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Using QEPCAD in a CoCalc Sage worksheet -- examples from the SageMath reference manual: http://doc.sagemath.org/html/en/reference/interfaces/sage/interfaces/qepcad.html
Project: ask-help-support
Path: qepcad.sagews
Views: 353(a, b, c, d, x, y, z)
(E y)[3 x^2 + 2 x y + y^2 - x + y - 7 = 0]
8 x^2 - 8 x - 29 <= 0
8 y^2 + 16 y - 85 <= 0
(X3 y)[(3 x^2 + 2 x y + y^2 - x + y - 7) (x^2 + y^2 - 3) = 0]
x^2 - 3 <= 0 /\ 8 x^2 - 8 x - 29 <= 0 /\ 8 x^4 - 26 x^2 - 4 x + 13 >= 0 /\ [ 8 x^4 - 26 x^2 - 4 x + 13 = 0 \/ x^2 - 3 = 0 \/ 8 x^2 - 8 x - 29 = 0 ]
(X3 y)[(3 x^2 + 2 x y + y^2 - x + y - 7) (x^2 + y^2 - 3) = 0]
8 x^2 - 8 x - 29 <= 0 /\ x^2 - 3 <= 0 /\ 8 x^4 - 26 x^2 - 4 x + 13 >= 0 /\ [ 8 x^4 - 26 x^2 - 4 x + 13 = 0 \/ x^2 - 3 = 0 \/ 8 x^2 - 8 x - 29 = 0 ]
x = _root_1 8 x^2 - 8 x - 29
\/
8 x^4 - 26 x^2 - 4 x + 13 = 0
\/
x = _root_-1 x^2 - 3
[3 x^2 + 2 x y + y^2 - x + y - 7 < 0 /\ x^2 + y^2 - 3 < 0]
y^2 + 2 x y + y + 3 x^2 - x - 7 < 0 /\ y^2 + x^2 - 3 < 0
[
[
x = _root_-2 8 x^4 - 26 x^2 - 4 x + 13
\/
x = _root_2 8 x^4 - 26 x^2 - 4 x + 13
\/
8 x^4 - 26 x^2 - 4 x + 13 < 0
]
/\
y^2 + 2 x y + y + 3 x^2 - x - 7 < 0
/\
y^2 + x^2 - 3 < 0
]
\/
[
x > _root_2 8 x^4 - 26 x^2 - 4 x + 13
/\
x < _root_-2 8 x^4 - 26 x^2 - 4 x + 13
/\
y^2 + x^2 - 3 < 0
]
(E y)[x^2 + y^2 - 3 = 0 /\ x + y > 0]
x^2 - 3 <= 0 /\ [ x > 0 \/ 2 x^2 - 3 < 0 ]
x^2 - 3 <= 0 /\ x > _root_1 2 x^2 - 3
{'y': 0.9685019685029527?, 'x': -1.468501968502953?}
True
[{'y': 0, 'x': 4}, {'y': 1, 'x': 2.468501968502953?}, {'y': -9, 'x': 2.468501968502953?}, {'y': 9, 'x': 1/2}, {'y': -1, 'x': 1/2}, {'y': -5, 'x': 1/2}, {'y': 3, 'x': -1.468501968502953?}, {'y': -1, 'x': -1.468501968502953?}, {'y': 0, 'x': -3}]
[True, True, True, True, True, True, True, True, True]
(X3 y)[(3 x^2 + 2 x y + y^2 - x + y - 7) (x^2 + y^2 - 3) = 0]
[{'x': 1.732050807568878?}, {'x': 1.731054913462534?}, {'x': 0.678911384208004?}, {'x': -0.9417727377417167?}, {'x': -1.468193559928821?}, {'x': -1.468501968502953?}]
y^4 + 4.464101615137755?*y^3 + 0.2679491924311227?*y^2
[(-4.403249005600958?, 1), (-0.06085260953679653?, 1), (0, 2)]
[3, 3, 3, 3, 3, 3]
QEPCAD object in phase 'Before Normalization'
Error GETCID: This command is not active here.
QEPCAD object has moved to phase 'Before Projection (x)'
QEPCAD object has moved to phase 'Before Choice'
QEPCAD object has moved to phase 'Before Solution'
4 c - b^2 > 0
QEPCAD object in phase 'EXITED'
QEPCAD object in phase 'Before Normalization'
QEPCAD object has moved to phase 'Before Projection (y)'
QEPCAD object has moved to phase 'Before Choice'
QEPCAD object has moved to phase 'Before Solution'
---------- Information about the cell (3,4) ----------
Level : 2
Dimension : 1
Number of children : 0
Truth value : T by trial evaluation.
