Robert Johansson
Source code listings for Numerical Python - Scientific Computing and Data Science Applications with Numpy, SciPy and Matplotlib (ISBN 978-1-484242-45-2).
import sympy
sympy.init_printing()
from sympy import I, pi, oo
x = sympy.Symbol("x")
y = sympy.Symbol("y", real=True)
y.is_real
True
x.is_real is None
True
sympy.Symbol("z", imaginary=True).is_real
False
x = sympy.Symbol("x")
y = sympy.Symbol("y", positive=True)
sympy.sqrt(x ** 2)
sympy.sqrt(y ** 2)
n1, n2, n3 = sympy.Symbol("n"), sympy.Symbol("n", integer=True), sympy.Symbol("n", odd=True)
sympy.cos(n1 * pi)
sympy.cos(n2 * pi)
sympy.cos(n3 * pi)
a, b, c = sympy.symbols("a, b, c", negative=True)
d, e, f = sympy.symbols("d, e, f", positive=True)
i = sympy.Integer(19)
"i = {} [type {}]".format(i, type(i))
"i = 19 [type <class 'sympy.core.numbers.Integer'>]"
i.is_Integer, i.is_real, i.is_odd
(True, True, True)
f = sympy.Float(2.3)
"f = {} [type {}]".format(f, type(f))
"f = 2.30000000000000 [type <class 'sympy.core.numbers.Float'>]"
f.is_Integer, f.is_real, f.is_odd
(False, True, False)
i, f = sympy.sympify(19), sympy.sympify(2.3)
type(i)
sympy.core.numbers.Integer
type(f)
sympy.core.numbers.Float
n = sympy.Symbol("n", integer=True)
n.is_integer, n.is_Integer, n.is_positive, n.is_Symbol
(True, False, None, True)
i = sympy.Integer(19)
i.is_integer, i.is_Integer, i.is_positive, i.is_Symbol
(True, True, True, False)
i ** 50
sympy.factorial(100)
"%.25f" % 0.3 # create a string represention with 25 decimals
'0.2999999999999999888977698'
sympy.Float(0.3, 25)
sympy.Float('0.3', 25)
sympy.Rational(11, 13)
r1 = sympy.Rational(2, 3)
r2 = sympy.Rational(4, 5)
r1 * r2
r1 / r2
x, y, z = sympy.symbols("x, y, z")
f = sympy.Function("f")
type(f)
sympy.core.function.UndefinedFunction
f(x)
g = sympy.Function("g")(x, y, z)
g
g.free_symbols
sympy.sin
sin
sympy.sin(x)
sympy.sin(pi * 1.5)
n = sympy.Symbol("n", integer=True)
sympy.sin(pi * n)
h = sympy.Lambda(x, x**2)
h
h(5)
h(1+x)
x = sympy.Symbol("x")
e = 1 + 2 * x**2 + 3 * x**3
e
e.args
e.args[1]
e.args[1].args[1]
e.args[1].args[1].args[0]
e.args[1].args[1].args[0].args
expr = 2 * (x**2 - x) - x * (x + 1)
expr
sympy.simplify(expr)
expr.simplify()
expr
expr = 2 * sympy.cos(x) * sympy.sin(x)
expr
sympy.trigsimp(expr)
expr = sympy.exp(x) * sympy.exp(y)
expr
sympy.powsimp(expr)
expr = (x + 1) * (x + 2)
sympy.expand(expr)
sympy.sin(x + y).expand(trig=True)
a, b = sympy.symbols("a, b", positive=True)
sympy.log(a * b).expand(log=True)
sympy.exp(I*a + b).expand(complex=True)
sympy.expand((a * b)**x, power_exp=True)
sympy.exp(I*(a-b)*x).expand(power_exp=True)
sympy.factor(x**2 - 1)
sympy.factor(x * sympy.cos(y) + sympy.sin(z) * x)
sympy.logcombine(sympy.log(a) - sympy.log(b))
expr = x + y + x * y * z
expr.factor()
expr.collect(x)
expr.collect(y)
expr = sympy.cos(x + y) + sympy.sin(x - y)
expr.expand(trig=True).collect([sympy.cos(x), sympy.sin(x)]).collect(sympy.cos(y) - sympy.sin(y))
sympy.apart(1/(x**2 + 3*x + 2), x)
sympy.together(1 / (y * x + y) + 1 / (1+x))
sympy.cancel(y / (y * x + y))
(x + y).subs(x, y)
sympy.sin(x * sympy.exp(x)).subs(x, y)
sympy.sin(x * z).subs({z: sympy.exp(y), x: y, sympy.sin: sympy.cos})
expr = x * y + z**2 *x
values = {x: 1.25, y: 0.4, z: 3.2}
expr.subs(values)
sympy.N(1 + pi)
sympy.N(pi, 50)
(x + 1/pi).evalf(7)
expr = sympy.sin(pi * x * sympy.exp(x))
[expr.subs(x, xx).evalf(3) for xx in range(0, 10)]
expr_func = sympy.lambdify(x, expr)
expr_func(1.0)
