sympy/polynomials/tests/test_polynomials.py

   1 import py
   2 from sympy.utilities.pytest import XFAIL
   3
   4 from sympy import Polynomial, pi, symbols, Symbol, Rational, S, sqf_part, sqf,\
   5         solve_system, Integer, I, roots, cos, sin, resultant, \
   6         count_real_roots, lcm, groebner, gcd, factor, div, Real, \
   7         PolynomialException, sqrt
   8
   9 def test_Polynomial():
  10     x = Symbol("x")
  11     y = Symbol("y")
  12     z = Symbol('z')
  13     f = Polynomial(x+2)
  14     g = Polynomial(y**2-1)
  15     h = f + g
  16     assert f.var == [x]
  17     assert f.coeffs == ((1, 1), (2, 0))
  18     assert str(f) == "2 + x"
  19     assert repr(f) == "Polynomial(2 + x, ((1, 1), (2, 0)), [x], 'grevlex')"
  20     assert f == 2 + x
  21     assert f == Polynomial(None, f.coeffs, f.var, f.order)
  22     assert f.nth_coeff(0) == 2
  23     assert f.nth_coeff(2) == 0
  24     assert h.var == [x, y]
  25     assert h.coeffs == ((1, 0, 2), (1, 1, 0), (1, 0, 0))
  26     h = f*Polynomial(y, var=x)
  27     assert h.var == [x]
  28     assert h.coeffs == ((y, 1), (2*y, 0))
  29     h = f*y
  30     assert h == (x + 2)*y
  31     assert not isinstance(h, Polynomial)
  32     h = Polynomial(h, var=y)
  33     assert h.var == [y]
  34     assert h.coeffs == ((2+x, 1),)
  35     assert Polynomial(1, var=x).diff(x) == Polynomial(0, var=x)
  36     assert Polynomial(x**3*y).diff(x) == Polynomial(3*x**2*y)
  37     assert Polynomial(coeffs=((Integer(1), Integer(1)),), var=x).diff(x) \
  38            == Polynomial(1, var=x)
  39     assert Polynomial(x**2 + y**2)(3,-4) == 25
  40     assert Polynomial(y*x)(-z, z) == -z**2
  41
  42     #TODO better test that differs between all orders ?
  43     from sympy import sin
  44     assert Polynomial(1).coeffs == ((1,),)
  45     assert Polynomial(x).coeffs == ((1, 1),)
  46     assert Polynomial(x**2+y**3, order='lex').coeffs \
  47                    == ((1, 2, 0), (1, 0, 3))
  48     assert Polynomial(x**2+y**3, var=[y, x]).coeffs == ((1,3,0), (1,0,2))
  49     assert Polynomial(x*y).coeffs == ((1, 1, 1),)
  50     assert Polynomial(x**2*y**4 + sin(z)*x**3 + x*y**5,
  51                       var=[x, y], order='lex').coeffs \
  52            == ((sin(z), 3, 0), (1, 2, 4), (1, 1, 5))
  53     assert Polynomial(x**2*y**4 + sin(z)*x**3 + x*y**5,
  54                       var=[x, y], order='grlex').coeffs \
  55            == ((1, 2, 4), (1, 1, 5), (sin(z), 3, 0))
  56     assert Polynomial(x**2*y**4 + sin(z)*x**3 + x**5*y,
  57                       var=[x, y], order='grevlex').coeffs \
  58                == ((1, 5, 1), (1, 2, 4), (sin(z), 3, 0))
  59     assert Polynomial(z*x + x**2*y**2 + x**3*y,
  60                       var=[z, x, y], order='1-el').coeffs \
  61            == ((1, 1, 1, 0), (1, 0, 3, 1), (1, 0, 2, 2))
  62
  63     py.test.raises(PolynomialException, "Polynomial(sqrt(x),var=x)")
  64     py.test.raises(PolynomialException, "Polynomial(sin(x),var=x)")
