#!/usr/bin/python
import polypy
import polypy_test
import sys
polypy_test.init()
x = polypy.Variable("x")
y = polypy.Variable("y")
polypy.variable_order.set([y, x])
polypy_test.start("model-based GCD")
# polypy.trace_enable("coefficient::gcd")
# polypy.trace_enable("polynomial")
# polypy.trace_enable("coefficient")
# polypy.trace_enable("coefficient::order")
# polypy.trace_enable("coefficient::reduce")
polypy.trace_enable("coefficient::mgcd")
x0 = polypy.Variable("x0")
x1 = polypy.Variable("x1")
x2 = polypy.Variable("x2")
x3 = polypy.Variable("x3")
x4 = polypy.Variable("x4")
polypy.variable_order.set([x0, x1, x2, x3, x4])
def check_resultant(p, q, expected):
resultant = p.resultant(q)
ok = resultant == expected
if (not ok):
print "p =", p
print "q =", q
print "resultant =", resultant
print "expected =", expected
polypy_test.init()
# polypy.trace_enable("coefficient::resultant")
# polypy.trace_enable("polynomial::expensive")
polypy_test.start("Speed")
x5 = polypy.Variable("x5")
x9 = polypy.Variable("x9")
x10 = polypy.Variable("x10")
x11 = polypy.Variable("x11")
x12 = polypy.Variable("x12")
polypy.variable_order.set([x9, x12, x11, x10, x5]);
A = (((1*x12**6 + (4*x9)*x12**5 + (10*x9**2)*x12**4 + (14*x9**3)*x12**3 + (13*x9**4)*x12**2 + (6*x9**5)*x12 + (1*x9**6))*x11**2)*x10**2 + (((2*x9**2)*x12**6 + (4*x9**3)*x12**5 + (6*x9**4)*x12**4 + (2*x9**5)*x12**3)*x11)*x10 + ((1*x9**4)*x12**6))*x5**2 + ((((-1*x9)*x12**3 + (-1*x9**2)*x12**2 + (-1*x9**3)*x12)*x11**4)*x10**4 + ((-2*x12**6 + (-9*x9)*x12**5 + (-17*x9**2)*x12**4 + (-21*x9**3)*x12**3 + (-12*x9**4)*x12**2 + (-4*x9**5)*x12)*x11**3)*x10**3 + (((-2*x9**2)*x12**6 + (-6*x9**3 - 2*x9)*x12**5 + (-4*x9**4 - 8*x9**2)*x12**4 + (-3*x9**5 - 16*x9**3)*x12**3 + (-1*x9**6 - 18*x9**4)*x12**2 + (-1*x9**7 - 10*x9**5)*x12 + (-2*x9**6))*x11**2)*x10**2 + (((-1*x9**5 - 2*x9**3)*x12**5 + (-1*x9**6 - 4*x9**4)*x12**4 + (-1*x9**7 - 2*x9**5)*x12**3)*x11)*x10)*x5 + ((((1*x9)*x12**3 + (1*x9**2)*x12**2)*x11**5)*x10**5 + ((1*x12**6 + (5*x9)*x12**5 + (7*x9**2)*x12**4 + (6*x9**3)*x12**3 + (3*x9**4)*x12**2 + (1*x9**3)*x12)*x11**4)*x10**4 + (((2*x9**3 + 2*x9)*x12**5 + (2*x9**4 + 7*x9**2)*x12**4 + (1*x9**5 + 13*x9**3)*x12**3 + (1*x9**6 + 8*x9**4)*x12**2 + (4*x9**5)*x12)*x11**3)*x10**3 + (((-1*x9**2)*x12**6 + (1*x9**5)*x12**5 + (1*x9**6 - 2*x9**4 + 1*x9**2)*x12**4 + (2*x9**5 + 4*x9**3)*x12**3 + (6*x9**4)*x12**2 + (1*x9**7 + 4*x9**5)*x12 + (1*x9**6))*x11**2)*x10**2 + (((-2*x9**4)*x12**6 + (-1*x9**6)*x12**4 + (1*x9**7)*x12**3)*x11)*x10 + ((-1*x9**6)*x12**6))
