{28} • Numerical Methods 0 16 18 20 24 32 19 x 32’s digit 0 a 0 = 0 2 b 0 = 100000 2 16’s digit 0 16’s digit 1 a 1 = 10000 2 b 1 = 100000 2 8’s digit 0 8’s digit 1 a 2 = 10000 2 b 2 = 11000 2 4’s digit 0 4’s digit 1 a 3 = 10000 2 b 3 = 10100 2 2’s digit 0 2’s digit 1 a 4 = 10010 2 b 4 = 10100 2 1’s digit 0 1’s digit 1 19 = 10011 2 This continues even when we consider fractions: we determine binary digits which appear after the “decimal" point. The bisection method is well suited to the binary, or base 2 number system, because it splits the interval at each stage into two pieces, which can align with binary digits. The bisection method is less well suited to the decimal number system. The intervals [a n ,b n ] do not correspond nicely to decimal digits. As we have found previously, the number of iterations required to find a solution to a desired number of decimal places can vary rather haphazardly. Exercise 16 Can you describe a “decimation method" for solving an equation f (x) = 0, analogous to the bisection method, but which at each stage produces an interval one tenth the size of the previous interval, and at each stage determines one decimal digit of the solution?

A guide for teachers – Years 11 and 12 • {29} Complex numbers and Newton fractals Newton’s method works just as well for complex numbers as for real numbers: sometimes finding a solution at blistering speed, and sometimes failing to work at all. For instance, suppose we want to solve the equation z 3 = 1. There is just one real solution z = 1, but over the complex numbers there are three solutions: z = 1, − 1 2 + 3 2 i , − 1 2 − 3 2 i . Starting from an initial point z 1 , Newton’s method works just as over the reals. We let f (z) = z 3 − 1, so f ′ (z) = 3z 2 , and then calculate z 2 = z 1 − f (z 1) f ′ (z 1 ) = z 1 − z3 − 1 3z 2 . From z 2 , we calculate z 3 , and then z 4 , and so on. We can ask: which choices of z 1 lead to which solutions? This leads to a very interesting picture. The picture below shows the complex plane, with real and imaginary axes labelled. For each point complex number z, we run Newton’s method with z 1 = z, and see where the z n go. If they approach − 1 2 + 3 2 i , we colour the point brown. If they approach − 1 2 − 3 2 i , we colour the point blue. And if they approach 1, we colour the point white.

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