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assuming that f(x0 + h) = 0. In other words, if you consider x0 as a good approximation for a zero point of f, and it holds that f(x0 + h) = 0, you can estimate the unknown h in accordance with Eq. 6.2. An iteration algorithm for determining a zero of f follows from this. |
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A. Start with a guess for x0. |
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B. If xn is an approximation of the zero point, let |
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C. If the approximation xn+1 is sufficiently good, or if the number of iterations is large enough, then stop. Otherwise, proceed to B. |
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Let's look at a special case. Given f(x) = x3 - 1, permitting not only real-valued x, but also complex numbers, the formula in Eq. 6.3 will then look like this: |
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Now we can experiment with this formula. You begin the iteration process using the starting value x0, which you can select with complete freedom. Remember that x0 is a complex number, so you can visualize properties of these values in the complex plane. One of these properties is whether a prescribed starting value converges toward the zero point x = 1 while iterating. You can't take this for grantedafter all, there are two other zero points, exp (2pi/3) and exp (4pi/3), and even there convergence isn't guaranteed. The question also remains as to which complex numbers x0 converge toward 1 in steps A through C of the iteration process. |
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The reality of this situation turns out to be very exciting. Based on symmetry, one third of the complex plane (initial values being intended) converges toward 1, one third converges toward exp (2pi/3), and one third converges toward exp (4pi/3). If you ignore rotations and displacements, the corresponding convergence zones should be identical. You would expect a simple threefold division of the complex plane into pie pieces, with 1, exp (2pi/3), and exp (4pi/3) somehow situated in the center of these parts. |
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This really happens, but you'll see a strange breakdown. Figure 6.11 shows the results obtained using LabVIEW to investigate a square of the complex plane starting at (2, 2i) and ending at (+2, +2i). You can see stable islands (for which the convergence proceeds undisturbed toward 1) among bizarre structures that have completely different convergence properties, creating a fractal structure with truly fascinating details. |
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