Degrees after substitution : Not known yet or No polynomial.
Multiplicities : ((1,1))
Signs of Projection Factors
Level 1 : (-)
Level 2 : (0)
---------- Sample point ----------
The sample point is in a PRIMITIVE representation.
alpha = the unique root of x^2 - 3 between 0 and 4
= 1.7320508076-
Coordinate 1 = 0
= 0.0000000000
Coordinate 2 = alpha
= 1.7320508076-
----------------------------------------------------
QEPCAD cell (3, 4)
2
(3, 4)
5
5
(0, 1.732050807568878?)
{'y': 1.732050807568878?, 'x': 0}
[QEPCAD cell (4, 2), QEPCAD cell (3, 4), QEPCAD cell (3, 2), QEPCAD cell (2, 2)]
QEPCAD cell (3, 1)
[QEPCAD cell (3, 1), QEPCAD cell (3, 2), QEPCAD cell (3, 3), QEPCAD cell (3, 4), QEPCAD cell (3, 5)]
(X2 y)[x^2 + y^2 - 3 = 0]
(X2 y)[3 x^2 + 2 x y + y^2 - x + y - 7 = 0]
Error in lines 1-1
Traceback (most recent call last):
File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 904, in execute
exec compile(block+'\n', '', 'single') in namespace, locals
File "", line 1, in <module>
File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/interfaces/qepcad.py", line 1962, in and_
formula_strs, vars = self._combine_formulas(formulas)
File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/interfaces/qepcad.py", line 1858, in _combine_formulas
raise ValueError("QEPCAD formulas must be in prenex"
ValueError: QEPCAD formulas must be in prenex (quantifiers outermost) form
QEPCAD object has moved to phase 'Before Projection (y)'
QEPCAD object has moved to phase 'Before Choice'
QEPCAD object has moved to phase 'Before Solution'
P_1,1 = fac(J_1,1) = fac(dis(A_2,1))
= 8 x^2 - 8 x - 29
P_1,2 = fac(J_1,2) = fac(dis(A_2,2))
= x^2 - 3
P_1,3 = fac(J_1,3) = fac(res(A_2,1|A_2,2))
= 8 x^4 - 26 x^2 - 4 x + 13
A_2,1 = input
= y^2 + 2 x y + y + 3 x^2 - x - 7
A_2,2 = input
= y^2 + x^2 - 3
0
8 x^2 - 8 x - 29 < 0
/\
x^2 - 3 < 0
'SINGULAR /projects/sage/sage-6.10/local/bin\n'
(a, b, c, d, x, y, z)
set(['x^2 - 3 = 0', '8 x^2 - 8 x - 29 <= 0', '8 x^4 - 26 x^2 - 4 x + 13 = 0', '8 x^2 - 8 x - 29 = 0', 'x^2 - 3 <= 0', '8 x^4 - 26 x^2 - 4 x + 13 >= 0'])
set(['x = _root_1 8 x^2 - 8 x - 29', '8 x^4 - 26 x^2 - 4 x + 13 = 0', 'x = _root_-1 x^2 - 3'])
set(['y^2 + 2 x y + y + 3 x^2 - x - 7 < 0', 'y^2 + x^2 - 3 < 0'])
set(['8 x^4 - 26 x^2 - 4 x + 13 < 0', 'x > _root_2 8 x^4 - 26 x^2 - 4 x + 13', 'x = _root_2 8 x^4 - 26 x^2 - 4 x + 13', 'y^2 + 2 x y + y + 3 x^2 - x - 7 < 0', 'x < _root_-2 8 x^4 - 26 x^2 - 4 x + 13', 'y^2 + x^2 - 3 < 0', 'x = _root_-2 8 x^4 - 26 x^2 - 4 x + 13'])
set(['x > 0', '2 x^2 - 3 < 0', 'x^2 - 3 <= 0'])
set(['x > _root_1 2 x^2 - 3', 'x^2 - 3 <= 0'])
QEPCAD object has moved to phase 'Before Projection (y)'
QEPCAD object has moved to phase 'Before Choice'
QEPCAD object has moved to phase 'Before Solution'
set(['x^2 - 3 < 0', '8 x^2 - 8 x - 29 < 0'])