  65
  66     assert 3*x**2 == Polynomial(coeffs=((Integer(3), Integer(2)),),
  67                                 var=x).sympy_expr
  68     assert 2*x + 3*x**2 - 5 \
  69            == Polynomial(coeffs=((Integer(-5), Integer(0)),
  70                                  (Integer(2), Integer(1)),
  71                                  (Integer(3), Integer(2))),
  72                          var=[x]).sympy_expr
  73     assert 2*x**100 + 3*x**2 - 5 \
  74            == Polynomial(coeffs=((Integer(-5), Integer(0)),
  75                                  (Integer(3), Integer(2)),
  76                                  (Integer(2), Integer(100))),
  77                          var=[x]).sympy_expr
  78
  79     assert sqrt(y)*x == Polynomial(coeffs=((sqrt(y), Integer(1)),),
  80                                    var=[x]).sympy_expr
  81
  82 def Polynomial1():
  83     x, y = symbols()
  84     p = Polynomial(x/3 + 12*y + x**2/8)
  85     assert p.as_integer() == (24, Polynomial(3*x**2 + 8*x + 288*y))
  86     assert p.as_monic() == (Rational(1,8), Polynomial(x**2 + 96*y + 8*x/3))
  87     assert p.as_primitive() == (0, p)
  88     p = Polynomial(100*x + 12*y + 8*x**2)
  89     assert p.as_primitive() == (4, Polynomial(2*x**2 + 3*y + 25*x))
  90     assert p.leading_coeff() == Rational(8)
  91     assert p.leading_term() == Polynomial(8*x**2)
  92
  93 def test_coeff_ring():
  94     from sympy.polynomials.base import coeff_ring
  95     x = Symbol("x")
  96     assert coeff_ring([Rational(2)]) == 'int'
  97     assert coeff_ring([Rational(2), Rational(1,2)]) == 'rat'
  98     assert coeff_ring([Rational(2)**Rational(1,2)]) == 'real'
  99     assert coeff_ring([pi]) == 'real'
 100     assert coeff_ring([Real(2.1), Rational(-1)**Rational(1,2)]) == 'cplx'
 101     assert coeff_ring([I, x]) == 'sym'
 102
 103
 104 ## sympy/modules/polynomials/wrapper.py
 105
 106 def test_div():
 107     x = Symbol("x")
 108     y = Symbol('y')
 109
 110     assert div(x**3-12*x**2-42, x-3, x) == (x**2-9*x-27, -123)
 111     assert div(x**3-12*x**2-42, x**2+x-3, x) == (x-13, 16*x-81)
 112
 113     assert div(2+2*x+x**2, 1, x) == (2+2*x+x**2, 0)
 114     assert div(2+2*x+x**2, 2, x, coeff='int') == (1+x, x**2)
 115
 116     assert div(3*x**3, x**2, x) == (3*x, 0)
 117
 118     assert div(1,1) == (1, 0)
 119     assert div(1,x,x) == (0, 1)
 120     assert div(x*y+2*x+y,x,x) == (2+y, y)
 121     assert div(x*y+2*x+y,x,y) == (2+(1+1/x)*y, 0)
 122
 123     assert div(x*y**2 + 1, [x*y+1, y+1], [x,y]) == ([y, -1], 2)
 124     assert div(x**2*y+x*y**2+y**2, [x*y-1, y**2-1], [x, y]) \
 125            == ([x+y, 1], 1+x+y)
 126     assert div(x**2*y+x*y**2+y**2, [y**2-1, x*y-1], [x, y]) \
 127            == ([1+x, x], 1+2*x)
 128
 129 def test_factor():
 130     x = Symbol("x")
 131     y = Symbol("y")
 132     z = Symbol("z")
 133     assert factor(Rational(3, 8)*x**2 - Rational(3, 2)) \
 134            == Rational(3, 8)*((x + 2)*(x - 2))
 135     assert factor(x**3-1) == (x-1)*(x**2+x+1)
