Lines Matching refs:Poly
5 Poly, Rational, exp, factor, log, sin, sqrt, symbols, tan)
29 assert gcdex_diophantine(Poly(x**4 - 2*x**3 - 6*x**2 + 12*x + 15),
30 Poly(x**3 + x**2 - 4*x - 4), Poly(x**2 - 1)) == \
31 (Poly((-x**2 + 4*x - 3)/5), Poly((x**3 - 7*x**2 + 16*x - 10)/5))
35 assert frac_in(Poly((x + 1)/x*t, t), x) == \
36 (Poly(t*x + t, x), Poly(x, x))
38 (Poly(t*x + t, x), Poly(x, x))
39 assert frac_in((Poly((x + 1)/x*t, t), Poly(t + 1, t)), x) == \
40 (Poly(t*x + t, x), Poly((1 + t)*x, x))
42 assert frac_in(Poly((2 + 2*x + x*(1 + x))/(1 + x)**2, t), x, cancel=True) == \
43 (Poly(x + 2, x), Poly(x + 1, x))
48 Poly(t + 2*z, t, z), Poly(t + 2*z, z, t)]
50 Poly(2*z + 3*z**2, t, z), Poly(2*z + 3*z**2, z, t)]
52 Poly(2/(exp(2) + 1)*z, t, z), Poly(2/(exp(2) + 1)*z, z, t)]
54 Poly(t + 2/(exp(2) + 1)*z, t, z), Poly(t + 2/(exp(2) + 1)*z, z, t)]
55 assert as_poly_1t(Integer(0), t, z) == Poly(0, t, z)
59 p = Poly(4*x**4*t**5 + (-4*x**3 - 4*x**4)*t**4 + (-3*x**2 + 2*x**3)*t**3 +
61 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t**2 - 3/(2*x)*t + 1/(2*x), t)]})
62 assert derivation(p, DE) == Poly(-20*x**4*t**6 + (2*x**3 + 16*x**4)*t**5 +
66 assert derivation(Poly(1, t), DE) == Poly(0, t)
67 assert derivation(Poly(t, t), DE) == DE.d
68 assert derivation(Poly(t**2 + 1/x*t + (1 - 2*x)/(4*x**2), t), DE) == \
69 … Poly(-2*t**3 - 4/x*t**2 - (5 - 2*x)/(2*x**2)*t - (1 - 2*x)/(2*x**3), t, domain=ZZ.inject(x).field)
70 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t1), Poly(t, t)]})
71 assert derivation(Poly(x*t*t1, t), DE) == Poly(t*t1 + x*t*t1 + t, t)
72 assert derivation(Poly(x*t*t1, t), DE, coefficientD=True) == \
73 Poly((1 + t1)*t, t)
74 DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
75 assert derivation(Poly(x, x), DE) == Poly(1, x)
78 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
84 p = Poly(4*x**4*t**5 + (-4*x**3 - 4*x**4)*t**4 + (-3*x**2 + 2*x**3)*t**3 +
86 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t**2 - 3/(2*x)*t + 1/(2*x), t)]})
87 assert splitfactor(p, DE) == (Poly(4*x**4*t**3 + (-8*x**3 - 4*x**4)*t**2 +
88 …(4*x**2 + 8*x**3)*t - 4*x**2, t), Poly(t**2 + 1/x*t + (1 - 2*x)/(4*x**2), t, domain=ZZ.inject(x).f…
89 assert splitfactor(Poly(x, t), DE) == (Poly(x, t), Poly(1, t))
90 …r = Poly(-4*x**4*z**2 + 4*x**6*z**2 - z*x**3 - 4*x**5*z**3 + 4*x**3*z**3 + x**4 + z*x**5 - x**6, t)
91 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
93 (Poly(x*z - x**2 - z*x**3 + x**4, t), Poly(-x**2 + 4*x**2*z**2, t))
95 (((Poly(x*z - x**2 - z*x**3 + x**4, t), 1),), ((Poly(-x**2 + 4*x**2*z**2, t), 1),))
96 assert splitfactor(Poly(0, t), DE) == (Poly(0, t), Poly(1, t))
97 assert splitfactor_sqf(Poly(0, t), DE) == (((Poly(0, t), 1),), ())
101 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
102 assert canonical_representation(Poly(x - t, t), Poly(t**2, t), DE) == \
103 (Poly(0, t), (Poly(0, t),
104 Poly(1, t)), (Poly(-t + x, t),