B = (((2*x12**6 + (8*x9)*x12**5 + (20*x9**2)*x12**4 + (28*x9**3)*x12**3 + (26*x9**4)*x12**2 + (12*x9**5)*x12 + (2*x9**6))*x11**2)*x10**2 + (((4*x9**2)*x12**6 + (8*x9**3)*x12**5 + (12*x9**4)*x12**4 + (4*x9**5)*x12**3)*x11)*x10 + ((2*x9**4)*x12**6))*x5 + ((((-1*x9)*x12**3 + (-1*x9**2)*x12**2 + (-1*x9**3)*x12)*x11**4)*x10**4 + ((-2*x12**6 + (-9*x9)*x12**5 + (-17*x9**2)*x12**4 + (-21*x9**3)*x12**3 + (-12*x9**4)*x12**2 + (-4*x9**5)*x12)*x11**3)*x10**3 + (((-2*x9**2)*x12**6 + (-6*x9**3 - 2*x9)*x12**5 + (-4*x9**4 - 8*x9**2)*x12**4 + (-3*x9**5 - 16*x9**3)*x12**3 + (-1*x9**6 - 18*x9**4)*x12**2 + (-1*x9**7 - 10*x9**5)*x12 + (-2*x9**6))*x11**2)*x10**2 + (((-1*x9**5 - 2*x9**3)*x12**5 + (-1*x9**6 - 4*x9**4)*x12**4 + (-1*x9**7 - 2*x9**5)*x12**3)*x11)*x10)
A.psc(B)
polypy.variable_order.set([w, z, y, x]);
import polypy
import polypy_test
polypy_test.init()
def check_str(o, expected):
ok = str(o) == expected
if (not ok):
print("o = {0}".format(o))
print("expected = {0}".format(expected))
polypy_test.check(ok)
x = polypy.Variable("x")
y = polypy.Variable("y")
polypy_test.start("Variable")
order = polypy.VariableOrder([x])
check_str(order, "[x]")
order = polypy.VariableOrder([x, y])
check_str(order, "[x, y]")
order = polypy.variable_order
check_str(order, "[]")
order.push(x)
check_str(order, "[x]")
order.push(y)
check_str(order, "[x, y]")
def check_comparison(a1, a2, cmp, result, expected):
if (result != expected):
print("a1 = {0}".format(a1))
print("a2 = {0}".format(a2))
print("cmp = {0}".format(cmp))
print("result = {0}".format(result))
print("expected = {0}".format(expected))
polypy_test.check(False)
else:
polypy_test.check(True)
polypy_test.init()
polypy_test.start("Construction and order")
a1 = polypy.AlgebraicNumber((x**2 - 2) * (x - 10), 1)
a2 = polypy.AlgebraicNumber((x**2 - 2), 1)
result = (a1 < a2)
check_comparison(a1, a2, "<", result, False)
result = (a1 == a2)
check_comparison(a1, a2, "==", result, True)
p = x**2
zero = polypy.AlgebraicNumber(p, 0)
result = zero.to_double() == 0.0
check_comparison(zero, 0, ".to_double ==", result, True)
polypy.variable_order.set([z, y, x])
def check_value(p, assignment, value, expected_value):
value_double = value.to_double()
ok = abs(value_double - expected_value) < 0.000001
if (not ok):
print "p =", p
print "assignment =", assignment
print "value =", value
print "value_double =", value_double
print "expected_value =", expected_value
polypy_test.check(ok)
polypy_test.start("Polynomial Evaluation")
# polypy.trace_enable("polynomial")
# polypy.trace_enable("factorization")
# polypy.trace_enable("algebraic_number")
# polypy.trace_enable("coefficient")
# polypy.trace_enable("coefficient::sgn")
# polypy.trace_enable("coefficient::roots")
# polypy.trace_enable("coefficient::arith")
sqrt2 = polypy.AlgebraicNumber(x**2 - 2, 1)
sqrt2_4 = polypy.AlgebraicNumber(x**4 - 2, 1)
sqrt3 = polypy.AlgebraicNumber(x**2 - 3, 1)
assignment = polypy.Assignment()
assignment.set_value(x, sqrt2)
check_gcd_single(-p, q)
check_gcd_single(-p, -q)
def check_extended_gcd_single(p, q):
global polypy_test
(gcd, u, v) = p.extended_gcd(q)
ok = polypy_test.sympy_checker.check_extended_gcd(p, q, gcd, u, v)