 136     assert factor(x**2+2*x+1) == (x+1)**2
 137     assert factor(x**3-3*x**2+3*x-1) == (x-1)**3
 138     assert factor(x**2+x-2) == (x-1)*(x+2)
 139     assert factor(x**3-x) == x*(x-1)*(x+1)
 140     assert factor(x**6-1) == (1+x**2-x)*(1+x)*(1+x+x**2)*(-1+x)
 141     assert factor(2*x**2+5*x+2) == (2+x)*(1+2*x)
 142
 143     #assert factor(x**2 + y**2) == x**2 + y**2
 144     #assert factor(x*y + x*z + y*z) == x*y + x*z + y*z
 145     #assert factor(x*(y+1) + x*z) == x*(z + y + 1)
 146     #assert factor(x**5 - y**2) == x**5 - y**2
 147
 148     assert factor(-2) == -2
 149     assert factor(-x) == -x
 150     assert factor(-2*x**2+x) == x*(1 - 2*x)
 151
 152 def test_gcd():
 153     x = Symbol("x")
 154     y = Symbol("y")
 155     z = Symbol("z")
 156
 157     assert gcd(x**2, x, x) == x
 158     assert gcd(3*x**2, x, x) == x
 159     assert gcd(3*x**2, 6*x, x, coeff='rat') == x
 160     assert gcd(3*x**2, 6*x, x) == 3*x
 161     assert gcd(x**2+2*x+1, x+1, x) == x+1
 162     assert gcd(x**2+2*x+2, x+1, x) == 1
 163
 164     assert gcd(x**2+2*x+1, 2+2*x, x) == 1+x
 165     assert gcd(x**2+2*x+2, 2+2*x, x) == 1
 166
 167     assert gcd(4, 6) == Rational(2)
 168     assert gcd(6, 4, coeff='rat') == Rational(1)
 169     assert gcd(x, y) == Rational(1)
 170     assert gcd(sin(z)*(x+y), x**2+2*x*y+y**2, [x, y]) == x+y
 171
 172 def test_groebner():
 173     x = Symbol('x')
 174     y = Symbol('y')
 175     z = Symbol('z')
 176
 177     assert groebner(y*x, x) == [x]
 178     assert groebner(y*x, x, reduced=False) == [x*y]
 179     assert groebner(x*y, z) == [1]
 180
 181     # This one already is a Groebner base.
 182     assert groebner([y-x**2, z-x**3], [y,z,x], 'lex', reduced=False) \
 183            == [-x**2+y, z-x**3]
 184
 185     assert groebner([x**3-2*x*y, x**2*y-2*y**2+x], [x,y], 'grlex',
 186                     reduced=False) \
 187            == [x**3-2*x*y, x+x**2*y-2*y**2, -x**2, -2*x*y, -2*y**2+x]
 188     assert groebner([x**3-2*x*y, x**2*y-2*y**2+x], [x,y], 'grlex') \
 189            == [x**2, x*y, Rational(-1,2)*x+y**2]
 190
 191 def test_lcm():
 192     x = Symbol('x')
 193     y = Symbol('y')
 194
 195     assert lcm(6, 4) == Rational(12)
 196     assert lcm(6, 4, coeff='rat') == Rational(1)
 197     assert lcm(4, y) == 4*y
 198     assert lcm(x, y) == x*y
 199     assert lcm(y*(x+1), x, x) ==x+x**2
 200     assert lcm(2*x, x**2) == 2*x**2
 201
 202 def test_count_real_roots():
 203     x = Symbol('x')
 204
 205     f = x-1
 206     assert count_real_roots(f) == 1
 207     assert count_real_roots(f, None, Rational(0)) == 0
 208     assert count_real_roots(f, Rational(0), Rational(1)) == 1
 209     assert count_real_roots(f, Rational(1), None) == 0
 210     f = x**2 - 4
 211     assert count_real_roots(f) == 2
 212     assert count_real_roots(f, None, Rational(0)) == 1
 213     assert count_real_roots(f, Rational(-1), Rational(1)) == 0
 214
 215 def test_resultant():