105 Poly(t**2, t)))
106 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
107 assert canonical_representation(Poly(t**5 + t**3 + x**2*t + 1, t),
108 Poly((t**2 + 1)**3, t), DE) == \
109 (Poly(0, t), (Poly(t**5 + t**3 + x**2*t + 1, t),
110 Poly(t**6 + 3*t**4 + 3*t**2 + 1, t)), (Poly(0, t), Poly(1, t)))
114 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
116 assert hermite_reduce(Poly(x - t, t), Poly(t**2, t), DE) == \
117 ((Poly(-x, t), Poly(t, t)), (Poly(0, t), Poly(1, t)), (Poly(-x, t), Poly(1, t)))
119 … DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t**2 - t/x - (1 - nu**2/x**2), t)]})
122 Poly(x**2*t**5 + x*t**4 - nu**2*t**3 - x*(x**2 + 1)*t**2 - (x**2 - nu**2)*t - x**5/4, t),
123 Poly(x**2*t**4 + x**2*(x**2 + 2)*t**2 + x**2 + x**4 + x**6/4, t), DE) == \
124 ((Poly(-x**2/4 - 1, t), Poly(t**2 + x**2/2 + 1, t)),
125 (Poly((-8*nu**2 - 4*x**4)*t - 8*x**3 - 8*x, t), Poly(8*x**2*t**2 + 4*x**4 + 8*x**2, t)),
126 (Poly(x*t + 1, t), Poly(x, t)))
128 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
131 Poly(-t**2 + 2*t + 2, t),
132 Poly(-x*t**2 + 2*x*t - x, t), DE) == \
133 ((Poly(3, t), Poly(t - 1, t)),
134 (Poly(0, t), Poly(1, t)),
135 (Poly(1, t), Poly(x, t)))
138 Poly(-x**2*t**6 + (-1 - 2*x**3 + x**4)*t**3 + (-3 - 3*x**4)*t**2 - 2*x*t - x - 3*x**2, t),
139 Poly(x**4*t**6 - 2*x**2*t**3 + 1, t), DE) == \
140 ((Poly(x**3*t + x**4 + 1, t), Poly(x**3*t**3 - x, t)),
141 (Poly(0, t), Poly(1, t)),
142 (Poly(-1, t), Poly(x**2, t)))
145 Poly((-2 + 3*x)*t**3 + (-1 + x)*t**2 + (-4*x + 2*x**2)*t + x**2, t),
146 Poly(x*t**6 - 4*x**2*t**5 + 6*x**3*t**4 - 4*x**4*t**3 + x**5*t**2, t), DE) == \
147 ((Poly(t**2 + t/3 + x, t), Poly(t**4 - 3*x*t**3 + 3*x**2*t**2 - x**3*t, t)),
148 (Poly(0, t), Poly(1, t)),
149 (Poly(0, t), Poly(1, t)))
152 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(2*log(2)/3*t, t)]})
154 assert (hermite_reduce(Poly(1, t), Poly(1 + t), DE) ==
155 ((Poly(0, t), Poly(1, t)), (Poly(1, t), Poly(1 + t)),
156 (Poly(0, t), Poly(1, t))))
160 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
161 assert polynomial_reduce(Poly(1 + x*t + t**2, t), DE) == \
162 (Poly(t, t), Poly(x*t, t))
163 assert polynomial_reduce(Poly(0, t), DE) == \
164 (Poly(0, t), Poly(0, t))
168 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1, t)]})
169 a = Poly(36, t)
170 d = Poly((t - 2)*(t**2 - 1)**2, t)
171 F = Poly(t**2 - 1, t)
174 (Poly(-3*t**3 + 3*t**2 - 6*t - 8, t), Poly(t**5 + t**4 - 2*t**3 - 2*t**2 + t + 1, t),
175 [Poly(-3*t**3 - 6*t**2, t), Poly(2*t**6 + 6*t**5 - 8*t**3, t)])
179 DE = DifferentialExtension(extension={'D': [Poly(1, t)]})
180 a = Poly(36, t)
181 d = Poly((t - 2)*(t**2 - 1)**2, t)
183 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
184 a = Poly(2, t)
185 d = Poly(t**2 - 1, t)
187 assert recognize_derivative(Poly(x*t, t), Poly(1, t), DE) is True
188 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
