By Abdelwahab Kharab

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4) ✐ ✐ ✐ ✐ ✐ ✐ “k” — 2011/11/22 — 10:14 — page 41 — ✐ ✐ THE BISECTION METHOD 41 Using the Intermediate Value Theorem, it follows that there exists at least one zero of f in (a, b). To simplify our discussion, we assume that f has exactly one root α. 3. 3 The bisection method and the ﬁrst two approximations to its zero α. The bisection method is based on halving the interval [a, b] to determine a smaller and smaller interval within which α must lie. The procedure is carried out by ﬁrst deﬁning the midpoint of [a, b], c = (a + b)/2 and then computing the product f (c)f (b).

Use the bisection method to ﬁnd both roots. 9. 1x2 − x ln x = 0 between 1 and 2. 10. Use the bisection method to ﬁnd the root of the equation x+cos x = 0 correct to two decimal places. 11. If the bisection method is applied on the interval from a = 14 to b = 16, how many iterations will be required to guarantee that a root is located to the maximum accuracy of the IEEE double-precision standard? 12. Let n be the number of iterations of the bisection method needed to ensure that a root α is bracketed in an interval [a, b] of length less than .

2. 8125. 8125)10 = (110100)2 . 011011101)2 × 25 . The value of c is c = 127 + 5 = (132)10 whose 8-bit form is (10000100)2 and s = 1 since the number is negative. f ) has a 1 in each bit position and the biased exponent c = (1111 1110)2 = (254)10 . The value of this number is therefore (2−2−23 )2127 ≈ 3×1038 . The smallest positive number is 2−126 ≈ 10−38 . The IEEE double-precision format employs 64 bits, of which 11 bits are reserved for the biased exponent, and 52 bits for the fractional part f of the normalized mantissa.