 216      x, a, b, c, = [Symbol(y) for y in ['x', 'a', 'b', 'c']]
 217
 218      s_res = resultant(x**2-1, x**3-x**2+2, x, method='sylvester').expand()
 219      b_res = resultant(x**2-1, x**3-x**2+2, x, method='bezout').expand()
 220
 221      assert b_res == s_res == 0
 222
 223      s_res = resultant(3*x**3-x, 5*x**2+1, x, method='sylvester').expand()
 224      b_res = resultant(3*x**3-x, 5*x**2+1, x, method='bezout').expand()
 225
 226      assert b_res == s_res == 64
 227
 228      s_res = resultant(x**2-2*x+7, x**3-x+5, x, method='sylvester').expand()
 229      b_res = resultant(x**2-2*x+7, x**3-x+5, x, method='bezout').expand()
 230
 231      assert b_res == s_res == 265
 232
 233      s_res = resultant((x-a)**2-2, a**2-3, a, method='sylvester').expand()
 234      b_res = resultant((x-a)**2-2, a**2-3, a, method='bezout').expand()
 235
 236      assert b_res == s_res == 1 - 10*x**2 + x**4
 237
 238      s_res = resultant((x-1)*(x-2)*(x-3), (x-4)*(x-5)*(x-6), x, method='sylvester').expand()
 239      b_res = resultant((x-1)*(x-2)*(x-3), (x-4)*(x-5)*(x-6), x, method='bezout').expand()
 240
 241      assert b_res == s_res == -8640
 242
 243      s_res = resultant((x-1)*(x-2)*(x-3), (x-4)*(x-5)*(x-1), x, method='sylvester').expand()
 244      b_res = resultant((x-1)*(x-2)*(x-3), (x-4)*(x-5)*(x-1), x, method='bezout').expand()
 245
 246      assert b_res == s_res == 0
 247
 248      s_res = resultant(x**3-1, x**3+2*x**2+2*x-1, x, method='sylvester').expand()
 249      b_res = resultant(x**3-1, x**3+2*x**2+2*x-1, x, method='bezout').expand()
 250
 251      assert b_res == s_res == 16
 252
 253      s_res = resultant(x**8-2, x-1, x, method='sylvester').expand()
 254      b_res = resultant(x**8-2, x-1, x, method='bezout').expand()
 255
 256      assert b_res == s_res == -1
 257
 258      s_res = resultant(3*x**2+2*a*x+3*a**2-2, 3*x**2-2*a*x+3*a**2-2, x, method='sylvester').expand()
 259      b_res = resultant(3*x**2+2*a*x+3*a**2-2, 3*x**2-2*a*x+3*a**2-2, x, method='bezout').expand()
 260
 261      assert b_res == s_res == 144*a**4 - 96*a**2
 262
 263      s_res = resultant((x-a)*(x-b), x-c, x, method='sylvester').expand()
 264      b_res = resultant((x-a)*(x-b), x-c, x, method='bezout').expand()
 265
 266      assert b_res == s_res == ((a-c)*(b-c)).expand()
 267
 268 def test_roots():
 269     a = Symbol("a")
 270     b = Symbol("b")
 271     c = Symbol("c")
 272     x = Symbol("x")
 273     assert roots(x**2-3*x+2) == [1, 2]
 274     assert roots(x**2-3*x/2+Rational(1,2)) == [Rational(1,2), 1]
 275     assert roots(2*x**2-3*x+1) == [Rational(1,2), 1]
 276     assert roots(x**2-1) == [1, -1]
 277     assert roots(x**2+1) == [I, -I]
 278     assert roots(x**3-1) == [1,
 279                              Rational(-1,2) + I*Rational(1,2)*3**Rational(1,2),
 280                              Rational(-1,2) - I*Rational(1,2)*3**Rational(1,2)]
 281     assert roots(x**3) == [0, 0, 0]
 282     assert roots(x**3-x) == [-1, 0, 1]
 283     assert roots(Rational(2),x) == []
