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2011-12-15 13:42:08 +0100	marked best answer	Newton's cubic Fractal help. Plotting `LA(x) = x - (x^3.0-1.0)/(3.0*x^2.0) S = solve(LA==0,x) n=0 for s in S: a=solve(LA == s.rhs(),x) n=n+1 point([a[i].rhs().n().real(),a[i].rhs().n().imag()] for i in range(n))` Does this help?
2011-12-14 21:44:19 +0100	asked a question	Newton's cubic Fractal help. Plotting I'm solving newton's cubic for all the black points that occur within the fractal: I have the code: LA(x) = x - (x^3.0-1)/(3.0*x^2.0) S = solve(LA==0,x) for s in S: solve(LA == s.rhs(),x) and i want to take all those points I'm generating and plot them in a scatter plot. I think I have to use an array? Can anyone tell me exactly how I tell sage to put these points in an array then plot them? Thanks everyone for your help
2011-12-13 20:23:42 +0100	commented answer	Newton's cubic Math Modeling thanks. i appreciate your help. I'm still looking for an algorithm but cant seem to come up with anything!
2011-12-13 20:22:57 +0100	marked best answer	Newton's cubic Math Modeling I don't think this is a built-in behavior. I'd be interested in a reference for an algorithm, even! Basically, this would just be 0, anything whose tangent line sends it to zero, anything whose tangent line sends it to one of those, etc., right? If you just wanted to visualize this, you could try using some of the ideas at this ticket about Newton basins, though it wouldn't give the "black points". For a graphical way of exploring this, I would search sagenb.org for implementations of Newton's method in interactive mode. You could use `solve` to try to get some of these points, though I'm not sure how useful this would be. sage: LA(x) = x - (x^3-1)/(3x^2) sage: LA(0) --------------------------------------------------------------------------- RuntimeError: power::eval(): division by zero sage: solve(LA==0,x) [x == 1/4(I(-1)^(1/3)sqrt(3) - (-1)^(1/3))2^(2/3), x == 1/4(-I(-1)^(1/3)sqrt(3) - (-1)^(1/3))2^(2/3), x == 1/2(-1)^(1/3)2^(2/3)] sage: S = solve(LA==0,x) sage: for s in S: solve(LA == s.rhs(),x) ....: [x == -1/576(-18Isqrt(3) - 18)(-Isqrt(3) + 1)2^(1/3)/(1/8sqrt(6) - 5/16)^(1/3) + 1/24(3Isqrt(3) - 3)(-1)^(1/3)2^(2/3) - 1/2(1/8sqrt(6) - 5/16)^(1/3)(Isqrt(3) + 1), x == -1/576(-18Isqrt(3) - 18)(Isqrt(3) + 1)2^(1/3)/(1/8sqrt(6) - 5/16)^(1/3) + 1/24(3Isqrt(3) - 3)(-1)^(1/3)2^(2/3) - 1/2(1/8sqrt(6) - 5/16)^(1/3)(-Isqrt(3) + 1), x == 1/24(3Isqrt(3) - 3)(-1)^(1/3)2^(2/3) + 1/288(-18Isqrt(3) - 18)2^(1/3)/(1/8sqrt(6) - 5/16)^(1/3) + (1/8sqrt(6) - 5/16)^(1/3)] [x == -1/576(-Isqrt(3) + 1)(18Isqrt(3) - 18)2^(1/3)/(1/8sqrt(6) - 5/16)^(1/3) - 1/24(3Isqrt(3) + 3)(-1)^(1/3)2^(2/3) - 1/2(1/8sqrt(6) - 5/16)^(1/3)(Isqrt(3) + 1), x == -1/576(Isqrt(3) + 1)(18Isqrt(3) - 18)2^(1/3)/(1/8sqrt(6) - 5/16)^(1/3) - 1/24(3Isqrt(3) + 3)(-1)^(1/3)2^(2/3) - 1/2(1/8sqrt(6) - 5/16)^(1/3)(-Isqrt(3) + 1), x == -1/24(3Isqrt(3) + 3)(-1)^(1/3)2^(2/3) + 1/288(18Isqrt(3) - 18)2^(1/3)/(1/8sqrt(6) - 5/16)^(1/3) + (1/8sqrt(6) - 5/16)^(1/3)] [x == -1/2(Isqrt(3) + 1)(1/8sqrt(2)sqrt(3) - 5/16)^(1/3) - 1/16(-Isqrt(3) + 1)*2 ... (more)
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2011-12-12 22:05:24 +0100	asked a question	Newton's cubic Math Modeling I need to find the "black points" (where the function becomes undefined) of an equation x^3 -1 using newtons method. So the function turns out to be F(x) = x - (x^3-1)/(3x^2). Is there a way to find a majority of the points (where x is undefined as in the Newton cubic fractal photo aka the black points) using Sage?