polypy_test.check(ok)
def check_extended_gcd(p, q):
check_extended_gcd_single(p, q)
check_extended_gcd_single(p, -q)
check_extended_gcd_single(-p, q)
check_extended_gcd_single(-p, -q)
polypy_test.start("GCD in Z_13 (Knuth)")
# Example from Knuth (p 424)
K = polypy.CoefficientRing(13)
x = polypy.x.to_ring(K);
p = x**8 + x**6 + 10*x**4 + 10*x**3 + 8*x**2 + 2*x + 8
q = 3*x**6 + 5*x**4 + 9*x**2 + 4*x + 8
check_gcd(p, q)
polypy_test.start("Regressions (modular)")
K = polypy.CoefficientRing(7);
x = polypy.x.to_ring(K)
p = 2*x
polypy_test.init()
def check_factorization(p, p_factors, expected_size):
mul = 1
for (factor, multiplicity) in p_factors:
mul = mul * (factor**multiplicity)
ok = (p == mul) and (expected_size == len(p_factors))
if (not ok):
print "p =", p
print "p_factors =", p_factors
print "expected_size =", expected_size
print "mul =", mul
polypy_test.check(ok)
polypy_test.start("Square-Free Factorization")
# polypy.trace_enable("polynomial")
# polypy.trace_enable("factorization")
# polypy.trace_enable("coefficient::gcd")
#
# x00 = polypy.Variable("x00")
# x01 = polypy.Variable("x01")
# x10 = polypy.Variable("x10")
# x11 = polypy.Variable("x11")
# x20 = polypy.Variable("x20")
# x21 = polypy.Variable("x21")
# x30 = polypy.Variable("x30")
# x31 = polypy.Variable("x31")
#
# polypy.variable_order.set([x31, x30, x21, x20, x11])
[x0, x1, x2, x3, x4, x5, x6, x7] = [polypy.Variable(name) for name in ['x0', 'x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'x7']]
# polypy_test.start("Polynomial Root Isolation: Speed")
#
# polypy.variable_order.set([y, x])
#
# y_value_poly = 8192*x**6 + (-14336*x**5) + 75376*x**4 + (-32736*x**3) + 109496*x**2 + 133752*x + (-32441)
# y_value = polypy.AlgebraicNumber(y_value_poly, 1)
#
# assignment = polypy.Assignment()
# assignment.set_value(y, y_value)
#
# p = (1*y**2 + 3)*x**6 + (4*y**2 + 24*y + 20)*x**5 + (36*y**2 + 112*y + 44)*x**4 + (8*y**3 + 114*y**2 + 144*y + 30)*x**3 + (1*y**4 + 24*y**3 + 142*y**2 + 40*y - 15)*x**2 + (2*y**4 + 40*y**3 + 64*y**2 - 32*y - 26)*x + (5*y**4 + 16*y**3 + 7*y**2 - 16*y - 8)
# roots = p.roots_isolate(assignment)
polypy_test.start("Polynomial Root Isolation: Bugs");
polypy.variable_order.set([x0, x1, x2, x3, x4, x5, x6, x7])
p = ((16*x0**2 - 32*x0 + 16)*x5**2 + ((32*x0 - 32)*x1)*x5 + (16*x1**2))*x7**2 + ((8*x0**2 - 16*x0 + 8)*x5**2 + ((16*x0**2 - 32*x0 + 16)*x2**2 + (-8*x1**2)))*x7 + ((1*x0**2 - 2*x0 + 1)*x5**2 + ((-2*x0 + 2)*x1)*x5 + (1*x1**2))
p_lc = ((16*x0**2 - 32*x0 + 16)*x5**2 + ((32*x0 - 32)*x1)*x5 + (16*x1**2))
assignment = polypy.Assignment()
v = polypy.AlgebraicNumber(32*x**2 + (-64*x) + 15, 0)
assignment.set_value(x0, v)
assignment.set_value(x1, 28);
assignment.set_value(x2, -1, 4)
assignment.set_value(x3, 3)
assignment.set_value(x4, 277771, 4096)
assignment.set_value(x5, 786743, 20480)