 284     assert roots(a*x**2 + b*x + c, var=[x]) == \
 285            [-b/(a*2)+(((b/a)**2-4*c/a)**Rational(1,2))/2,
 286             -b/(a*2)-(((b/a)**2-4*c/a)**Rational(1,2))/2]
 287     assert roots(x**3 + x**2 + x + 1) == [-1, I, -I]
 288     assert roots(x**3 + x**2 - x + 1) == \
 289            [Rational(-1,3) - (Rational(19,27) + Rational(1,9)*3**Rational(1,2)*11**Rational(1,2))**Rational(1,3) - Rational(4,9)*(Rational(19,27) + Rational(1,9)*3**Rational(1,2)*11**Rational(1,2))**Rational(-1,3),
 290             Rational(-1,3) + Rational(1,2)*(Rational(19,27) + Rational(1,9)*3**Rational(1,2)*11**Rational(1,2))**Rational(1,3) + Rational(4,9)/(Rational(1,2) + Rational(1,2)*I*3**Rational(1,2))*(Rational(19,27) + Rational(1,9)*3**Rational(1,2)*11**Rational(1,2))**Rational(-1,3) + Rational(1,2)*I*3**Rational(1,2)*(Rational(19,27) + Rational(1,9)*3**Rational(1,2)*11**Rational(1,2))**Rational(1,3),
 291             Rational(-1,3) + Rational(1,2)*(Rational(19,27) + Rational(1,9)*3**Rational(1,2)*11**Rational(1,2))**Rational(1,3) + Rational(4,9)/(Rational(1,2) - Rational(1,2)*I*3**Rational(1,2))*(Rational(19,27) + Rational(1,9)*3**Rational(1,2)*11**Rational(1,2))**Rational(-1,3) - Rational(1,2)*I*3**Rational(1,2)*(Rational(19,27) + Rational(1,9)*3**Rational(1,2)*11**Rational(1,2))**Rational(1,3)]
 292     assert roots(x**4 - 1) == [1, I, -1, -I]
 293     assert roots(x**4 + 1) == [((-1)**Rational(1,4)).expand(complex=True),
 294                                ((-1)**Rational(3,4)).expand(complex=True),
 295                                (-(-1)**Rational(1,4)).expand(complex=True),
 296                                (-(-1)**Rational(3,4)).expand(complex=True)]
 297     assert roots(x**8 - 1) == [1, 2**Rational(1,2)/2 + I*2**Rational(1,2)/2,
 298                                I, -2**Rational(1,2)/2 + I*2**Rational(1,2)/2,
 299                                -1, -2**Rational(1,2)/2 - I*2**Rational(1,2)/2,
 300                                -I, 2**Rational(1,2)/2 - I*2**Rational(1,2)/2]
 301     assert roots(x**5 - Rational(3,2)) == \
 302            [Rational(1,2)**Rational(1,5)*3**Rational(1,5),
 303             Rational(1,2)**Rational(1,5)*3**Rational(1,5)*cos(2*pi/5)
 304             + I*Rational(1,2)**Rational(1,5)*3**Rational(1,5)*sin(2*pi/5),
 305             Rational(1,2)**Rational(1,5)*3**Rational(1,5)*cos(4*pi/5)
 306             + I*Rational(1,2)**Rational(1,5)*3**Rational(1,5)*sin(4*pi/5),
 307             Rational(1,2)**Rational(1,5)*3**Rational(1,5)*cos(6*pi/5)
 308             + I*Rational(1,2)**Rational(1,5)*3**Rational(1,5)*sin(6*pi/5),
 309             Rational(1,2)**Rational(1,5)*3**Rational(1,5)*cos(8*pi/5)
 310             + I*Rational(1,2)**Rational(1,5)*3**Rational(1,5)*sin(8*pi/5)]
 311
 312 def test_solve_system():
 313     x = Symbol("x")
 314     y = Symbol("y")
 315     z = Symbol("z")
 316     assert solve_system(x-1) == [(S.One,)]
 317     assert solve_system([2*x - 3, 3*y/2 - 2*x, z - 5*y]) \
