How many steps of the bisection method are needed to determine the root of accur

Question

How many steps of the bisection method are needed to determine the root of accurate to machine precision, in single precision?

Explanation / Answer

It may help you a lot If a = 0.1 and b = 1.0, how many steps of the bisection method are needed to determine the root with an error of at most 1 × 10-8 . 2 Solution: Note that the bisection theorem gives that |b - a| 1 < × 10-8 . n+1 2 2 now, using logarithms we obtain log(0.9) - (n + 1) log(2) < - log(2) - 8 (n + 1) log(2) > log(2) + 8 - log(0.9) log(0.9) 8 - n+1 > 1+ log(2) log(2.0) 8 log(0.9) n > - ˜ 26.727 log(2) log(2.0) Thus, at least 27 steps of the bisection method are needed. 2. If the bisection method is applied with starting interval [a, a + 1] and a = 2m , where m = 0, what is the correct number of steps to compute the root to full machine precision on a 32-bit word-length computer? Solution: Recall that machine precision is 2-24 . Thus, our error bound becomes |b - a| |a + 1 - a| 1 1 = = n+1 = 2-24 = 24 . n+1 n+1 2 2 2 2 Thus, we need to choose n = 23.

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