assignment.set_value(x6, -6437)
import polypy
import polypy_test
polypy_test.init()
def check_str(o, expected):
ok = str(o) == expected
if (not ok):
print "o =", o
print "expected =", expected
polypy_test.check(ok)
x = polypy.Variable("x")
y = polypy.Variable("y")
polypy_test.start("Variable")
order = polypy.VariableOrder([x])
check_str(order, "[x]")
order = polypy.VariableOrder([x, y])
check_str(order, "[x, y]")
order = polypy.variable_order
check_str(order, "[]")
order.push(x)
check_str(order, "[x]")
order.push(y)
check_str(order, "[x, y]")
polypy_test.init()
[x, y, z] = [polypy.Variable(name) for name in ['x', 'y', 'z']]
polypy.variable_order.set([z, y, x])
def check_sgn(p, assignment, expected_sgn):
sgn = p.sgn(assignment)
ok = (sgn > 0 and expected_sgn > 0) or (sgn < 0 and expected_sgn < 0) or (sgn == 0 and expected_sgn == 0)
if (not ok):
print("p = {0}".format(p))
print("assignment = {0}".format(assignment))
print("sgn = {0}".format(sgn))
print("expected_sgn = {0}".format(expected_sgn))
polypy_test.check(ok)
polypy_test.start("Sign Determination")
# polypy.trace_enable("polynomial")
# polypy.trace_enable("factorization")
# polypy.trace_enable("algebraic_number")
# polypy.trace_enable("coefficient")
# polypy.trace_enable("coefficient::sgn")
# polypy.trace_enable("coefficient::arith")
sqrt2 = polypy.AlgebraicNumber(x**2 - 2, 1)
sqrt2_4 = polypy.AlgebraicNumber(x**4 - 2, 1)
sqrt3 = polypy.AlgebraicNumber(x**2 - 3, 1)
assignment = polypy.Assignment()
assignment.set_value(x, sqrt2_4)
assignment.set_value(y, sqrt2)
#!/usr/bin/python
import polypy
import polypy_test
import sys
polypy_test.init()
x = polypy.Variable("x");
y = polypy.Variable("y");
polypy.variable_order.set([y, x])
polypy_test.start("model-based GCD")
# polypy.trace_enable("coefficient::gcd")
# polypy.trace_enable("polynomial")
# polypy.trace_enable("coefficient")
# polypy.trace_enable("coefficient::order")
# polypy.trace_enable("coefficient::reduce")
polypy.trace_enable("coefficient::mgcd")
x0 = polypy.Variable("x0");
x1 = polypy.Variable("x1");
x2 = polypy.Variable("x2");
x3 = polypy.Variable("x3");
x4 = polypy.Variable("x4");
polypy.variable_order.set([x0, x1, x2, x3, x4])
polypy_test.init()
[x, y, z] = [polypy.Variable(name) for name in ['x', 'y', 'z']]
polypy.variable_order.set([z, y, x])
def check_sgn(p, assignment, expected_sgn):
sgn = p.sgn(assignment)
ok = (sgn > 0 and expected_sgn > 0) or (sgn < 0 and expected_sgn < 0) or (sgn == 0 and expected_sgn == 0)
if (not ok):
print "p =", p
print "assignment =", assignment
print "sgn =", sgn
print "expected_sgn =", expected_sgn
polypy_test.check(ok)
polypy_test.start("Sign Determination")
# polypy.trace_enable("polynomial")
# polypy.trace_enable("factorization")
# polypy.trace_enable("algebraic_number")
# polypy.trace_enable("coefficient")
# polypy.trace_enable("coefficient::sgn")
# polypy.trace_enable("coefficient::arith")
sqrt2 = polypy.AlgebraicNumber(x**2 - 2, 1)
