Import the library.



In [1]:

    
import polypy

Create variables $x$, $y$, $z$.



In [2]:

    
x = polypy.Variable('x')



In [3]:

    
[y, z] = [polypy.Variable(s) for s in ['y', 'z']]

Variable Ordering:



In [4]:

    
order = polypy.variable_order



In [5]:

    
order.push(z)



In [6]:

    
order.push(y)



In [7]:

    
order









    Out[7]:





polypy.VariableOrder([z, y])



In [8]:

    
order.pop()



In [9]:

    
order.push(x)



In [10]:

    
order









    Out[10]:





polypy.VariableOrder([z, x])

Creating Polynomials:

$$f = (x^2 - 2x + 1)(x^2 - 2), g = z(x^2 - y^2)$$



In [11]:

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



In [12]:

    
g = z*(x**2 - y**2)

Variable order & Polynomials



In [13]:

    
order.set([x, y, z])



In [14]:

    
print g









    



(-1*y**2 + (1*x**2))*z



In [15]:

    
order.set([z, y])



In [16]:

    
print g









    



(1*z)*x**2 + ((-1*z)*y**2)

Basic operations



In [17]:

    
g.var()









    Out[17]:





polypy.Variable('x')



In [18]:

    
g.degree()









    Out[18]:





2



In [19]:

    
g.coefficients()









    Out[19]:





[(-1*z)*y**2, 0, 1*z]



In [20]:

    
g.derivative()









    Out[20]:





(2*z)*x



In [21]:

    
f.factor_square_free()









    Out[21]:





[(1*x**2 - 2, 1), (1*x - 1, 2)]



In [22]:

    
g.factor_square_free()









    Out[22]:





[(1*z, 1), (1*x**2 + (-1*y**2), 1)]



In [23]:

    
f









    Out[23]:





1*x**4 - 2*x**3 - 1*x**2 + 4*x - 2



In [24]:

    
f.reductum()









    Out[24]:





-2*x**3 - 1*x**2 + 4*x - 2

Root isolation:



In [25]:

    
m = polypy.Assignment()



In [26]:

    
r = f.roots_isolate(m)



In [27]:

    
r









    Out[27]:





[<1*x**2 + (-2), (-3/2, -5/4)>, 1, <1*x**2 + (-2), (5/4, 3/2)>]



In [28]:

    
print r[0]









    



<1*x**2 + (-2), (-3/2, -5/4)>



In [29]:

    
print r[1]



In [30]:

    
print r[2]









    



<1*x**2 + (-2), (5/4, 3/2)>

Values can be integers, rationals, or algebraic numbers.



In [31]:

    
r[0].to_double()









    Out[31]:





-1.414213562373095



In [32]:

    
r[0] > r[1]









    Out[32]:





False



In [33]:

    
r[0].get_value_between(r[1])









    Out[33]:





-1



In [34]:

    
m.set_value(x, 0)



In [35]:

    
f.sgn(m)









    Out[35]:





-1



In [36]:

    
m.set_value(x, r[0])



In [37]:

    
f.sgn(m)









    Out[37]:





0



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