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Question: newtons method allows the calculation of increasingly good approximations to...

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Newtons method allows the calculation of increasingly good approximations to a root of an equation f (z) 0, where f(z) is analytic on a domain containing the root. Given an initial point zo, the sequence zn+ln-f(En)/f (Zn) for 0, 1, 2, generally converges to a root of f (z) 0, but the root to which it converges is sometimes unpredictable whern there is more than one root. Use this method with the function f() 23 -1 and the initial value zo - i to calculate zj and 2 exactly (i.e. as complex numbers a + bi where a and b are exact rational numbers). You will see that 21 and 22 approach one of the cube roots of unity. Find the distance (correct to 4 significant digits) between 22 and this cube root. [Aside We already know, of course, that the cube roots of unity are 1, exp(i2T/3) and exp(i4x/3) but this technique is used to produce the dazzling Newtonfr rectangular region of starting points (represented by a grid of pixels on a computer screen) are coloured according to which root they approach under Newton iteration actals (seen in class) in which a
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