sqrt2_4 = polypy.AlgebraicNumber(x**4 - 2, 1)
sqrt3 = polypy.AlgebraicNumber(x**2 - 3, 1)
assignment = polypy.Assignment()
assignment.set_value(x, sqrt2_4)
assignment.set_value(y, sqrt2)
polypy_test.init()
def check_factorization(p, p_factors, expected_size):
mul = 1
for (factor, multiplicity) in p_factors:
mul = mul * (factor**multiplicity)
ok = (p == mul) and (expected_size == len(p_factors))
if (not ok):
print("p = {0}".format(p))
print("p_factors = {0}".format(p_factors))
print("expected_size = {0}".format(expected_size))
print("mul = {0}".format(mul))
polypy_test.check(ok)
polypy_test.start("Square-Free Factorization")
# polypy.trace_enable("polynomial")
# polypy.trace_enable("factorization")
# polypy.trace_enable("coefficient::gcd")
#
# x00 = polypy.Variable("x00")
# x01 = polypy.Variable("x01")
# x10 = polypy.Variable("x10")
# x11 = polypy.Variable("x11")
# x20 = polypy.Variable("x20")
# x21 = polypy.Variable("x21")
# x30 = polypy.Variable("x30")
# x31 = polypy.Variable("x31")
#
# polypy.variable_order.set([x31, x30, x21, x20, x11])
def cyclotomic(n):
if (n == 1):
P1 = polypy.x - 1
return [P1]
else:
# Get up to 1 polynomials
L = cyclotomic(n-1)
# Compute Pn
Pn = polypy.x**n - 1
for d, Pd in enumerate(L):
if (n % (d+1) == 0):
Pn = Pn / Pd
L.append(Pn)
return L
polypy_test.start("Factorization in Z (Regressions)")
# polypy.trace_enable("factorization")
# polypy.trace_enable("hensel")
# polypy.trace_enable("arithmetic")
x = polypy.x
p = (x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5);
check_factorization(p)
p = 1*x**4 + 94*x**3 + 107*x**2 + 771*x + (-690)
check_factorization(p)
p= x**4 + 1
check_factorization(p)
from polypy import x
def check_comparison(a1, a2, cmp, result, expected):
if (result != expected):
print "a1 =", a1
print "a2 =", a2
print "cmp =", cmp
print "result =", result
print "expected =", expected
polypy_test.check(False)
else:
polypy_test.check(True)
polypy_test.init()
polypy_test.start("Construction and order")
a1 = polypy.AlgebraicNumber((x**2 - 2)*(x - 10), 1)
a2 = polypy.AlgebraicNumber((x**2 - 2), 1)
result = (a1 < a2)
check_comparison(a1, a2, "<", result, False)
result = (a1 == a2)
check_comparison(a1, a2, "==", result, True)
p = x**2
zero = polypy.AlgebraicNumber(p, 0);
result = zero.to_double() == 0.0
check_comparison(zero, 0, ".to_double ==", result, True)
]
# polypy_test.start("Polynomial Root Isolation: Speed")
#
# polypy.variable_order.set([y, x])
#
# y_value_poly = 8192*x**6 + (-14336*x**5) + 75376*x**4 + (-32736*x**3) + 109496*x**2 + 133752*x + (-32441)
# y_value = polypy.AlgebraicNumber(y_value_poly, 1)
#
# assignment = polypy.Assignment()
# assignment.set_value(y, y_value)
#
# p = (1*y**2 + 3)*x**6 + (4*y**2 + 24*y + 20)*x**5 + (36*y**2 + 112*y + 44)*x**4 + (8*y**3 + 114*y**2 + 144*y + 30)*x**3 + (1*y**4 + 24*y**3 + 142*y**2 + 40*y - 15)*x**2 + (2*y**4 + 40*y**3 + 64*y**2 - 32*y - 26)*x + (5*y**4 + 16*y**3 + 7*y**2 - 16*y - 8)
# roots = p.roots_isolate(assignment)
polypy_test.start("Polynomial Root Isolation: Bugs")