expr_func = sympy.lambdify(x, expr, 'numpy')
import numpy as np
xvalues = np.arange(0, 10)
expr_func(xvalues)
array([ 0. , 0.77394269, 0.64198244, 0.72163867, 0.94361635, 0.20523391, 0.97398794, 0.97734066, -0.87034418, -0.69512687])
f = sympy.Function('f')(x)
sympy.diff(f, x)
sympy.diff(f, x, x)
sympy.diff(f, x, 3)
g = sympy.Function('g')(x, y)
g.diff(x, y)
g.diff(x, 3, y, 2) # equivalent to s.diff(g, x, x, x, y, y)
expr = x**4 + x**3 + x**2 + x + 1
expr.diff(x)
expr.diff(x, x)
expr = (x + 1)**3 * y ** 2 * (z - 1)
expr.diff(x, y, z)
expr = sympy.sin(x * y) * sympy.cos(x / 2)
expr.diff(x)
expr = sympy.special.polynomials.hermite(x, 0)
expr.diff(x).doit()
d = sympy.Derivative(sympy.exp(sympy.cos(x)), x)
d
d.doit()
a, b = sympy.symbols("a, b")
x, y = sympy.symbols('x, y')
f = sympy.Function('f')(x)
sympy.integrate(f)
sympy.integrate(f, (x, a, b))
sympy.integrate(sympy.sin(x))
sympy.integrate(sympy.sin(x), (x, a, b))
sympy.integrate(sympy.exp(-x**2), (x, 0, oo))
a, b, c = sympy.symbols("a, b, c", positive=True)
sympy.integrate(a * sympy.exp(-((x-b)/c)**2), (x, -oo, oo))
sympy.integrate(sympy.sin(x * sympy.cos(x)))
expr = sympy.sin(x*sympy.exp(y))
sympy.integrate(expr, x)
expr = (x + y)**2
sympy.integrate(expr, x)
sympy.integrate(expr, x, y)
sympy.integrate(expr, (x, 0, 1), (y, 0, 1))
x = sympy.Symbol("x")
f = sympy.Function("f")(x)
sympy.series(f, x)
x0 = sympy.Symbol("{x_0}")
f.series(x, x0, n=2)
f.series(x, x0, n=2).removeO()
sympy.cos(x).series()
sympy.sin(x).series()
sympy.exp(x).series()
(1/(1+x)).series()
expr = sympy.cos(x) / (1 + sympy.sin(x * y))
expr.series(x, n=4)
expr.series(y, n=4)
expr.series(y).removeO().series(x).removeO().expand()
sympy.limit(sympy.sin(x) / x, x, 0)
f = sympy.Function('f')
x, h = sympy.symbols("x, h")
diff_limit = (f(x + h) - f(x))/h
sympy.limit(diff_limit.subs(f, sympy.cos), h, 0)
sympy.limit(diff_limit.subs(f, sympy.sin), h, 0)
expr = (x**2 - 3*x) / (2*x - 2)
p = sympy.limit(expr/x, x, oo)
q = sympy.limit(expr - p*x, x, oo)
p, q
n = sympy.symbols("n", integer=True)
x = sympy.Sum(1/(n**2), (n, 1, oo))
x
x.doit()
x = sympy.Product(n, (n, 1, 7))
x
x.doit()
x = sympy.Symbol("x")
sympy.Sum((x)**n/(sympy.factorial(n)), (n, 1, oo)).doit().simplify()
x = sympy.symbols("x")
sympy.solve(x**2 + 2*x - 3)
a, b, c = sympy.symbols("a, b, c")
sympy.solve(a * x**2 + b * x + c, x)
sympy.solve(sympy.sin(x) - sympy.cos(x), x)
sympy.solve(sympy.exp(x) + 2 * x, x)
sympy.solve(x**5 - x**2 + 1, x)
1 #s.solve(s.tan(x) - x, x)
eq1 = x + 2 * y - 1
eq2 = x - y + 1
sympy.solve([eq1, eq2], [x, y], dict=True)
eq1 = x**2 - y
eq2 = y**2 - x
sols = sympy.solve([eq1, eq2], [x, y], dict=True)
sols
[eq1.subs(sol).simplify() == 0 and eq2.subs(sol).simplify() == 0 for sol in sols]
[True, True, True, True]
sympy.Matrix([1,2])
sympy.Matrix([[1,2]])
sympy.Matrix([[1, 2], [3, 4]])
sympy.Matrix(3, 4, lambda m,n: 10 * m + n)
a, b, c, d = sympy.symbols("a, b, c, d")
M = sympy.Matrix([[a, b], [c, d]])
M
M * M
x = sympy.Matrix(sympy.symbols("x_1, x_2"))
M * x
p, q = sympy.symbols("p, q")
M = sympy.Matrix([[1, p], [q, 1]])
M
b = sympy.Matrix(sympy.symbols("b_1, b_2"))
b
x = M.solve(b)
x
x = M.LUsolve(b)
x
x = M.inv() * b
x
%reload_ext version_information
%version_information sympy, numpy
Software | Version |
---|---|
Python | 3.6.8 64bit [GCC 4.2.1 Compatible Clang 4.0.1 (tags/RELEASE_401/final)] |
IPython | 7.5.0 |
OS | Darwin 18.2.0 x86_64 i386 64bit |
sympy | 1.4 |
numpy | 1.16.3 |
Mon May 06 14:25:08 2019 JST |