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**Additional resources for An Introduction to Numerical Methods : A MATLAB Approach, Third Edition**

**Sample text**

3 1. A=B If A = [a1 , a2 ], B = [b1 , b2 ] ∈ I ( ), then ⇐⇒ a1 = b1 and a2 = b2 . 2. The absolute value or “magnitude” of A is deﬁned as | A| = max{ | a1 |, | a2 |}. 3. The width of A is deﬁned to be d(A) = a2 − a1 ≥ 0. 4. A partial ordering of the elements of I ( ) is deﬁned by A**
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**If b1 = 0, or else b1 = b2 = · · · = bk = 0 such a ﬂoating-point number is said to be normalized. 1b2 . . 2) where the mantissa is expressed as a sequence of zeros and ones. As an abbreviation, the two binary digits 0 and 1 are usually called bits and the ﬁxed-length group of binary bits is generally called a computer word. As an example, let us consider the ﬂoating-point number system of a 32-bit word length microcomputer. 1. The leftmost bit is used for the sign of the mantissa, where the zero bit corresponds to a plus sign, and the unit bit to a negative sign. **

N + 1)! which yields n = 7 for the smallest n that approximates e to three decimal places. 718253968254. 1! 2! 3! 4! 5! 6! 7! 718281828459. 4 Derive the Maclaurin series for f (x) = sin x. The derivatives of sin x are f (x) = cos x, f (x) = − sin x, f (x) = − cos x, ··· = ··· Therefore, n sin x = (−1)k k=0 f (0) = 1, f (0) = 0, f (0) = −1, ··· = ···. x2k+1 x2n+3 + (−1)n cos(ξ). (2k + 1)! (2n + 3)! ✐ ✐ ✐ ✐ ✐ ✐ “k” — 2011/11/22 — 10:14 — page 17 — ✐ TAYLOR SERIES ✐ 17 Another useful form of the Taylor series is obtained by replacing in Eqn.