polypy.variable_order.set([x0, x1, x2, x3, x4, x5, x6, x7])
p = ((16 * x0**2 - 32 * x0 + 16) * x5**2 + ((32 * x0 - 32) * x1) * x5 +
(16 * x1**2)) * x7**2 + ((8 * x0**2 - 16 * x0 + 8) * x5**2 + (
(16 * x0**2 - 32 * x0 + 16) * x2**2 +
(-8 * x1**2))) * x7 + ((1 * x0**2 - 2 * x0 + 1) * x5**2 +
((-2 * x0 + 2) * x1) * x5 + (1 * x1**2))
p_lc = ((16 * x0**2 - 32 * x0 + 16) * x5**2 + ((32 * x0 - 32) * x1) * x5 +
(16 * x1**2))
assignment = polypy.Assignment()
v = polypy.AlgebraicNumber(32 * x**2 + (-64 * x) + 15, 0)
assignment.set_value(x0, v)
assignment.set_value(x1, 28)
[x, y, z] = [polypy.Variable(name) for name in ['x', 'y', 'z']]
polypy.variable_order.set([z, y, x])
def check_value(p, assignment, value, expected_value):
value_double = value.to_double()
ok = abs(value_double - expected_value) < 0.000001
if (not ok):
print "p =", p
print "assignment =", assignment
print "value =", value
print "value_double =", value_double
print "expected_value =", expected_value
polypy_test.check(ok)
polypy_test.start("Polynomial Evaluation")
# polypy.trace_enable("polynomial")
# polypy.trace_enable("factorization")
# polypy.trace_enable("algebraic_number")
# polypy.trace_enable("coefficient")
# polypy.trace_enable("coefficient::sgn")
# polypy.trace_enable("coefficient::roots")
# polypy.trace_enable("coefficient::arith")
sqrt2 = polypy.AlgebraicNumber(x**2 - 2, 1)
sqrt2_4 = polypy.AlgebraicNumber(x**4 - 2, 1)
sqrt3 = polypy.AlgebraicNumber(x**2 - 3, 1)
assignment = polypy.Assignment()
assignment.set_value(x, sqrt2)
polypy_test.check(ok)
def check_unary(op, op_name, p, expected):
result = op(p)
ok = result == expected
if (not ok):
print("p = {0}".format(p))
print("{0} = {1}".format(op_name, result))
print("expected = {0}".format(expected))
polypy_test.check(ok)
polypy_test.init()
polypy_test.start("Addition")
x = polypy.Variable("x")
y = polypy.Variable("y")
z = polypy.Variable("z")
polypy.variable_order.set([z, y, x])
# polypy.trace_enable("polynomial")
# polypy.trace_enable("coefficient")
# polypy.trace_enable("coefficient::reduce")
def poly_plus(p, q):
return p + q
# All signs
sgns = [
polypy.SGN_LT_0, polypy.SGN_LE_0, polypy.SGN_EQ_0, polypy.SGN_NE_0,
polypy.SGN_GT_0, polypy.SGN_GE_0
]
sgn_name = {
polypy.SGN_LT_0: "< 0",
polypy.SGN_LE_0: "<= 0",
polypy.SGN_EQ_0: "== 0",
polypy.SGN_NE_0: "!= 0",
polypy.SGN_GT_0: "> 0",
polypy.SGN_GE_0: ">= 0"
}
polypy_test.start("Polynomial Feasibility Integer")
assignment = polypy.Assignment()
p1 = x**2 - 7
S1 = p1.feasible_set(assignment, polypy.SGN_GE_0) # (-inf, -2.6], [2.6, +inf)