189 assert recognize_derivative(Poly(t, t), Poly(1, t), DE) is True
194 a = Poly(2*x**2 + 4*x*t - 2*t - x**2*t, t)
195 d = Poly((2*x + t)*(t + x**2), t)
196 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
198 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
199 assert recognize_log_derivative(Poly(t + 1, t), Poly(t + x, t), DE) is True
200 assert recognize_log_derivative(Poly(2, t), Poly(t**2 - 1, t), DE) is True
201 DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
202 assert recognize_log_derivative(Poly(1, x), Poly(x**2 - 2, x), DE) is False
203 assert recognize_log_derivative(Poly(1, x), Poly(x**2 + x, x), DE) is True
204 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
205 assert recognize_log_derivative(Poly(1, t), Poly(t**2 - 2, t), DE) is False
206 assert recognize_log_derivative(Poly(1, t), Poly(t**2 + t, t), DE) is False
210 a = Poly(2*t**2 - t - x**2, t)
211 d = Poly(t**3 - x**2*t, t)
212 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)], 'Tfuncs': [log]})
214 ([(Poly(z**2 - Rational(1, 4), z), Poly((1 + 3*x*z - 6*z**2 -
217 ([(Poly(z**2 - Rational(1, 4), z), Poly(t + 2*x*z, t))], False)
218 assert residue_reduce(Poly(-2/x, t), Poly(t**2 - 1, t,), DE, z, invert=False) == \
219 ([(Poly(z**2 - 1, z), Poly(-2*z*t/x - 2/x, t))], True)
220 ans = residue_reduce(Poly(-2/x, t), Poly(t**2 - 1, t), DE, z, invert=True)
221 assert ans == ([(Poly(z**2 - 1, z), Poly(t + z, t))], True)
224 … DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t**2 - t/x - (1 - nu**2/x**2), t)]})
226 assert residue_reduce(Poly((-2*nu**2 - x**4)/(2*x**2)*t - (1 + x**2)/x, t),
227 Poly(t**2 + 1 + x**2/2, t), DE, z) == \
228 ([(Poly(z + Rational(1, 2)), Poly(t**2 + 1 + x**2/2, t, domain=EX))], True)
229 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
230 assert residue_reduce(Poly(-2*x*t + 1 - x**2, t),
231 Poly(t**2 + 2*x*t + 1 + x**2, t), DE, z) == \
232 ([(Poly(z**2 + Rational(1, 4), z), Poly(t + x + 2*z, t))], True)
233 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
234 assert residue_reduce(Poly(t, t), Poly(t + sqrt(2), t), DE, z) == \
235 ([(Poly(z - 1, z), Poly(t + sqrt(2), t))], True)
240 a = Poly((1 + 2*t1 + t1**2 + 2*t1**3)*t**2 + (1 + t1**2)*t + 1 + t1**2, t)
241 d = Poly(1, t)
242 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t1**2, t1),
243 … Poly(t*(1 + t1**2), t)], 'Tfuncs': [tan, Lambda(i, exp(tan(i)))]})
246 a = Poly((t1**3 + (x + 1)*t1**2 + t1 + x + 2)*t, t)
250 a = Poly(t, t)
251 d = Poly(1, t)
252 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(2*x*t, t)],
258 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)], 'Tfuncs': [exp]})
261 a = Poly(25*t**6 - 10*t**5 + 7*t**4 - 8*t**3 + 13*t**2 + 2*t - 1, t)
262 d = Poly(25*t**6 + 35*t**4 + 11*t**2 + 1, t)
265 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t0, t0), Poly(t0*t, t)],
267 … assert integrate_hyperexponential(Poly(2*t0*t**2, t), Poly(1, t), DE) == (exp(2*exp(x)), 0, True)
