simple demo of Newtons method root finding
evaluate p(z) at z = (2,0.2)
polynomial p of degree 6
p^6 = (1,0)
p^5 = (0,0)
p^4 = (0,0)
p^3 = (0,0)
p^2 = (0,0)
p^1 = (-1,0)
p^0 = (-1,0)
(51.4959,36.9238) = value of function
polynomial der of degree 5
der^5 = (6,0)
der^4 = (0,0)
der^3 = (0,0)
der^2 = (0,0)
der^1 = (0,0)
der^0 = (-1,0)
remove a root at z=1.1
polynomial reduced of degree 5
reduced^5 = (1,0)
reduced^4 = (1.1,0)
reduced^3 = (1.21,0)
reduced^2 = (1.331,0)
reduced^1 = (1.4641,0)
reduced^0 = (0.61051,0)
find 6th root
guess x0= (2,0) y0= (61,0)  dy0= (191,0)
 next x1= (1.68063,0) y1= (19.8529,0)
guess x0= (1.68063,0) y0= (19.8529,0)  dy0= (79.4469,0)
 next x1= (1.43074,0) y1= (6.1468,0)
guess x0= (1.43074,0) y0= (6.1468,0)  dy0= (34.9711,0)
 next x1= (1.25497,0) y1= (1.65166,0)
guess x0= (1.25497,0) y0= (1.65166,0)  dy0= (17.6775,0)
 next x1= (1.16154,0) y1= (0.29431,0)
guess x0= (1.16154,0) y0= (0.29431,0)  dy0= (11.6858,0)
 next x1= (1.13635,0) y1= (0.0168261,0)
guess x0= (1.13635,0) y0= (0.0168261,0)  dy0= (10.3689,0)
 next x1= (1.13473,0) y1= (6.57384e-05,0)
guess x0= (1.13473,0) y0= (6.57384e-05,0)  dy0= (10.2879,0)
 next x1= (1.13472,0) y1= (1.01541e-09,0)
found root= (1.13473,0) remove root from p
polynomial red of degree 5
red^5 = (1,0)
red^4 = (1.13473,0)
red^3 = (1.28761,0)
red^2 = (1.46109,0)
red^1 = (1.65795,0)
red^0 = (0.881324,0)
find 5th root
guess x0= (-2,-0) y0= (-20.7354,0)  dy0= (54.9536,0)
 next x1= (-1.62267,-0) y1= (-6.84624,0)
guess x0= (-1.62267,-0) y0= (-6.84624,0)  dy0= (22.3596,0)
 next x1= (-1.31649,-0) y1= (-2.25289,0)
guess x0= (-1.31649,-0) y0= (-2.25289,0)  dy0= (9.1683,0)
 next x1= (-1.07076,-0) y1= (-0.715408,0)
guess x0= (-1.07076,-0) y0= (-0.715408,0)  dy0= (3.95821,0)
 next x1= (-0.890019,-0) y1= (-0.191134,0)
guess x0= (-0.890019,-0) y0= (-0.191134,0)  dy0= (2.05441,0)
 next x1= (-0.796983,-0) y1= (-0.0275327,0)
guess x0= (-0.796983,-0) y0= (-0.0275327,0)  dy0= (1.50217,0)
 next x1= (-0.778654,-0) y1= (-0.000766781,0)
guess x0= (-0.778654,-0) y0= (-0.000766781,0)  dy0= (1.41981,0)
 next x1= (-0.778114,-0) y1= (-6.23494e-07,0)
found root= (-0.778654,-0) remove root from p
polynomial red of degree 4
red^4 = (1,0)
red^3 = (0.356076,0)
red^2 = (1.01035,0)
red^1 = (0.674378,0)
red^0 = (1.13284,0)
find 4th root
guess x0= (2,0) y0= (25.3716,0)  dy0= (40.9887,0)
 next x1= (1.38101,0) y1= (8.56631,0)
guess x0= (1.38101,0) y0= (8.56631,0)  dy0= (16.0377,0)
 next x1= (0.846873,0) y1= (3.15921,0)
guess x0= (0.846873,0) y0= (3.15921,0)  dy0= (5.58128,0)
 next x1= (0.280836,0) y1= (1.41602,0)
guess x0= (0.280836,0) y0= (1.41602,0)  dy0= (1.41471,0)
 next x1= (-0.720091,0) y1= (1.30705,0)
guess x0= (-0.720091,0) y0= (1.30705,0)  dy0= (-1.72036,0)
 next x1= (0.0396595,0) y1= (1.1612,0)
guess x0= (0.0396595,0) y0= (1.1612,0)  dy0= (0.756448,0)
 next x1= (-1.49541,0) y1= (6.19382,0)
guess x0= (-1.49541,0) y0= (6.19382,0)  dy0= (-13.335,0)
 next x1= (-1.03093,0) y1= (2.25086,0)
guess x0= (-1.03093,0) y0= (2.25086,0)  dy0= (-4.65626,0)
 next x1= (-0.547527,0) y1= (1.09791,0)
guess x0= (-0.547527,0) y0= (1.09791,0)  dy0= (-0.768335,0)
 next x1= (0.881426,0) y1= (3.35964,0)
guess x0= (0.881426,0) y0= (3.35964,0)  dy0= (6.02457,0)
 next x1= (0.323769,0) y1= (1.48017,0)
guess x0= (0.323769,0) y0= (1.48017,0)  dy0= (1.57636,0)
 next x1= (-0.615211,0) y1= (1.1607,0)
guess x0= (-0.615211,0) y0= (1.1607,0)  dy0= (-1.09587,0)
 next x1= (0.44395,0) y1= (1.70136,0)
guess x0= (0.44395,0) y0= (1.70136,0)  dy0= (2.13201,0)
 next x1= (-0.354061,0) y1= (1.02064,0)
guess x0= (-0.354061,0) y0= (1.02064,0)  dy0= (-0.0847029,0)
 next x1= (11.6956,0) y1= (19427.3,0)
guess x0= (11.6956,0) y0= (19427.3,0)  dy0= (6569.57,0)
 next x1= (8.7384,0) y1= (6152.55,0)
guess x0= (8.7384,0) y0= (6152.55,0)  dy0= (2768.94,0)
 next x1= (6.51641,0) y1= (1950.12,0)
guess x0= (6.51641,0) y0= (1950.12,0)  dy0= (1166.04,0)
 next x1= (4.84399,0) y1= (619.147,0)
guess x0= (4.84399,0) y0= (619.147,0)  dy0= (490.169,0)
 next x1= (3.58086,0) y1= (197.27,0)
guess x0= (3.58086,0) y0= (197.27,0)  dy0= (205.27,0)
 next x1= (2.61983,0) y1= (63.3447,0)
guess x0= (2.61983,0) y0= (63.3447,0)  dy0= (85.2251,0)
 next x1= (1.87657,0) y1= (20.7104,0)
no convergence, quitting