2017-12-24 10:45:32 +0200 | received badge | ● Good Question
(source)
|
2017-11-25 02:12:34 +0200 | received badge | ● Nice Question
(source)
|
2015-07-09 19:18:02 +0200 | received badge | ● Popular Question
(source)
|
2015-07-09 19:18:02 +0200 | received badge | ● Notable Question
(source)
|
2015-07-09 19:18:02 +0200 | received badge | ● Famous Question
(source)
|
2015-03-24 20:15:29 +0200 | received badge | ● Famous Question
(source)
|
2014-03-31 18:04:04 +0200 | received badge | ● Notable Question
(source)
|
2014-01-14 08:25:21 +0200 | asked a question | Bound box for parametric 3D plots? I am trying to plot a parametric surface but only in a certain box in 3D space. Namely, I would like to get JMol to plot this: var('s t')
p = [s^2, s^2+t^2, s^2+s*t+s+t]; q = s^2+t^2+s-t+1
parametric_plot3d((p[0]/q,p[1]/q,p[2]/q),(s,-5,5),(t,-5,5), points=[100,100])
but restricted to [0,1]x[0,1]x[0,1] (for example). I see that implicit plots work exactly like that, but the singular part of this surface looks quite ugly: http://imgur.com/5IxilIz. I would be fine with tachyon too, but I haven't been able to figure out how to rotate the camera (I basically want to cut out a chunk of that surface to see a section). |
2014-01-14 08:09:44 +0200 | received badge | ● Supporter
(source)
|
2014-01-14 08:09:35 +0200 | received badge | ● Scholar
(source)
|
2014-01-14 08:09:35 +0200 | marked best answer | Partial fraction decomposition over the reals or complex Expanding on Snark's comment: use algebraic numbers to get an exact answer. Algebraic numbers can be explored, you can ask their minimal polynomial or a radical expression. sage: R.<x> = QQbar['x']
sage: ff = x^3 - 2
sage: factor(ff)
(x - 1.259921049894873?) * (x + 0.6299605249474365? - 1.091123635971722?*I) * (x + 0.6299605249474365? + 1.091123635971722?*I)
sage: (1/ff).partial_fraction_decomposition()
(0,
[(0.2099868416491456? + 0.?e-19*I)/(x - 1.259921049894873?),
(-0.10499342082457277? + 0.1818539393286203?*I)/(x + 0.6299605249474365? - 1.091123635971722?*I),
(-0.10499342082457277? - 0.1818539393286203?*I)/(x + 0.6299605249474365? + 1.091123635971722?*I)])
The entries are exact; they live in QQbar . To investigate the entries: sage: for q in factor(ff):
....: q = q[0]
....: print 'x - a =', q
....: a = QQbar(x - q)
....: print ' a =', a
....: print ' minpoly:', a.minpoly()
....: print ' radical:', a.radical_expression()
....:
x - a = x - 1.259921049894873?
a = 1.259921049894873?
minpoly: x^3 - 2
radical: 2^(1/3)
x - a = x + 0.6299605249474365? - 1.091123635971722?*I
a = -0.6299605249474365? + 1.091123635971722?*I
minpoly: x^3 - 2
radical: 1/2*I*sqrt(3)*2^(1/3) - 1/2*2^(1/3)
x - a = x + 0.6299605249474365? + 1.091123635971722?*I
a = -0.6299605249474365? - 1.091123635971722?*I
minpoly: x^3 - 2
radical: -1/2*I*sqrt(3)*2^(1/3) - 1/2*2^(1/3)
and sage: for q in (1/ff).partial_fraction_decomposition()[1]:
....: print 'a / (x - b) =', q
....: a = QQbar(q.numerator())
....: print ' a =', a
....: print ' minpoly:', a.minpoly()
....: print ' radical:', a.radical_expression()
....: b = QQbar(x - q.denominator())
....: print ' b =', b
....: print ' minpoly:', b.minpoly()
....: print ' radical:', b.radical_expression()
....:
a / (x - b) = 0.2099868416491456?/(x - 1.259921049894873?)
a = 0.2099868416491456?
minpoly: x^3 - 1/108
radical: 1/12*4^(2/3)
b = 1.259921049894873?
minpoly: x^3 - 2
radical: 2^(1/3)
a / (x - b) = (-0.10499342082457277? + 0.1818539393286203?*I)/(x + 0.6299605249474365? - 1.091123635971722?*I)
a = -0.10499342082457277? + 0.1818539393286203?*I
minpoly: x^3 - 1/108
radical: 1/24*4^(2/3)*(I*sqrt(3) - 1)
b = -0.6299605249474365? + 1.091123635971722?*I
minpoly: x^3 - 2
radical: 1/2*I*sqrt(3)*2^(1/3) - 1/2*2^(1/3)
a / (x - b) = (-0.10499342082457277? - 0.1818539393286203?*I)/(x + 0.6299605249474365? + 1.091123635971722?*I)
a = -0.10499342082457277? - 0.1818539393286203?*I
minpoly: x^3 - 1/108
radical: 1/24*4^(2/3)*(-I*sqrt(3) - 1)
b = -0.6299605249474365? - 1.091123635971722?*I
minpoly: x^3 - 2
radical: -1/2*I*sqrt(3)*2^(1/3) - 1/2*2^(1/3)
|
2013-09-21 22:08:01 +0200 | received badge | ● Popular Question
(source)
|
2012-11-20 17:10:21 +0200 | received badge | ● Student
(source)
|
2012-11-20 15:29:04 +0200 | asked a question | Partial fraction decomposition over the reals or complex Hi, I am trying to get partial fraction decomposition (for integration) but over the complex numbers. For example, the denominator (x^2+1)*(x^3-2) should produce five fractions. I have experimented a bit with coercion but didn't manage to get anything. For example: f=x^3-2
R=CC['x']
ff=R(f)
factor(f); factor(ff)
produces x^3 - 2
(x - 1.25992104989487) * (x + 0.629960524947437 - 1.09112363597172*I) * (x + 0.629960524947437 + 1.09112363597172*I)
so far so good (except for decimals). But (1/f).partial_fraction() produces 1/(x^3 - 2) , and (1/ff).partial_fraction() produces Traceback (click to the left of this block for traceback)
...
AttributeError: 'FractionFieldElement_1poly_field' object has no attribute 'partial_fraction'
On the other hand, (1/ff).partial_fraction_decomposition() gives (0, [0.209986841649145/(x - 1.25992104989487), (-0.104993420824573 +
0.181853939328620*I)/(x + 0.629960524947437 - 1.09112363597172*I),
(-0.104993420824573 - 0.181853939328620*I)/(x + 0.629960524947437 +
1.09112363597172*I)])
which is correct but not exact. Is there a command to get an exact decomposition over C, or am I stuck with having to set and solve a linear system? Thanks |