)}, {x: -(b + sqrt(-4*a*c + b**2))/(2*a)}]
_images/chapter4-2.png 
   from sympy import Symbol, solve, pprint
   s = Symbol('s')
   u = Symbol('u')
   t = Symbol('t')
   a = Symbol('a')
   expr = u*t + (1/2)*a*t*t - s
   t_expr = solve(expr,t, dict=True)
   pprint(t_expr)


.. image:: ./img/chapter4-3.png

Solving a System of Linear Equations

2x + 3y = 6 3x + 2y = 12

#-*-coding=utf-8
from sympy import Symbol,solve

x = Symbol('x')
y = Symbol('y')
expr1 = 2*x + 3*y - 6
expr2 = 3*x + 2*y - 12

print(solve((expr1, expr2), dict=True))


soln = solve((expr1, expr2), dict=True)
soln = soln[0]
print(expr1.subs({x:soln[x], y:soln[y]}))
#0
print(expr2.subs({x:soln[x], y:soln[y]}))
#0

Plotting Using SymPy

from sympy.plotting import plot
from sympy import Symbol
x = Symbol('x')
#plot(2*x+3)

#plot((2*x + 3), (x, -5, 5))

#plot(2*x + 3, (x, -5, 5), title='A Line', xlabel='x', ylabel='2x+3')

p = plot(2*x + 3, (x, -5, 5), title='A Line', xlabel='x', ylabel='2x+3', show=False)
p.save('line.png')

Plotting Expressions Input by the User

y = 2x + 3,

from sympy import Symbol,solve,sympify
expr = input('Enter an expression: ')
#Enter an expression: 2*x + 3*y - 6
expr = sympify(expr)
y = Symbol('y')
print(solve(expr, y))

'''
Plot the graph of an input expression
'''
from sympy import Symbol, sympify, solve
from sympy.plotting import plot
def plot_expression(expr):
    y = Symbol('y')
    solutions = solve(expr, y)
    expr_y = solutions[0]
    plot(expr_y)

if __name__=='__main__':
    expr = input('Enter your expression in terms of x and y: ')
    try:
        expr = sympify(expr)
    except SympifyError:
        print('Invalid input')
    else:
        plot_expression(expr)

Plotting Multiple Functions

from sympy.plotting import plot
from sympy import Symbol
x = Symbol('x')
plot(2*x+3, 3*x+1)

from sympy.plotting import plot
from sympy import Symbol
x = Symbol('x')
p = plot(2*x+3, 3*x+1, legend=True, show=False)
p[0].line_color = 'b'
p[1].line_color = 'r'
p.show()