Let p(x) = – x. The roots of pare -1, 0, and 1. Let’s see how fast we can find them using Newton’s..

Let p(x) = – x. The roots of pare -1, 0, and 1. Let’s see how fast we can find them using Newton’s method. We begin by finding N(x): It converges, but the convergence is not as quick and it is to the root that is furthest away from our initial estimate. This illustrates one of the problems with Newton’s method. While it often finds a root, it may not find the closest one. It is also worth noting that the values given for the previous calculations Were rounded to the nearest thousandth after the calculation. The actual calculations were done with 16 significant digits. Given our previous work,

View complete question »Let p(x) = – x. The roots of pare -1, 0, and 1. Let’s see how fast we can find them using Newton’s method. We begin by finding N(x): It converges, but the convergence is not as quick and it is to the root that is furthest away from our initial estimate. This illustrates one of the problems with Newton’s method. While it often finds a root, it may not find the closest one. It is also worth noting that the values given for the previous calculations Were rounded to the nearest thousandth after the calculation. The actual calculations were done with 16 significant digits. Given our previous work, we should not be too surprised to find out that one needs to be careful when rounding, as small errors can make a difference. For example, if we had started the last calculation at .46 rather than .45, then the sequence would have converged to 1. Similarly, had we rounded XL down to -.47, the sequence would again have converged to 1. To get some idea of the fragility of these calculations, the reader is encouraged to try other initial values for this function. Values between .4472 and .46 should prove to be especially interesting. The unpredictable behavior of Newton’s method in the preceding example for initial values near .45 leads one to ask what else might go wrong. The answer is plenty. We will see that there are cubic polynomials and intervals of real numbers such that points in the intervals not only take a long time to converge but never converge to a root at all. However, we will also outline a procedure for determining intervals where the convergence is quick and to the nearest root in Theorem 12.15, and we will show in Theorem 12.2 that a number is an attracting fixed point of Newton’s function for a polynomial if and only if it is a root of the polynomial.

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