By Abdelwahab Kharab
Praise for earlier Editions
Kharab and Guenther provide an enticing, transparent, error-free, and well-written advent to numerical tools ... hugely recommended.
-J.H. Ellison, CHOICE
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Extra info for An Introduction to Numerical Methods : A MATLAB Approach, Third Edition
Xn ), called interval evaluation for f , will compute the actual range of values of f for xi ∈ Xi (see ). Here we assume that all occurring expressions are composed of ﬁnitely many operations for which the interval arithmetic operations in F are deﬁned. 7 Consider the two expressions for the function f (x, a) = 1 − a/x, with x ∈ [2, 3] a x x−a f2 (x, a) = . x f1 (x, a) = 1 − Using interval arithmetic, the interval evaluations for f1 and f2 when a ∈ [0, 1] are F1 ([2, 3], [0, 1]) = 1 − F2 ([2, 3], [0, 1]) = [0, 1] 1 = ,1 , [2, 3] 2 [2, 3] − [0, 1] 1 3 = , .
Use the bisection algorithm to approximate the zero of f to within 10−4 . 4) is satisﬁed. 125. 5]. 609375. 5]. Next we set a2 = c1 and b2 = b1 . 465454. 1. 000122 Solution of x3 − x2 − 1 = 0 in [1, 2] using the bisection method. ✐ ✐ ✐ ✐ ✐ ✐ “k” — 2011/11/22 — 10:14 — page 43 — ✐ THE BISECTION METHOD ✐ 43 Some of the features of MATLAB can be used to view, for example, step-by-step calculations involved in the function bisect. This can be done by using the pause function of the form pause(s). If we add the line pause(1); just before the end statement of the while loop, execution is held up for approximately 1 second at the end of each output line.
Determine the representation of the following decimal numbers in the ﬂoatingpoint number system of a 32-bit word length microcomputer. 84375)10 . 3. On a computer, ﬂoating-point numbers are represented in the following way t e u m t = sign of exponent, 1 bit e = absolute value of exponent, 3 bits u = sign of mantissa, 1 bit ✐ ✐ ✐ ✐ ✐ “k” — 2011/11/22 — 10:14 — page 27 — ✐ FLOATING-POINT ARITHMETIC ✐ 27 m = mantissa, 5 bits The base is 2 and the mantissa is normalized. (a) Give the largest number that can be represented exactly.
An Introduction to Numerical Methods : A MATLAB Approach, Third Edition by Abdelwahab Kharab