p2 = x * (x - 4)
S2 = p2.feasible_set(assignment, polypy.SGN_LT_0) # (0, 4)
S = S1.intersect(S2)
v = S.pick_value()
polypy_test.check(v.to_double() == 3)
polypy_test.start("Feasibility Intervals Intersection")
assignment = polypy.Assignment()
polypy_test.init()
def check_comparison(x, expected, precision=0.0000001):
diff = x.to_double() - expected
if (diff < -precision or diff > precision):
print("x = {0}".format(x))
print("expected = {0}".format(expected))
print("precision = {0}".format(precision))
polypy_test.check(False)
else:
polypy_test.check(True)
polypy_test.start("Construction")
zero_int = polypy.Value(0)
one_int = polypy.Value(1)
two_int = polypy.Value(2)
three_int = polypy.Value(3)
check_comparison(zero_int, 0)
check_comparison(one_int, 1)
check_comparison(two_int, 2)
check_comparison(three_int, 3)
p = x**2 - 2
sqrt2_neg = polypy.Value(polypy.AlgebraicNumber(p, 0))
sqrt2_pos = polypy.Value(polypy.AlgebraicNumber(p, 1))
print("lb = {0}".format(lb))
print("ub = {0}".format(ub))
print("count = {0}".format(count))
print("expected = {0}".format(expected))
else:
polypy_test.check(True)
# polypy.trace_enable("roots")
# polypy.trace_enable("algebraic_number")
# polypy.trace_enable("division")
# polypy.trace_enable("sturm_sequence_check")
x = polypy.x
polypy_test.start("Root isolation")
p = x**2 - 1
check_roots_isolate(p, 2)
p = (x - 1) * (x + 1) * (x - 2)
check_roots_isolate(p, 3)
p = 1 * x**5 + (-23 * x**4) + (-57 * x**3) + 85 * x**2 + 64 * x + (-63)
check_roots_isolate(p, 5)
p = 41 * x**5 + (-79 * x**4) + 44 * x**3 + 56 * x**2 + (-10 * x) + 41
check_roots_isolate(p, 1)
p = 9 * x**13 - 18 * x**11 - 33 * x**10 + 102 * x**8 + 7 * x**7 - 36 * x**6 - 122 * x**5 + 49 * x**4 + 93 * x**3 - 42 * x**2 - 18 * x**+9
check_roots_isolate(p, 6)
print("gcd = {0}".format(gcd))
polypy_test.check(ok)
def check_lcm(p, q, expected):
gcd = p.lcm(q)
ok = gcd == expected
if (not ok):
print("p = {0}".format(p))
print("q = {0}".format(q))
print("expected = {0}".format(expected))
print("gcd = {0}".format(gcd))
polypy_test.check(ok)
polypy_test.start("GCD")
# polypy.trace_enable("coefficient::gcd")
# polypy.trace_enable("polynomial")
# polypy.trace_enable("coefficient")
# polypy.trace_enable("coefficient::order")
# polypy.trace_enable("coefficient::reduce")
polypy.variable_order.set([z, y, x])
# p = ((1*z**68)*y**71 + (-2*z**48)*y**52 + (26896*z**118 + 26896*z**116 + 6724*z**114)*y**48 + (1*z**28)*y**33 + (-172200*z**59 - 86100*z**57)*y**24 + 275625)
# q = ((-328*z**129 + 164*z**127 + 82*z**125)*y**95 + (656*z**109 - 328*z**107 - 164*z**105)*y**76 + (4410944*z**177 + 6616416*z**175 + 3308208*z**173 + 551368*z**171)*y**72 + (-525*z**68)*y**71 + (-328*z**89 + 164*z**87 + 82*z**85)*y**57 + (1050*z**48)*y**52 + (-42361200*z**118 - 42361200*z**116 - 10590300*z**114)*y**48 + (-525*z**28)*y**33 + (135607500*z**59 + 67803750*z**57)*y**24 - 144703125)
# p.gcd(q)
# Wrong gcd
p = 4 * z**4 + 4 * z**2 + 1