 318            == [(Rational(3, 2), Integer(2), Integer(10))]
 319     assert solve_system([y - x, y - x - 1]) == []
 320     assert solve_system([y - x**2, y + x**2]) == [(S.Zero, S.Zero)]
 321     assert solve_system([y - x**2, y + x**2 + 1]) == \
 322            [(I*Rational(1,2)**Rational(1,2), Rational(-1,2)),
 323             (-I*Rational(1,2)**Rational(1,2), Rational(-1,2))]
 324
 325 def test_sqf():
 326     x = Symbol("x")
 327     assert sqf(3*x**2, x) == 3*x**2
 328     assert sqf(x**2+2*x+1, x) == (x+1)**2
 329     assert sqf(x**5 - x**4 - x + 1) == (x-1)**2*(x**3 + x**2 + x + 1)
 330
 331 def test_sqf_part():
 332     x = Symbol('x')
 333     assert sqf_part(3*x**2, x) == 3*x
 334     assert sqf_part(x**2 + 2*x + 1, x) == x+1
 335     assert sqf_part(x**5 - x**4 - x + 1) == x**4 - 1
 336
 337 ## sympy/modules/polynomials/ideals.py
 338
 339 ## def test_Ideal():
 340 ##     x = Symbol('x')
 341 ##     y = Symbol('y')
 342 ##     z = Symbol('z')
 343
 344 ##     # TODO: more complete tests?
 345 ##     assert len(Ideal()) == 1
 346 ##     assert not x in Ideal()
 347 ##     I = Ideal([x,y], [x,y])
 348 ##     assert x*y**2 in I
 349 ##     assert z*x in I
 350 ##     assert not z in I
 351 ##     assert I == I + Ideal()
 352 ##     assert Ideal() == Ideal()*I
 353 ##     assert I + I == I
 354 ##     assert I == I % Ideal()
 355 ##     assert Ideal() == Ideal(x*y, [x,y]) % I
 356 ##     assert Ideal(z, [x,y,z]) == Ideal([x,z], [x,y,z]) % Ideal([x,y], [x,y,z])
 357
 358 ## sympy/modules/polynomials/roots_.py
 359
 360 def test_sturm():
 361     from sympy.polynomials import roots_
 362     x = Symbol('x')
 363
 364     f = Polynomial(Rational(5), var=x)
 365     assert roots_.sturm(f) == [f]
 366     f = Polynomial(2*x)
 367     assert roots_.sturm(f) == [f, Polynomial(2, var=x)]
 368     f = Polynomial(x**3 - 2*x**2 + 3*x -5)
 369     assert roots_.sturm(f) == \
 370            [Polynomial(-5-2*x**2+x**3+3*x), Polynomial(3+3*x**2-4*x),
 371             Polynomial(Rational(13,3)-Rational(10,9)*x),
 372             Polynomial(Rational(-3303,100), var=x)]
 373
 374
 375 ## Issues
 376
 377 def test_poly_content():
 378     x, y = symbols("xy")
 379     assert Polynomial(y**2*x + y, var=x).content()  == y
 380     assert Polynomial(y**2*x + 2, var=x).content()  == 1
 381
 382 @XFAIL  # see #442
 383 def test_poly_content_0():
 384     assert Polynomial(sin(y)*x, var=x).content()    == sin(y)
 385
 386 @XFAIL  # see #442
 387 def test_poly_content_1():
 388     assert Polynomial(sqrt(2)*x, var=x).content()   == sqrt(2)
 389
 390 def test_poly_integrate():
 391     x, y, z = symbols("xyz")
 392     assert Polynomial(x**2+1).integrate(x) == x**3/3 + x
 393     assert Polynomial(x**2+y*x).integrate(y) == x**2*y + y**2*x/2
 394     assert Polynomial(x**2+x).integrate(y) == x**2*y + y*x
 395     assert Polynomial(x*(y+x+z)).integrate(x) == x**2*y/2 + x**3/3 + x**2*z/2
 396     assert Polynomial(x*(y+x+z)).integrate(z) == x*y*z + x**2*z + x*z**2/2