Comment #8 on issue 51 by mattpap: RootOf for polynomial equations
http://code.google.com/p/sympy/issues/detail?id=51
I made some progress with RootOf implementation. A few examples follow:
In [1]: f = cyclotomic_poly(20, x)
In [2]: RootOf(f, 0)
Out[2]:
⎛ 2 4 6 8⎞
ζ₀⎝1 - x + x - x + x ⎠
In [3]: for i in xrange(0, 8): RootOf(f, i).evalf()
...:
Out[3]: -0.951056516295154 + 0.309016994374947⋅ⅈ
Out[3]: -0.951056516295154 - 0.309016994374947⋅ⅈ
Out[3]: -0.587785252292473 + 0.809016994374947⋅ⅈ
Out[3]: -0.587785252292473 - 0.809016994374947⋅ⅈ
Out[3]: 0.951056516295154 + 0.309016994374947⋅ⅈ
Out[3]: 0.951056516295154 - 0.309016994374947⋅ⅈ
Out[3]: 0.587785252292473 + 0.809016994374947⋅ⅈ
Out[3]: 0.587785252292473 - 0.809016994374947⋅ⅈ
In [4]: f = expand((x-1)**2*(x**2 - 2)*(x**3 + 2)**2)
In [5]: for i in xrange(0, 10): RootOf(f, i)
...:
Out[5]:
⎛ 2⎞
ζ₀⎝-2 + x ⎠
Out[5]:
⎛ 3⎞
ζ₀⎝2 + x ⎠
Out[5]:
⎛ 3⎞
ζ₀⎝2 + x ⎠
Out[5]: 1
Out[5]: 1
Out[5]:
⎛ 2⎞
ζ₁⎝-2 + x ⎠
Out[5]:
⎛ 3⎞
ζ₁⎝2 + x ⎠
Out[5]:
⎛ 3⎞
ζ₁⎝2 + x ⎠
Out[5]:
⎛ 3⎞
ζ₂⎝2 + x ⎠
Out[5]:
⎛ 3⎞
ζ₂⎝2 + x ⎠
In [6]: for i in xrange(0, 10): RootOf(f, i).evalf()
...:
Out[6]: -1.41421356237310
Out[6]: -1.25992104989487
Out[6]: -1.25992104989487
Out[6]: 1.00000000000000
Out[6]: 1.00000000000000
Out[6]: 1.41421356237310
Out[6]: 0.629960524947437 + 1.09112363597172⋅ⅈ
Out[6]: 0.629960524947437 + 1.09112363597172⋅ⅈ
Out[6]: 0.629960524947437 - 1.09112363597172⋅ⅈ
Out[6]: 0.629960524947437 - 1.09112363597172⋅ⅈ
This is all quite slow as complex root isolation and refinement algorithms
are slow
at the moment. To improve speed I use caching, so that after univariate
factoring
over integers, unique (irreducible, primitive) polynomials are obtained and
root
isolation is done only once per polynomial and stored in cache. Each
successive call
to RootOf uses previously computed intervals. If new polynomials are
encountered,
then root isolation is performed only for those polynomials. If some of the
real or
complex intervals aren't disjoint, then they are refined until they are
disjoint and
new smaller intervals are stored in cache. evalf() uses mpmath's findroot()
and root
refinement, if the current interval doesn't make findroot() successful.
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