By Abdelwahab Kharab
Praise for past Editions
Kharab and Guenther provide an enticing, transparent, error-free, and well-written advent to numerical equipment ... hugely recommended.
-J.H. Ellison, CHOICE
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Extra info for An Introduction to Numerical Methods : A MATLAB Approach, Third Edition
The exponent is positive and is equal to 0 × 20 + 1 × 21 + 0 × 22 + 0 × 23 + 0 × 24 + 0 × 25 = (2)10 . 8125)10 . 25)10 . 78125 in the normalized binary form. 110010)2 . 101010110010)2 × 26 . 78125 in normalized binary form is (00000110101010110010000000000000)2 . The largest normalized positive machine number is 0 0111111 111111111111111111111111 ≈ 263 ≈ 1018 . Thus, the range of numbers that can be stored is from about 10−19 to almost 1018 in decimal magnitude. Numbers occurring in calculations that have a magnitude outside the above computer range result in what is called an underﬂow or an overﬂow.
Intel processors also have an 80-bit extended format that is used within a computer’s hardware arithmetic unit, but which is not always available to the programmer. 1 1. 875)10 . 2. 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.
Log 2 So one needs at most 19 iterations to obtain an approximation accurate to 10−6 . 1 1. Verify that the function f (x) = x2 sin x + 2x − 3 has exactly one root in (0, 2). Find this root by using the bisection method with an error of no more than 10−5 . 1. 2. Find a root of f (x) = x3 + 2x − 3 in the range 0 ≤ x ≤ 7/5 using the bisection method. 3. 31 has one root in the interval [3, 4]. How many steps of the bisection method are needed to locate the root with an error of at most 10−5 . 4. 6 has exactly one root in (1, 2).
An Introduction to Numerical Methods : A MATLAB Approach, Third Edition by Abdelwahab Kharab