if (n == 1):
P1 = polypy.x - 1
return [P1]
else:
# Get up to 1 polynomials
L = cyclotomic(n - 1)
# Compute Pn
Pn = polypy.x**n - 1
for d, Pd in enumerate(L):
if (n % (d + 1) == 0):
Pn = Pn / Pd
L.append(Pn)
return L
polypy_test.start("Factorization in Z (Regressions)")
# polypy.trace_enable("factorization")
# polypy.trace_enable("hensel")
# polypy.trace_enable("arithmetic")
x = polypy.x
p = (x - 1) * (x - 2) * (x - 3) * (x - 4) * (x - 5)
check_factorization(p)
p = 1 * x**4 + 94 * x**3 + 107 * x**2 + 771 * x + (-690)
check_factorization(p)
p = x**4 + 1
check_factorization(p)
print "p =", p
print "lb =", lb
print "ub =", ub
print "count =", count
print "expected = ", expected
else:
polypy_test.check(True)
# polypy.trace_enable("roots")
# polypy.trace_enable("algebraic_number")
# polypy.trace_enable("division")
# polypy.trace_enable("sturm_sequence_check")
x = polypy.x
polypy_test.start("Root isolation")
p = x**2 - 1
check_roots_isolate(p, 2)
p = (x - 1)*(x + 1)*(x - 2)
check_roots_isolate(p, 3)
p = 1*x**5 + (-23*x**4) + (-57*x**3) + 85*x**2 + 64*x + (-63)
check_roots_isolate(p, 5)
p = 41*x**5 + (-79*x**4) + 44*x**3 + 56*x**2 + (-10*x) + 41
check_roots_isolate(p, 1)
p = 9*x**13 - 18*x**11 - 33*x**10 + 102*x**8 + 7*x**7 - 36*x**6 - 122*x**5 + 49*x**4 + 93*x**3 - 42*x**2 - 18*x** + 9
check_roots_isolate(p, 6)
check_gcd_single(-p, q)
check_gcd_single(-p, -q)
def check_extended_gcd_single(p, q):
global polypy_test
(gcd, u, v) = p.extended_gcd(q)
ok = polypy_test.sympy_checker.check_extended_gcd(p, q, gcd, u, v)
polypy_test.check(ok)
def check_extended_gcd(p, q):
check_extended_gcd_single(p, q)
check_extended_gcd_single(p, -q)
check_extended_gcd_single(-p, q)
check_extended_gcd_single(-p, -q)
polypy_test.start("GCD in Z_13 (Knuth)")
# Example from Knuth (p 424)
K = polypy.CoefficientRing(13)
x = polypy.x.to_ring(K);
p = x**8 + x**6 + 10*x**4 + 10*x**3 + 8*x**2 + 2*x + 8
q = 3*x**6 + 5*x**4 + 9*x**2 + 4*x + 8
check_gcd(p, q)
polypy_test.start("Regressions (modular)")
K = polypy.CoefficientRing(7);
x = polypy.x.to_ring(K)
p = 2*x
polypy.SGN_EQ_0,
polypy.SGN_NE_0,
polypy.SGN_GT_0,
polypy.SGN_GE_0
]
sgn_name = {
polypy.SGN_LT_0 : "< 0",
polypy.SGN_LE_0 : "<= 0",
polypy.SGN_EQ_0 : "== 0",
polypy.SGN_NE_0 : "!= 0",
polypy.SGN_GT_0 : "> 0",
polypy.SGN_GE_0 : ">= 0"
}
polypy_test.start("Polynomial Feasibility Integer")
assignment = polypy.Assignment()
p1 = x**2 - 7
S1 = p1.feasible_set(assignment, polypy.SGN_GE_0) # (-inf, -2.6], [2.6, +inf)
p2 = x*(x - 4)
S2 = p2.feasible_set(assignment, polypy.SGN_LT_0) # (0, 4)
S = S1.intersect(S2)
v = S.pick_value();
polypy_test.check(v.to_double() == 3)
polypy_test.start("Feasibility Intervals Intersection")
assignment = polypy.Assignment()
