The polynomial
z4 - 1 has roots at 1,
i, -1, and -
i. White dots are placed around the roots.
The polynomial
z3 - 1 has the three roots of unity as its roots. The three roots are 1 (of course), -0.5 + 0.86603
i, and -0.5 - 0.86603
i.
This and the last example are specific cases of the
n-th roots of unity. They are the roots of the polynomial
zn - 1. The
n roots are equally spaced around the unit circle (that is, the circle whose equation is
x2 +
y2 = 1), so their coordinates can easily be given in terms of cosines and sines. The angles for these
n points as measured counterclockwise from the positive
x-axis are
0°, 360°/n, 360°·2/n, ..., 360° (n-1)/n,
the k-th of these angles being 360° k/n. Therefore, the k-th root is
cos (360° k/n) + i sin (360° k/n).
In particular, the 0-th root is 1 itself.
Even when the roots are all real, the Newton basins in the complex plane are interesting. In this example, the polynomial is
z3 -
z, and its roots are -1, 0, and 1.
Here's an example where all the roots are complex but come in pairs.
The six roots are ±(1+3
i), ±(5+
i), and ±(3-2
i). The sixth degree polynomial is (
z2 - (1+3
i)
2)(
z2 - (5+
i)
2)(
z2 - (3-2
i)
2).
The polynomial (
z4 - 1)(
z4 + 16), that is,
z8 + 15
z4 - 16, has eight roots: ±1, ±
i, ±(1+
i), and ±(1-
i).
The roots are at the corners and the midpoints of the sides of the displayed square.
This polynomial (
z4 - 1)(
z2 - (1+
i)
2) has six roots: ±1, ±
i, and ±(1+
i).
The lighter colors indicate fewer iterations of Newton's method before the approximations are close to the roots.
The polynomial (
z2 + 4
z + 1)(
z2 +4)(
z - 2), that is,
z5 + 2
z4 - 3
z3 + 6
z2 - 28
z - 8,
has five roots: 2+
i, 2-
i, 2
i, -2
i, and -2.
The polynomial (
z4 - 1)(
z4 - 16), that is,
z8 - 17
z4 + 16, has eight roots: ±1, ±
i, ±2, and ±2
i.
Looking at the examples so far you'll get the impression that Newton's method almost always works. But sometimes that isn't so. Here's an example of a cubic polynomial for which lots of initial guesses never get close to any solution. Those initial guesses are colored black.
The cubic polynomial for this image has two of the same roots as the last polynomial, but the third is slightly moved. The black area is a much more interesting shape.
