I’m trying to approximate Julia sets using roots of polynomials in Python. In particular I want to find the roots of the nth iterate of the polynomial q(z) = z^2-0.5. In other words I want to find the roots of q(q(q(..))) composed n times. In order to get many sample points I would need to compute the roots of polynomials of degree >1000.
I’ve tried solving this problem using both the built in polynomial class of numpy which has a root function and also the function solver of sympy. In the first case precision is lost when I choose degrees larger than 100. The sympy computation simply takes to long time. Here is my code:
p = P([-0.5,0,1]) for k in range(9): p = p**2-0.5 roots = p.roots() plt.plot([np.real(r) for r in roots], [np.imag(r) for r in roots],'x') plt.show() abs_vector = [np.abs(p(r)) for r in roots] max = 0 for a in abs_vector: if a > max: max = a print(max)
The max value above gives the largest value of p at a supposed root. However running this code gives me 7.881370400084486e+296 which is very large.
How would one go about computing roots of high degree polynomials with good accuracy in a short amount of time?
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Answer
For the n-times composition of a polynomial q
you can reconstruct the roots iteratively
q = [1,0,-0.5] n = 9 def q_preimage(w): c = q.copy() c[-1] -= w return np.roots(c) rts = [0] for k in range(n): rts = np.concatenate([q_preimage(w) for w in rts])
which returns
array([ 1.36444432e+00+0.00095319j, -1.36444432e+00-0.00095319j, 1.40104860e-03-0.92828301j, -1.40104860e-03+0.92828301j, 8.82183775e-01-0.52384727j, -8.82183775e-01+0.52384727j, 8.78972436e-01+0.52576116j, -8.78972436e-01-0.52576116j, 1.19545693e+00-0.21647154j, -1.19545693e+00+0.21647154j, 3.61362916e-01+0.71612883j, -3.61362916e-01-0.71612883j, 1.19225541e+00+0.21925381j, -1.19225541e+00-0.21925381j, 3.66786415e-01-0.71269419j, -3.66786415e-01+0.71269419j, ...
or plotted
plt.plot(rts.real, rts.imag,'ob', ms=2); plt.grid(); plt.show()