269 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t0, t0), Poly(-t0*t, t)],
271 assert integrate_hyperexponential(Poly(-27*exp(9) - 162*t0*exp(9) +
272 … 27*x*t0*exp(9), t), Poly((36*exp(18) + x**2*exp(18) - 12*x*exp(18))*t, t), DE) == \
275 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)], 'Tfuncs': [exp]})
276 assert integrate_hyperexponential(Poly(x**2/2*t, t), Poly(1, t), DE) == \
278 assert integrate_hyperexponential(Poly(1 + t, t), Poly(t, t), DE) == \
280 assert integrate_hyperexponential(Poly(x, t), Poly(t + 1, t), DE) == \
283 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t0), Poly(2*x*t1, t1)],
286 elem, nonelem, b = integrate_hyperexponential(Poly((8*x**7 - 12*x**5 + 6*x**3 - x)*t1**4 +
295 … 4*x**5 - 4*x**4 - x**3 + x**2, t1), Poly((8*x**7 - 12*x**5 + 6*x**3 -
310 p = Poly((-28*x**11*t0 - 6*x**8*t0 + 6*x**9*t0 - 15*x**8*t0**2 +
327 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t0), Poly(2*x*t1, t1)]})
329 Poly((x - t0)*t1**2 + (-2*t0 + 2*x)*t1, t1), Poly(-2*x*t0 + x**2 +
332 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t0, t0)]})
333 assert integrate_hyperexponential_polynomial(Poly(0, t0), DE, z) == (
334 Poly(0, t0), Poly(1, t0), True)
367 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)],
369 assert integrate_primitive(Poly(t, t), Poly(1, t), DE) == (x*log(x), -1, True)
370 …assert integrate_primitive(Poly(x, t), Poly(t, t), DE) == (0, NonElementaryIntegral(x/log(x), x), …
372 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t1), Poly(1/(x + 1), t2)],
374 assert integrate_primitive(Poly(t1, t2), Poly(t2, t2), DE) == \
377 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t1), Poly(1/(x*t1), t2)],
379 assert integrate_primitive(Poly(t2, t2), Poly(t1, t2), DE) == \
382 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t0)],
384 assert integrate_primitive(Poly(x**2*t0**3 + (3*x**2 + x)*t0**2 + (3*x**2
385 … + 2*x)*t0 + x**2 + x, t0), Poly(x**2*t0**4 + 4*x**2*t0**3 + 6*x**2*t0**2 +
391 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
392 assert integrate_hypertangent_polynomial(Poly(t**2 + x*t + 1, t), DE) == \
393 (Poly(t, t), Poly(x/2, t))
394 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(a*(t**2 + 1), t)]})
395 assert integrate_hypertangent_polynomial(Poly(t**5, t), DE) == \
396 (Poly(1/(4*a)*t**4 - 1/(2*a)*t**2, t), Poly(1/(2*a), t))
400 a, d, = Poly(x**2*t**5 + x*t**4 - nu**2*t**3 - x*(x**2 + 1)*t**2 - (x**2 -
401 … nu**2)*t - x**5/4, t), Poly(x**2*t**4 + x**2*(x**2 + 2)*t**2 + x**2 + x**4 + x**6/4, t)
405 DE = DifferentialExtension(extension={'D': [Poly(1, x),
406 … Poly(-t**2 - t/x - (1 - nu**2/x**2), t)], 'Tfuncs': [f]})
409 assert integrate_nonlinear_no_specials(Poly(t, t), Poly(1, t), DE) == \
421 (Poly(t1 + t0, t1), Poly(1, t1), [Poly(1, x,), Poly(t0, t0),
422 Poly(2*x*t1, t1)], [x, t0, t1], [Lambda(i, exp(i)),
425 (Poly(t0**2 + t0, t0), Poly(1, t0), [Poly(1, x), Poly(t0, t0)], [x, t0],
428 (Poly(t0**2 + t0, t0), Poly(1, t0), [Poly(1, x), Poly(t0/2, t0)],
432 (Poly((1 + t0)*t1 + t0, t1), Poly(1, t1), [Poly(1, x), Poly(t0, t0),
433 … Poly(2*x*t1, t1)], [x, t0, t1], [Lambda(i, exp(i)),
437 (Poly((1 + E*t0)*t1 + t0, t1), Poly(1, t1),
438 [Poly(1, x), Poly(t0, t0), Poly(2*x*t1, t1)], [x, t0, t1],
443 (Poly((t0 + 1)*t1 + t0**2, t1), Poly(1, t1), [Poly(1, x),
444 … Poly(t0/2, t0), Poly(2*x*t1, t1)], [x, t0, t1],
449 (Poly((t0*exp(3) + 1)*t1 + t0**2, t1), Poly(1, t1), [Poly(1, x),
450 … Poly(t0/2, t0), Poly(2*x*t1, t1)], [x, t0, t1], [Lambda(i, exp(i/2)),
454 (Poly(t0, t0), Poly(1, t0), [Poly(1, x), Poly(t0/2, t0)], [x, t0],
458 (Poly(t0, t0), Poly(1, t0), [Poly(1, x), Poly(t0/2, t0)], [x, t0],
465 (Poly(t0*t1**2 + (t0*log(2) + t0**2)*t1, t1), Poly(1, t1),
466 [Poly(1, x), Poly(1/x, t0),
467 Poly(1/(x + 1), t1, expand=False)], [x, t0, t1],
471 (Poly(t0*t1, t1), Poly(1, t1), [Poly(1, x), Poly(1/x, t0),
472 Poly((1 + t0)*t1, t1)], [x, t0, t1], [Lambda(i, log(i)),
479 (Poly(t0*x, t1), Poly(1, t1), [Poly(1, x), Poly(1/x, t0), Poly((t0 +
483 (Poly(y*t0, t0), Poly(1, t0), [Poly(1, x), Poly(1/x, t0)], [x, t0],
486 (Poly(t0, t0), Poly(2, t0), [Poly(1, x), Poly(1/x, t0)], [x, t0],
493 (Poly(t0*t1, t1), Poly(1, t1), [Poly(1, x), Poly(1/x, t0),
494 … Poly(t1, t1)], [x, t0, t1], [Lambda(i, log(i)), Lambda(i, exp(i))],
498 (Poly(t0*t1, t1), Poly(1, t1), [Poly(1, x), Poly(t0, t0),
499 … Poly(1/x, t1)], [x, t0, t1], [Lambda(i, exp(i)), Lambda(i, log(i))],
508 (Poly((-1 + x - x*t0**2)*t1, t1), Poly(x, t1),
509 [Poly(1, x), Poly(1/x, t0), Poly((1 + t0)*t1, t1)], [x, t0, t1],
528 assert DE.d == Poly(1/x, t1) == DE.D[DE.level]
533 assert DE.d == Poly(t0, t0) == DE.D[DE.level]
538 assert DE.d == Poly(1, x) == DE.D[DE.level]
545 assert DE.d == Poly(1/x, t1) == DE.D[DE.level]
551 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
552 assert DE._important_attrs == (None, None, [Poly(1, x), Poly(t, t)], [x, t],
554 assert DE.d == Poly(t, t)
561 DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)],
563 assert DE._important_attrs == (None, None, [Poly(1, x), Poly(t, t)], [x, t],
571 (Poly(sin(y)*t0, t0, domain=ZZ.inject(sin(y))), Poly(1, t0),
572 [Poly(1, x), Poly(t0)], [x, t0],
576 (Poly(t0, t0), Poly(1, t0), [Poly(1, x), Poly(log(10)*t0, t0)], [x, t0],
580 (Poly(3*t0, t0), Poly(2, t0), [Poly(1, x), Poly(2/x, t0)], [x, t0],
582 (Poly(3*t0, t0), Poly(1, t0), [Poly(1, x), Poly(1/x, t0)], [x, t0],
585 (Poly(0, x), Poly(1, x), [Poly(1, x)], [x], [], [], [], [], [], [])
593 (Poly((1757211400 + 2581284541*t0)*t1, t1), Poly(39916800 +
595 [Poly(1, x), Poly(t0, t0), Poly(-(10 + 21*t0 + 10*t0**2)/(1 + 2*t0 +
608 assert DE.d == Poly(t0/(t0 + 1), t1)
614 assert DE.d == Poly(t0, t0)
620 assert DE.d == Poly(1, x)
625 assert DE.d == Poly(t0, t0)
630 assert DE.d == Poly(t0/(t0 + 1), t1)
644 assert DE.d == Poly(t0/(t0 + 1), t1)